Lecrure Notes

Thermodynamics for Molecular Engineering, Wu and Prausnitz, 3/2010

1 | T h e m a t e r i a l i s s u b j e c t t o c o p y r i g h t

Contents

BASICS OF STATISTICAL THERMODYNAMICS .................................................................................................. 2

1.1 Macroscopic systems and microstates ..................................................................................................................... 5

1.2 Ensemble average .................................................................................................................................................... 8

1.3 Entropy and internal energy ................................................................................................................................... 11

1.4 Microcanonical ensemble ...................................................................................................................................... 16

1.5 Canonical ensemble ............................................................................................................................................... 18

1.6 Ideal monatomic gas .............................................................................................................................................. 24

Internal energy ......................................................................................................................................................... 24

Microstates and entropy ........................................................................................................................................... 25

Ideal-gas law ............................................................................................................................................................ 29

1.7 Diatomic and polyatomic ideal gases .................................................................................................................... 31

1.8 Ideal solids ............................................................................................................................................................. 37

1.9 Photons and thermal radiation ............................................................................................................................... 45

Planck distribution ................................................................................................................................................... 45

Thermal radiation .................................................................................................................................................... 46

1.10 Ising model .......................................................................................................................................................... 51

Two-dimensional Ising model ................................................................................................................................. 61

Three-dimensional Ising model ............................................................................................................................... 66

1.11 Helix-coil transition of polypeptides ................................................................................................................... 71

1.12 Grand canonical ensemble ................................................................................................................................... 80

1. 13 Gas adsorption .................................................................................................................................................... 84

Langmuir adsorption isotherm ................................................................................................................................. 84

Brunauer-Emmett-Teller (BET) adsorption isotherm .............................................................................................. 87

Gas adsorption in porous materials .......................................................................................................................... 90

1. 14 Isobaric-isothermal systems ................................................................................................................................ 94

1.15 Gas hydrates ...................................................................................................................................................... 100

1.16 Partition functions for classical systems ............................................................................................................ 108

Summary.................................................................................................................................................................... 111

Concepts checklist ..................................................................................................................................................... 111

Problems .................................................................................................................................................................... 112



BASICSOFSTATISTICALTHERMODYNAMICS

As discussed in numerous textbooks, classical thermodynamics is concerned with two fundamental laws

of nature that are applicable to a large variety of phenomena. The first law asserts conservation of energy; the

second law asserts that spontaneous events in nature always proceed in a particular direction. The laws of classical

thermodynamics are derived from repeated observations of how macroscopic systems in nature behave. Because

of extensive experience over many years, we are confident that these laws are permanent, unlikely to be refuted

by future scientific developments.

Closely affiliated with the fundamental laws are two key concepts of thermodynamics: internal energy

and entropy. Internal energy is the total energy of the individual elements or particles of a macroscopic system. By

macroscopic we mean that the system contains a very large number of elements or particles (e.g., photons,

electrons, molecules, colloidal particles, stars etc.) such that it follows the thermodynamic laws. Entropy is related

to the number of ways or the number of microstates of a thermodynamic system may exist. Here a microstate

means a particular specification of the individual elements (or particles) of the thermodynamic system (e.g., the

instantaneous positions and momenta of classical particles). Entropy provides a measure of the probability

distribution of microstates in a thermodynamic system.

In addition to internal energy and entropy, auxiliary thermodynamic functions and thermodynamic

relations are introduced to describe the macroscopic properties of a thermodynamic system at equilibrium. For a

system at equilibrium, there are no net fluxes of energy or mass across the system’s boundary. In an equilibrium

system, the macroscopic properties of interest are independent of time1. More precisely, time independence

means that the duration of observation is long compared to the time scale pertinent to the dynamics of individual

elements or particles in the system.

Whereas both internal energy and entropy depend on the properties of individual elements, classical

thermodynamics provides no information on microscopic details. Toward that end, we rely on statistical

mechanics, which quantitatively connects macroscopic properties of a thermodynamic system with the

1 The macroscopic properties of a steady‐state system are also independent of time but that system is not necessary at equilibrium. In a steady‐state system, there are net fluxes of energy or mass, independent of time. However, for equilibrium, it is necessary that there is no net flux of mass or energy.



microscopic details of individual elements. Formally, statistical mechanics is concerned with the statistical

distribution of microstates in a macroscopic system. Statistical mechanics relates the dynamics of individual

elements to the macroscopic properties, such as internal energy, entropy, and other thermodynamic properties.

While in principle statistical mechanics is applicable to macroscopic systems at or away from equilibrium, our

concern here is primarily directed at thermodynamic properties of equilibrium systems, i.e., statistical

thermodynamics. When applied to a molecular system, statistical thermodynamics connects macroscopic

properties with the molecular structure, with the interaction forces within a molecule and with the interaction

forces between molecules, and with the spatial arrangement of molecules.

What is the advantage of statistical thermodynamics? What can statistical thermodynamics do that

classical thermodynamics cannot? Statistical thermodynamics provides a powerful tool to describe the properties

of a complex system from a microscopic perspective. For example, consider a copolymer containing segments A

and B. How do the properties of this copolymer depend on the chain length? Or geometric arrangement? How do

the properties change when a random sequence of A and B is replaced by a regular arrangement (‐A‐B‐A‐B‐) or by

a block arrangement (‐A‐A‐A‐B‐B‐B‐)? Or, how do the properties of a colloidal crystal change with electric charge or

say, when we change the crystal structure from face‐centered cubic to body‐centered cubic? Or, how do the

properties of a fluid (say, oil or water) change when the fluid is confined to very small pores in rocks or sands?

Finally, how does a change in a protein’s tertiary structure or electric charge affect its tendency to form crystals or

fibrils (as found in Alzheimer’s disease)? Satisfactory answers to these and many other questions in nature and

modern technology require consideration of microscopic details.

Classical thermodynamics ignores microscopic details. Therefore, by itself, classical thermodynamics is not

able to answer such questions. However, when classical thermodynamics is augmented by pertinent insights from

physics or physical chemistry, it is often able to provide useful results. Because such help is usually based on

phenomenological grounds, the utility of augmented classical thermodynamics (sometimes called molecular

thermodynamics) is limited. Statistical thermodynamics provides more fundamental (and often more rigorous)

answers to questions concerning how macroscopic properties are related to the microscopic details of a system’s

individual elements. Because statistical thermodynamics provides a link between macroscopic thermodynamic

properties and molecular properties, it is possible to “engineer” the structures of molecules and their spatial



arrangements for some particular application. Statistical thermodynamics enables us to predict the macroscopic

consequences of microscopic engineering.

This introductory chapter presents the essential concepts of statistical thermodynamics and the

mathematical framework for describing thermodynamic properties of a system in terms of the microscopic details

of that system’s individual elements. We begin with the basic assumptions of statistical thermodynamics and

discuss the definitions of entropy and internal energy. We then consider different specifications (or “constraints”)

of thermodynamic systems and how each specification2 leads to corresponding relations between microscopic and

macroscopic properties. These concepts are illustrated with a few applications, first to idealized systems and then

to a few systems of practical interest. The examples in this chapter include properties of ideal gases and ideal

solids, non‐interacting photons and radiation, helix‐coil transition of a polypeptide, gas adsorption on a solid

substrate, vacancies in crystalline solids (Schottky effect), and phase equilibria in natural gas/hydrate systems.

2 Here specification means that the system of interest is kept at a particular macroscopic condition (called a “constraint”). For example, the system may be constrained to constant temperature and constant pressure. In that case, we say that the system is isothermal and isobaric.



1.1Macroscopicsystemsandmicrostates

Classical thermodynamics is concerned with quantitative relations that connect the macroscopic

properties of an equilibrium system, i.e., the properties that can be measured over length and time scales much

larger than those characterizing the dynamics of the individual elements. Some thermodynamic variables, such as

temperature, pressure, volume and composition, can be measured directly; while others, such as entropy, internal

energy, are determined indirectly from measurements of other, directly measurable, properties.

From a macroscopic point of view, an equilibrium system, regardless of its complexity, can be defined by a

few thermodynamic variables. For example, the properties of a glass of pure liquid water are completely specified

by its temperature, pressure, and total volume. Similarly, the properties of a tank of gasoline can be specified by

the total volume, temperature, and composition. These variables are both necessary and sufficient to characterize

a macroscopic system at equilibrium. In other words, once these variables are fixed, all other thermodynamic

properties can be calculated, in principle, using thermodynamic relations derived from the two laws of

thermodynamics.

The minimum number of thermodynamic variables to define an equilibrium system, or the degrees of

freedom, is provided by the Gibbs phase rule. For example, to define a system free of chemical reactions, we use

fN intensive variables:

2f c pN N N (1)

where cN is the number of components, and pN is the number of coexisting phases. An intensive variable is

independent of system size; examples are temperature, pressure, density, and chemical potential. Conversely, an

extensive variable depends on the system size; examples are internal energy, entropy, and volume. The Gibbs

phase rule is not concerned with the system size, which can be fixed by an extensive variable.

Equilibrium systems are thermodynamically equivalent if they exhibit the same macroscopic properties.

To illustrate, Figure 1 shows two tanks of argon gas, A and B, at the same temperature, mass, and total volume. If

the two containers are identical, we assert that all macroscopic properties of A are identical to those of B, i.e., any

macroscopic measurement for A is equivalent to that for B.



Figure 1 Two tanks of argon gas are equivalent from a thermodynamic point of view if they have the same temperature, mass and total volume and if the containers are also the same. But they are not identical from a microscopic perspective because at any instant, the relative positions and momenta of argon molecules in tank A are not the same as those in tank B. The systems are identical only on a macroscopic scale, not on a microscopic scale.

From a microscopic point of view, however, systems with the same macroscopic properties may be quite

different. For the two systems shown in Figure 1, argon molecules are in motion and interact with each other and

with the container. A complete description of the microscopic details of the molecules in each tank would require

an enormous number of variables to specify the positions and momenta of individual molecules. At any instant, it

is extremely unlikely that all molecules in one tank have the same positions and momenta in that tank as those in

the other. In other words, while the macroscopic state of argon in tank A is the same as that in tank B, at any

instant the microstate in tank A is most likely not the same as that in tank B.

To describe a macroscopic system in terms of its many possible microstates, we must rely on statistics.

We need to know the statistical distribution of microstates because, for a system with a large number of individual

elements (e.g., >1023), it is impossible to specify the microscopic details at any instant. To obtain a statistical

description, statistical thermodynamics uses the concept of an ensemble, defined as an assembly of a large

number of microstates for systems that are macroscopically equivalent. At any instant, each system in the

ensemble has a particular microstate. Because the number of thermodynamically equivalent systems (and thus the

number of microstates) in the ensemble can be arbitrarily large, the ensemble includes not only all of the possible

microstates of the real system under study but also those corresponding to the “mental copies” of the real system

that have identical thermodynamic properties. Figure 2 shows a schematic picture of the ensemble for a box of

A

P T

B

P T



argon gas at given temperature T. All boxes have the same volume V and each contains N identical argon

molecules.

N, T, V

N, T, V

N,T, V

N,T, V

N, T, V

N, T, V

N, T, V

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N, T, V

Figure 2 An ensemble for a system of argon gas at given temperature (T ), number of molecules ( N ) and

volume (V ) includes the microstates of an arbitrary large number of argon‐gas systems at the same T , N , and V . Here the box in solid lines represents the real system under study; all boxes in dashed lines

are mental copies of the real system. While all boxes have the same T , N , and V , at any moment, each

box may (but need not) have a different microstate.



1.2Ensembleaverage

In statistical thermodynamics, the macroscopic properties of an equilibrium system are defined in terms

of ensemble averages of dynamic variables, i.e., properties that vary with time. To illustrate, suppose that M is a

dynamic property, e.g., the instantaneous energy of a macroscopic system. The ensemble average is defined as

v

M p Μ (1)

where subscript denotes a microstate, p is the probability of microstate in the ensemble, and M

represents the value of M at microstate . Because at any moment the microstate of an equilibrium system is

unknown, an ensemble average provides the expectation (i.e. expected value) for a dynamic property of a

macroscopic system specified by a few thermodynamic variables. The expected value is indicated by brackets

.

For example, the internal energy is defined as the ensemble average of the system energy over all

microstates

U p E

(2)

where E is the total energy of the system at microstate . Internal energy is affiliated with the energies of the

individual elements (e.g., molecules) of a macroscopic system; it does not include the kinetic energies due to the

overall motion or potential energies related to the position of the entire system3. For a molecular system, vE

includes the kinetic energies arising from the translational, vibrational and rotational motions of individual

molecules, the potential energy due to intra‐ and inter‐ molecular interactions, and the energies arising from

interactions of the molecules with an external field (e.g., an electric, gravitational, or magnetic potential, or the

interaction of a wall with molecules near a surface).

To calculate ensemble average M , we need information concerning the property vM for all

microstates and the probability distribution of microstates p . The former is specified by physical laws such as

Newtonian mechanics or quantum mechanics that dictate the dynamics of individual elements. To calculate

3 If the entire system is in motion relative to some frame of reference, the energy associated with that motion is not included in internal energy. Neither does the internal energy include the gravitational potential related to the elevation of the entire system relative to some frame of reference.



probability p , we need to have a relation between p and entropy as defined by Boltzmann or by Gibbs (see

next section). For molecular systems, the microscopic details (e.g., variations of the momenta and positions of

individual argon molecules with time) can be calculated (in principle) by solving the Schrödinger equation or, if we

are content with classical mechanics, Newton’s equations. In that case, we may calculate the thermodynamic

properties by solving the equations of motion for a large number of molecules or by sampling a large number of

microstates. Molecular simulation provides a computational procedure to calculate thermodynamic properties

directly from the properties of microstates of an ensemble.

Figure 3 Time variance of dynamic property M of an equilibrium system [e.g., ( )M t could be the

instantaneous total energy of a tank of argon gas at fixed N , V and T ]. While M changes with time,

at equilibrium, time‐average M (dashed line) is constant.

How do we know that the thermodynamic properties obtained from the ensemble average are equivalent

to those defined in classical thermodynamics? As illustrated schematically in Figure 3, the instantaneous value for

any measurable property M of an equilibrium system varies with time because of the motion or dynamics of

individual elements. In classical thermodynamics, a thermodynamic property M refers to the time average of that

property M over a time period that is much larger than the time scale relevant to the dynamics of the

individual elements in the system of interest

0

1( )M M t dt

. (3)

M(t)

Time, t



For a simple molecular system (e.g., argon gas), the time scale relevant to the dynamics of the individual molecules

is very small, of the order of 10‐15 seconds. Over the duration of a typical thermodynamic measurement (of the

order of minutes or hours), the system visits a very large number of microstates. As a result, it is reasonable to

hypothesize that the distribution of microstates in a specific system under consideration is the same as that in the

ensemble. In other words, the ensemble average and the time average should yield identical results. The

equivalence of time average and ensemble average is called ergodicity, a key assumption of statistical mechanics.

In this book, we are concerned exclusively with ergodic systems, i.e., equilibrium systems that follow the ergodic

hypothesis.

We can illustrate ergodicity and the conceptual difference between time average and ensemble average

by using an analogy. Suppose that we have two distinguishable dice, each with the 6 facets labeled with numbers

1 to 6. If we roll the dice once, we want to know: What is the probability that the face values of the two dice give a

seven?

We can answer this question in two ways. In the first way, we have one person roll the dice repeatedly,

say n times. For each roll, we record the sum of the face values of the dice. For two different dice, there are 36

different outcomes (or microstates). We can calculate the desired probability by noting how often the two dice

gives seven and divide that by n, the total number of rolls. This would give a time‐average result because it comes

from a large number of rolls, each with the same dice at a different time. Alternatively, we can assemble a large

number of persons and give each one two distinguishable dice. The dice given to any one person are identical to

those given to any other person. There are n persons in the assembly. At a fixed instant, everyone in the assembly

rolls his or her dice. We can then calculate the desired probability by looking at all the rolls and counting the

number of rolls that result in seven. We take that number and divide it by n, the number of persons who, at the

same time, rolled his or her two dice. In this case, the calculated probability is not the average of many rolls taken

by one person over a long period of time. Instead, it is the average of many rolls, each taken by one of many

persons (an ensemble) at a fixed time. Both methods give the same result. Of the 36 possible outcomes, six

outcomes give seven; they are: (1,6), (6,1), (2,5), (5,2), (3,4), (4,3). The desired probability is 6/36 = 1/6.

The ergodicity assumption ensures that the definitions of thermodynamic properties in statistical

thermodynamics are equivalent to those used in classical thermodynamics.



1.3Entropyandinternalenergy

In classical thermodynamics, the lack of a physically comprehensible definition of entropy tends to make

this important function mysterious and esoteric. Entropy is often interpreted as a measure of disorder,

randomness or chaos; the higher the entropy, the greater the disorder. This interpretation may trace back to

Rudolf Clausius, the German physicist who first introduced the concept of entropy and articulated the two laws of

thermodynamics in the middle of the nineteenth century. Hermann von Helmholtz, another founding father of

thermodynamics, also used the word “Unordnung” (disorder) to describe entropy.

If entropy is affiliated with disorder, the relentless increase of entropy would ultimately degrade our

universe to a state of complete randomness or chaos. As enunciated by Clausius in 1865, “The entropy of the

universe tends to a maximum.” This bleak prediction of classical thermodynamics has raised concerns among

philosophers and laypersons alike. For example, Bertrand Russell, a British mathematician and philosopher, wrote

pessimistically that “all the labors of the ages, all the devotion, all the inspiration, all the noonday brightness of

human genius, are destined to extinction.” Yet life appears on earth, beginning from simple inorganic forms to

biomacromolecules and their self‐assembly to cells and eventually evolution to human beings with incremental

ordered structures. Is entropy necessarily affiliated with disorder?

In searching for a mechanical theory of heat, Ludwig Boltzmann, a professor of mathematical physics in

Germany and Austria, concluded in 1877 that entropy is an intrinsic property of an equilibrium system reflecting

the statistical distribution of the system’s elements in different microstates. According to Boltzmann’s statistical

interpretation, entropy is defined as

lnBS k W (1)

where 23

B 1.381 10 J/Kk is the Boltzmann constant, and W stands for the average number of accessible

microstates, i.e., the average number of microstates that the system may exist at any moment. Boltzmann’s

definition of entropy marks the beginning of statistical mechanics. This definition gives a clear physical

interpretation: entropy provides a measure of microscopic freedom or the number of choices available to the

individual elements to exist in different microstates.



To illustrate the meaning of the total number of microstates, consider a simple example. Suppose that we

have a box containing four different sub-boxes. We have four balls and we ask: In how many different ways

(microstates) can we place these four balls into the box such that one ball is in each sub-box?

Figure 4 Six different ways to place two identical white balls and two identical black balls into four different sub‐boxes such that one ball is in each sub‐box.

When we place the first ball, we have four possibilities because all four sub-boxes are initially empty.

When we place the second ball, we have three possibilities because one sub-box is already occupied. When we

place the third ball, we have two possibilities and when we place the fourth ball, we have only one possibility.

Therefore, the number of different ways to place the balls is

W 4 3 2 1 4! . (2)

If the four balls are identical, i.e., indistinguishable, the final arrangements of balls in the sub-boxes are

equivalent, independent of the sequence of filling. In that case, it makes no difference which ball we place first,

second, third or fourth. Therefore, we must divide the apparent W by 4! giving W 1 . There is only one way to

place four identical balls into a box containing four different sub-boxes such that one ball is in each sub-box. For

identical balls, the number of possible microstates is unity.



But now suppose we have four balls where two are black and two are white; otherwise, the balls are

identical. Again, we ask, in how many different ways (microstates) can we place the four balls into the four sub-

boxes? For this case, we must divide the apparent W (Eq.2) by 2 2! ! , because two black balls and two white balls

are indistinguishable. For this case, W=6.

Figure 4 shows the six ways of filling four different sub-boxes with two black balls and two white balls

such that one ball is in each sub-box. Because entropy is related to the number of microstates, the entropy of two

black balls and two white balls in four sub-boxes is larger than that of four white balls (or four black balls) in four

sub-boxes.

Why is it larger? When filling the four sub-boxes with four identical balls, information is complete; there

is only one possible arrangement. But when filling four sub-boxes with two white balls and two black balls,

information is incomplete. As shown in Figure 4, that case has six possible arrangements. We know that there are

two black balls and two white balls in four sub-boxes, but we do not know the detailed arrangement; information

concerning the state of the system is incomplete. We see here a simple representative illustration of how entropy is

a measure of missing information.

While Boltzmann’s definition often provides a link between entropy and disorder or randomness, a

system in a state of higher entropy is not necessarily equivalent to that in a state of greater disorder4. By using

idealized models, scientists have realized for many years that increasing entropy sometimes serve as a driving

force for molecular ordering rather than disordering. For example, Lars Onsager predicted in the 1940s that at

sufficiently high density, a system of rod‐like particles exhibits a phase transition from an isotropic phase to a

lamellar phase (where the particles are aligned in the axial direction). The entropy of the ordered lamellar phase is

higher than that the disordered isotropic state. Onsager’s idea was used by chemical engineers to fabricate ultra‐

strong fibers such as Kevlar. For another example, in the 1950’s, Alder and coworkers found by using computer

models that at sufficiently high density, hard spherical particles may crystallize without any attractive force. In

other words, at sufficiently high density, the entropy of spherical particles in an ordered state may exceed that in a

random state.

4 As commonly perceived, we define “order” in terms of spatial organization of individual elements. For example, a crystal is referred to as an ordered phase because of the organized arrangement of individual atoms.



The connection between entropy and information (or missing information) was first identified by J.

Willard Gibbs who provided an alternate definition of entropy

lnB v vv

S k p p . (3)

Gibbs’ definition of entropy consists of an important cornerstone of information theory for signal process and data

analysis. In the context of ensemble average, Eq.(3) suggests that the microscopic counterpart of entropy,

lnB vk p , is related to “uncertainty” or “incomplete information.” Entropy gives the ensemble average of

“uncertainty” that exists when only a few macroscopic variables (e.g., temperature and density of a pure gas) are

used to describe a macroscopic system. When 1vp , the system is unique at the microscopic level, i.e., the

system has no “uncertainty.” For this case, ln 0B vk p and 0S . For 1vp , the microstate of a system at

any instant is not certain. We may regard “uncertainty” as given by lnB vk p . The smaller p , the larger the

“uncertainty.” Therefore, entropy can also be understood as a measure of incomplete information; when the

information is complete ( 1)vp , entropy is zero. Entropy falls as the available information becomes more

complete5.

Another key concept in statistical thermodynamic is internal energy, which is defined as the ensemble

average of the total energy over all microstates

U p E

. (3)

Once we have both internal energy and entropy, we can derive all other thermodynamic variables of an

equilibrium system following the standard relations of classical thermodynamics. For example, the temperature

can be derived from

,V N

UT

S

(4)

where N is the number of elements (or particles) in the system, and V stands for the total volume. Because

both entropy and total energy are extensive properties, temperature is intensive independent of the system size.

5 Conceptually it is preferable to use a quantity that rises (rather than falls) as information increases. For that purpose, negentropy is defined as the reciprocal of entropy.



Box1Boltzmann’sentropyandGibbs’entropy

We may show that the two definitions of entropy are equivalent. Consider an equilibrium system with a large number of individual elements. A microstate of the system, denoted by , is defined by the microscopic

details of all elements. Now imagine that we follow the microscopic details of the system in consecutive steps,

with very much larger than unity. We designate the total number of microstates for the entire system as n. Within the duration of observation, the multiplicity of the system in different microstates is equal to the number of

different outcomes to toss “a super coin” with n facets times

1 2

!

! ! !n

W

(B1)

where is the number of times that the system is in microstate , satisfying the normalization condition

1

n

.

Because the multiplicity of the outcomes involves steps, the average number of microstates accessible to the system at each step is

1/W W .

In a special case where all microstates are equally accessible (e.g., a fair coin), W n , i.e., the average number

of accessible microstates at each step is n. In general, evaluation of 1/W requires some specific knowledge on the

statistical distribution of the microstates.

According to Boltzmann, the entropy is given by

ln lnBB

kS k W W

.

By using Eq.(B1) for W and Stirling’s approximation ln ! ln , we find that Boltzmann’s definition of

entropy is equivalent to that by Gibbs

1 1 1

/ (ln ! ln !) / ( / ) ln( / ) lnn n n

BS k p p

where /p is the probability that the system is in microstate .



1.4Microcanonicalensemble

In classical thermodynamics, an equilibrium system is specified by macroscopic variables such as constant

temperature and constant pressure or constant volume. Because equilibrium systems fixed at the same

macroscopic thermodynamic variables are equivalent macroscopically but not microscopically, one essential task

of statistical thermodynamics is to obtain the distribution of microstates for equilibrium systems that are fixed (or

specified) in different macroscopic ways. To specify equilibrium systems microscopically, we apply different

constraints to all possible microstates. By a constraint we mean specification of a macroscopic variable such that

constrains the dynamics of all individual elements of the system. For example, in an isolated system, the

constraints of constant internal energy U , volume V , and number of individual elements N mean that these

conditions are satisfied in all microstates. In statistical mechanics, microstates conforming to these particular

constraints constitute a microcanonical ensemble.

From a macroscopic point of view, an isolated system is fully defined by the total energy, the total

volume, and the number of individual elements in the system. When energy transfer is prohibited, the first law of

thermodynamics requires that an isolated system must have constant total energy6. In other words, the internal

energy U of an isolated system is the same as the energy for any microstate, which is identical to the total energy

TE

TU E . (1)

The second law asserts that, at equilibrium, the entropy of an isolated system reaches a maximum.

According to Gibbs, entropy is given by

lnBS k p p

. (2)

Because p is normalized by

6 Assume no nuclear reactions within the system and no external energy, i.e., no energy arising from overall motion or from any potential (e.g., gravitational or electric field) imposed on the entire system.



1p

, (3)

We may find the distribution of microstates corresponding to the maximum entropy by using the Lagrange‐

multiplier method

ln 1 0Bk p p pp

(4)

where is the Lagrange multiplier.

After some algebra, Eq. (4) becomes

(ln 1) 0Bk p (5)

or

exp( / 1)Bp k . (6)

Because Bk and are constants, Eq.(6) indicates that in the microcanonical ensemble, vp is the same for all

microstates. In other words, the microstates are uniformly distributed in the microcanonical ensemble7.

From Eqs.(3) and (6), we find that vp is equal to the reciprocal of the total number of accessible

microstates

1/vp W . (7)

Substitution of Eq.(6) into Eq. (2) gives

lnBS k W (8)

which is the equation for entropy first proposed by Boltzmann. Eq.(8) indicates that if an isolated system has only

one microstate ( 1W ), its entropy is zero. This statement is essentially equivalent to the 3rd law of

thermodynamics, i.e., the entropy is zero for a perfect crystal at zero temperature.

7 Many statistical mechanics texts say that, in an isolated equilibrium system, equal probability of microstates is a fundamental postulate of statistical mechanics. As shown here, this postulate is equivalent to postulating the second law of thermodynamics.



1.5Canonicalensemble

A key task of statistical thermodynamics is to find the distribution of microstates for macroscopic systems

at equilibrium. The probability distribution depends on the choice of constraints, i.e., thermodynamic variables

imposed to define the equilibrium system. The microcanonical ensemble discussed in Section 1.4 is applicable to

systems with fixed total energy, volume and number of particles. While the microcanonical ensemble is useful to

illustrate the basic concepts of statistical thermodynamics, the constraints are often not convenient for realistic

systems. Because energy transfer between a system and its environment is common in real situations, a more

useful set of constraints for an equilibrium system is to specify the system volume, temperature, and number of

particles. An ensemble corresponding to those constraints is called a canonical8 ensemble.

Toward establishing the distribution function for microstates in the canonical ensemble, we consider a

closed system with fixed volume and total number of particles in contact with a large heat bath at constant

temperature. At equilibrium, the temperature of the system is identical to that of the heat bath. Because there is

no net transfer of energy between the system and the heat bath at equilibrium, the constraint of fixed

temperature is equivalent to the fact that the average internal energy of the system is constant.

The internal energy of the system is given by

U p E

(1)

where E stands for the total energy of the system at microstate , and p denotes its probability in the

canonical ensemble. To derive p , we again use the Lagrange multiplier method. As discussed earlier (Section

1.3), entropy is defined as

lnBS k p p

. (2)

The second law of thermodynamics asserts that the entropy is at a maximum, subject to the constraints that

conform to the macroscopic specifications of the system. In the canonical ensemble, the probability distribution of

microstates is subject to the constraint of fixed temperature (Eq.1) and the normalization condition

8 Canonical means according to the canon, that is, the conventional, well‐known and generally accepted code of procedure.



1p

. (3)

Within the constraints of Eqs.(1) and (3), maximizing the entropy S gives the distribution of microstates in the

canonical ensemble

exp( 1 )p E (4)

where and are yet‐to‐be‐determined Lagrange multipliers.

Substitution of Eq.(4) into (2) gives

/ ln 1 1B vS k p p E U

. (5)

From the thermodynamic identity, ,/

V NU S T , we find

1

,

/( )B

BV N

S kk T

U

. (6)

And from the normalization condition (Eq.3), we have

1 ln exp[ / ]BE k T

. (7)

With Eqs.(6) and (7), we can rewrite for the distribution of microstates in the canonical ensemble

exp( )E

pQ

(8)

where Q is the canonical partition function

vE

v

Q e . (9)

Eq.(8) is known as the Boltzmann distribution or the canonical distribution of microstates. It indicates that low

energy states have higher probability, i.e., for a closed system at constant temperature, volume and number of

particles, the equilibrium is biased to microstates with low energy.

The partition function plays a central role in connecting the microscopic properties of a molecular system

and its macroscopic thermodynamic variables. As mentioned earlier, Q is related to the distribution of



microstates in terms of the Boltzmann distribution law (Eq.8). On the other hand, Q is directly related to the

internal energy, entropy, and Helmholtz energy:

exp[ ]exp[ ]

ln( )

exp[ ] exp[ ]

vv v

vv

v vv v

EE E

QU

E E

, (10)

/ 1 lnB vS k E U Q , (11)

lnBF U TS k T Q , (12)

where F is the Helmholtz energy. With an expression for the partition function, Eqs.(8‐12) allow us to calculate

both the thermodynamic properties and the distribution of microstates.

Unlike that in a microcanonical ensemble, the total energy of microstates in a canonical ensemble

fluctuates around its mean value due to instantaneous exchange of energy between the system and the thermal

bath. From the partition function, we can also derive the mean‐square deviation of the total energy

2 22 2 2

2

1 1 ln( ) ( ) v vE E

v vv v

QE E E E e E e

Q Q

.

(13)

Upon substituting Eq.(10) and the definition of the constant‐volume heat capacity ( / )V VC U T

2 2( ) B VE k T C , (14)

Eq.(14) suggests that the constant‐volume heat capacity reflects the energy fluctuation in a canonical ensemble.

Because the left side is a square average, Eq.(14) indicates that for an equilibrium system, the constant‐volume

heat capacity is always non‐negative, as required by thermodynamic stability.

The relative mean‐square‐root deviation of the total energy is

2 2( ) / /B VE E k T C U . (15)

Because both heat capacity and internal energy are extensive variables proportional to the number of particles of

the system, the relative mean‐square‐root deviation of energy scales as

22( ) ~ 1/E E N . (16)



In a typical macroscopic system, the number of particles is on the order of 1023. As a result, the probability that an

isothermal closed system has energy appreciably different from its mean energy is extremely small.

We expect that for macroscopic systems, different ensembles, i.e., different ways to specify the

constraints of a macroscopic system, are equivalent in the thermodynamic limit (i.e., infinitely large systems).

However, for systems that are small (on the order of molecular dimensions), thermodynamic properties may

depend on the choice of a particular ensemble. To illustrate the equivalency of different ensembles, we may

consider an idealized system that consists of N distinguishable but non‐interacting particles (e.g., nuclear spins9 in

a uniform magnetic field). We assume that each particle can be in one of two possible microstates: one with

energy , and one with energy . We may derive the thermodynamic properties of this system using first, a

microcanonical ensemble, and second, a canonical ensemble.

In a microcanonical ensemble, the system energy is fixed; the internal energy U is the same as total

energy TE

( ) ( 2 )T o o oU E n N n N n (17)

where on is the number of particles with energy . Because the total energy is fixed, on must be a constant,

and the number of particles with energy is oN n . Because all microstates have equal probability in the

microcanonical ensemble, we can calculate the system entropy from the total number of microstates W using

the Boltzmann equation

!

ln ln ln!( )!B B B

o o o

N NS k W k k

n n N n

. (18)

In Eq.(18), the combinatorial number o

N

n

stands for the number of ways that the system can have on particles

with energy . Based on Stirling’s approximation for factorials, the entropy is

( )

ln ( ) lno oB o o

n N nS k n N n

N N

. (19)

9 The spin of a nucleus is its angular momentum.



From the entropy and internal energy, we derive the system temperature from the thermodynamic identity

1

,N V o oN N

U U ST

S n n

. (20)

Differentiation of U and S with respect to on and substitution into Eq.(20) gives

1

2ln o

B o

nT

k N n

. (21)

Now we consider the same system, but use the canonical ensemble, i.e., an ensemble corresponding to

systems at fixed T and total number of spins N . For a given microstate v ( specified by the particle energy

states), the total energy is

1

N

v ii

E s

(22)

where 1is stands for the energy state of particle i . The canonical partition function is

11

( )v i

i

NE s N

v si

Q e e e e

(23)

where 1

Bk T . Because the particles are independent, the partition function of the total system can be

alternatively written as the thN power of the partition function for an individual particle,

1

i

i

s

s

q e e e

. In that case, we have

( )N NQ q e e . (24)

Whereas each particle can be in either one of two energy states so that the total number of microstates remains

2N, the constraint of constant temperature makes most of these microstates have only negligible probability.

From the partition function, we can derive the Helmholtz energy

ln ln( )B BF k T Q Nk T e e , (25)

and the internal energy



N

F e eU N

e e

. (26)

The probability that a particle is in energy state is

o

ep

q

, (27)

and the average total numbers of particles in this state is

o

en N

e e

. (28)

With Eq.(28), the internal energy becomes

( ) ( 2 )o o oU n N n N n . (29)

Eq.(29) is identical to that from the microcanonical ensemble (Eq.17).

Finally, we derive the entropy from the thermodynamic identity, ( ) /S U F T ,

.

( 2 ) ln( )

[( 2 ) ln( )]

[ ( ) ln ln ln( )]

ln ( ) ln

ln ( ) ln

o B

B o

B o o

B o o

o oB o o

S N n Nk e eT

k N n N e e

k N n e n e N e e

e ek n N n

e e e e

n N nk n N n

N N

(30)

The entropy from Eq.(30) is also the same as that from the microcanonical ensemble (Eq.19). To evaluate S from

Eq.(30), we use Eq.(28) for on .

This example illustrates that different ensembles lead to the same thermodynamic properties. However,

the microstates and their distributions are different. In practical applications, the choice of ensemble depends on

how we specify an equilibrium system. That specification, in turn, depends on what information is available

concerning the system of interest.



1.6Idealmonatomicgas

We now consider the thermodynamic properties of an ideal monatomic gas (e.g., argon) and their

connections to the microscopic details of an ensemble of molecules. An ideal gas is an idealized model of a real gas

where intermolecular interactions are neglected (e.g., real gas at low density). This limiting case provides a first

step for subsequent application of statistical thermodynamics to realistic systems.

Internalenergy

In the absence of intermolecular interaction, the total energy of an ideal gas is equal to the sum of the

total energy of individual molecules. For an ideal monatomic gas, the molecular energy consists of a translational

kinetic energy due to molecular motion, and the energy of sub‐atomic particles such as electrons and protons. In

most cases the latter energy remains constant and it is neglected in our discussion. However, the kinetic energy of

each individual gas molecule is not fixed because of collisions of gas molecules with the system boundary. If the

motion of each gas molecule can be described by Newtonian mechanics, the velocity distribution of the gas

molecules is given by (see Box 2)

3/ 2 2

( ) exp2 2B B

m mvp

k T k T

v

(1)

where ( )p v

is the probability density of gas molecules with velocity v, m is the molecular mass, | |v v

stands

for the magnitude of velocity, and T is the temperature in Kelvin. Because ( )p v

depends only on the magnitude

of velocity, it satisfies the normalization condition 2

0( ) 4 ( ) 1d p dv v p v

v v

.

Equation (1) is known as the Maxwell velocity distribution, which provides the basis of the kinetic theory

of gases. According to Eq.(1), the average kinetic energy per molecule u is given by

2 2 2

0/ 2 4 ( )( / 2) 3 / 2Bu mv dv v p v mv k T

. (2)

Eq.(2) indicates that at a fixed temperature, each molecule has an average translational energy of 3 / 2Bk T .



If the ideal‐gas system contains N molecules, the internal energy, i.e., average total energy of the

molecular system, is then given by

3 / 2BU Nu Nk T . (3)

At room temperature ( 298T K), the internal energy of an ideal monatomic gas is 3.716 KJ/mol, independent of

the molecular mass.

Microstatesandentropy

The connection of the microscopic behavior of an ideal gas and its entropy is obtained within the context

of quantum mechanics. As shown in Figure 5, we consider N non‐interacting monatomic molecules in a cubic box

of volume V. We assume that the box exerts no force to the ideal gas molecules other than elastic collisions that

maintain an average total energy (or temperature) and that prevent the molecules from escaping. For an ideal‐gas

system with N identical molecules, a microstate of the system is defined by the quantum states, i.e., the energy

states of individual molecules.

Figure 5 A) A cubic box contains N non‐interacting monatomic molecules. B) According to quantum

mechanics, the kinetic energy of each molecule ( i ) is discretised.

According to elementary quantum mechanics, a monatomic molecule in a rigid cubic box may exist in any

one of a large number of quantum states that are related to its instantaneous energy i by

2

2 2 22

( ), ( , , )8i x y z x y z

hn n n i n n n

ma (4)

A)

n n

1

2

3

1 2

3 B)



where 346.626 10 J sh is the Planck constant, m is the molecular mass, 1/ 3a V is the side length of the

box, and , , 1, 2, 3,x y zn n n are quantum numbers affiliated with the translational motion of each molecule in

x, y and z directions.

As discussed in Section 1.3, imagine that we keep track of the quantum states that a given molecule visits

in consecutive steps, with arbitrarily large. We designate the total number of possible quantum states for

each molecule as n. Within the duration of observation, the multiplicity (or the number of ways) for the molecule

in different microstates is the same as the number of different outcomes to roll “a super die” with n facets

times

1 2

!

! ! !n

W

(5)

where i is the number of times that the molecule is in microstate i , satisfying the normalization condition

1

n

ii

. (6)

Because the total multiplicity of the outcomes involves steps, the average number of microstates accessible to

the molecule at each step is 1/W where W is the multiplicity.

For a system containing N identical molecules, the average number of microstates accessible to the

system is

/1

!NW W

N

(7)

where the factorial !N takes into account that the ideal‐gas molecules are indistinguishable. According to the

definition by Boltzmann, the entropy of the ideal gas system is thus given by

ln ln ln !B B

NS k W k W N

. (8)

With Eq.(5) for W and Stirling’s approximation, Eq.(8) can be rewritten as



1

/( ) ln ln 1n

B i ii

S Nk p p N

, (9)

where /i ip stands for the probability that a single ideal‐gas molecule is in quantum state i.

The second law of thermodynamics asserts that at equilibrium, the ideal gas system is in a state of

maximum entropy subject to the constraints of constant internal energy, volume, and total number of molecules.

The distribution of microstates that gives the constrained maximum can be obtained from the Lagrange‐multiplier

method

1 1

/( ) ( 1) ( ) 0n n

B i i ii ii

S Nk p p up

(10)

where and stand for the Lagrange multipliers, arising from

1

1n

ii

p

, (11)

1

n

i ii

p u

. (12)

Eq.(11) imposes a normalization condition for the probability distribution, and Eq.(12) gives the average energy per

molecule u . Inserting Eq.(9) into (10), we obtain

1 i

ip e . (13)

We now fix the Lagrange multipliers and using Eqs.(11) and (12).


1

1

i

n

i

e e

. (14)

The summation q is the single‐molecule partition function; it can be evaluated analytically



22 2 2

21

3 3/ 22 2 2

2 2 30

exp[ ( )]8

2exp( )

8

i

x y z

n

x y zi n n n

hq e n n n

ma

h x ma Vdx

ma h

(15)

where 2 / 2h m is the thermal de Broglie wavelength, and 3V a is the cubic‐box volume. In Eq.(15),

replacement of the summation with integration is justified because near room temperature, the energy gap

between different quantum states is exceedingly small in comparison with Bk T .

In terms of the single‐molecule partition function, the average energy per molecule is given by

1 1

ln/i

n n

i i ii i

qu p e q

. (16)

Upon substitution of q from Eq.(15) into (16), we find the Lagrange multiplier

3/(2 )u . (17)

Comparing Eq.(17) with Eq.(2) for the kinetic energy derived from Maxwell’s velocity distribution, we find

1( )Bk T . (18)

Using the probability distribution of quantum states (Eq.13) and the expression for the single molecule

partition function (Eq.15), we obtain a concise expression for the entropy

3/( ) ln( ) 5/ 2BS Nk (19)

where /N V stands for the molecular number density. Eq.(19) is known as the Sackur–Tetrode equation,

named after two physicists who derived it independently in 1912. It indicates that for a given monatomic ideal gas,

the entropy is solely determined by the molecular density and temperature.



Ideal‐gaslaw

Upon examining extensive experimental data, Robert Boyle, Jacques Charles, and Joseph‐Louis Gay‐Lussac

established the ideal gas law in the early eighteenth century. We can derive the ideal‐gas law using a simple

mechanical interpretation of pressure and the connection between molecular kinetic energy and temperature.

Figure 6 An ideal-gas pressure can be interpreted as the average collision force per unit area on a testing surface.

Here v stands for the magnitude of velocity.

Consider an elastic collision of an ideal‐gas molecule of mass m and velocity v perpendicular to a surface

(Figure 6). The collision results in a change of molecular momentum 2mv . At an infinitesimal time dt , an ideal‐gas

molecule reaches the surface only if it is located within distance vdt . Assume that the ideal‐gas molecules are

uniformly distributed in space; the likelihood of the gas molecule within the collision region is /vAdt V , where A

vdt

surface

Box2Maxwellequation

We can derive the Maxwell velocity distribution from the distribution of microstates. In Newtonian

mechanics, the energy of an ideal gas molecule is represented by the kinetic energy, i.e.,

2 / 2mv .

In the framework of Newtonian mechanics, a microstate of the system is specified by the position and momentum

of individual molecules. Because there is no interaction energy between ideal‐gas molecules, Eq.(13) indicates that

the distribution of microstates is related to the velocity distribution

2( ) ~ exp( / 2)p mvv

which is identical to the Maxwell equation (Eq.1). Here v stands for the magnitude of velocity.



is the surface area, and V is the system volume. According to Newton’s law, the collision force F on the surface

is

2( / )2 (2 / )dt vAdt V mv Adt V mv F . (20)

On average, each ideal‐gas molecule has a translational energy 3 / 2Bk T , i.e., / 4Bk T in each direction of

the coordinate (i.e., x, y, z, ‐x, ‐y, ‐z). The ensemble average of 2 / 2mv is / 4Bk T . From Eq.(20), the force per unit

area due to the collision of a single molecule with the wall is

/ /BA k T VF . (21)

For a system containing N ideal‐gas molecules, the total force per unit area, i.e., pressure P , is then

given by

/BP Nk T V . (22)

Eq.(22) is the familiar ideal gas law. It indicates that the Boltzmann constant is related to the gas constant by

/B Ak R N (23)

where 236.02 10AN is the Avogadro constant. As discussed in numerous classical thermodynamics texts, the

ideal‐gas law provides a basis for the definition of an absolute temperature scale.



1.7Diatomicandpolyatomicidealgases

We now consider the thermodynamic properties of a diatomic ideal gas based on the same equations for

monatomic systems. The energy of each diatomic molecule includes a translational term identical to that for a

monatomic ideal gas, a vibrational energy due to the bond stretching, and a rotational energy due to rotations of the

two atoms around the center of mass:

22 2

2/3 20

( 1)( , , ) ( 1/ 2)

8 8R R

v R v

n n hh nn n n n h

mV I

(1)

where 1 2m m m stands for the total atomic mass per molecule; 1m is the mass of one atom and 2m is the mass

of the other. Here, ( , , )V Rn n n

are, respectively, quantum numbers affiliated with translational, vibrational, and

rotational motions of each molecule, 0I stands for the moment of inertia. For a diatomic molecule with bond length

b , the moment of inertia is 2

0 1 2 1 2/ ( )I m m b m m . In Eq.(1), we assume that the electronic energy is at the

ground state such that its contribution to the partition function can be neglected.

Summation of all the quantum states leads to the single-molecule canonical partition function

/ 2

/30 0

exp[ ( , , ) / ]1

v

v

V R

T

V R B Tn n n R

V e Tq n n n k T

e

(2)

where 2 /(2 )Bh mk T is the thermal wave length, /v Bh k is the bond vibrational temperature, is the

vibrational frequency, 2 20/(8 )R Bh k I is the rotational temperature related to the molecular moment of inertia

0I , and is a symmetry factor that accounts for the number of equivalent orientation of the molecule ( 2 if

the two atoms are the same and 1 otherwise). Summation of the rotational degrees of freedom takes the

continuum limit, i.e.,0

0R

Rn

dn

, which is accurate for RT .



Table 1 presents the vibrational and rotational temperatures, v and R , along with the dissociation

energy b for some diatomic molecules. As for the ideal gas of atomic molecules, the canonical partition function

for an ideal gas of N identical diatomic molecules can be calculated from the single-molecule partition function

!

NIG q

N . (3)

Table 1 Parameters for some diatomic molecules. Here b stands for the bond dissociation energy.

V , K R , K b, kJ/mol

H2 6210 85.4 2 430

N2 3340 2.86 2 942

O2 2230 2.07 2 490

CO 3070 2.77 1 882

NO 2690 2.42 1 510

HCl 4140 15.2 1 427

HBr 3700 12.1 1 347

HI 3200 9.0 1 265

Cl2 810 0.346 2 239

Br2 470 0.116 2 190

I2 310 0.054 2 148

From T. L. Hill, An Introduction to Statistical Thermodynamics, Dover (1986) p153.

From the canonical partition function IG , we can derive the internal energy, entropy and heat capacity of

a diatomic ideal gas following the standard statistical-thermodynamic relations:

/

/ln 5/

2 2 1v

IGIG v v

T

TU N

N T e

(4)



3

//

1 (ln / )/( ) ln( ) 7 / 2 ln( / )

(1/ )

/ln(1 )

1v

v

IGIG

B R

TvT

S Nk TN

Te

e

(5)

2 /

/ 2

5

2 ( 1)

v

v

TV v

TB

C e

Nk T e

.

(6)

In addition to the translational energy as appeared in a monatomic ideal gas, the internal energy of a diatomic ideal

gas includes contributions due to the molecular rotation and bond vibration. As a result, the molar heat capacity of a

diatomic ideal gas is larger than that of a monatomic ideal gas and it depends on temperature. At low temperature,

/ 1v T , 5 / 2V BC Nk ; and at high temperature, / 1V T , 7 / 2V BC Nk . The form of VC curves

between two limits has been verified experimentally in a number of cases. At very low temperature, the

approximation for rotation (diatomic molecule as a rigid rotator) is no longer valid. In that case, we need to use a

more accurate procedure that gives precise energy levels10. At very low temperature, the heat capacity of a diatomic

gas becomes similar to that for a monatomic gas (i.e., 3 / 2V BC Nk ).

Similar procedures can be applied to a polyatomic ideal gas. For a molecule with only a few atoms (e.g.,

H2O, CO2, NH3), the intramolecular interactions can be represented by semi-empirical bond potentials. In that case,

the single-molecule partition function can be evaluated by using equations similar to those for a diatomic molecule.

In general, we need to use 3n coordinates to specify the positions of all atoms in a polyatomic molecule with n

atoms. Alternatively, the molecular degrees of freedom can be specified by the center of mass (3 spatial

coordinates), the molecular orientation (2 for a linear molecule, e.g., CO2 and 3 for a nonlinear molecule, e.g., H2O),

and 3n-5 (for a linear molecule) or 3n-6 (for a nonlinear molecule) intramolecular vibrations. With a microstate of a

single molecule specified by the translational motion of the molecular center of mass, molecular rotation (as a rigid

rotator), and 3n-5 or 3n-6 vibrational motions of individual atoms relative to the center of mass, we can write the

partition function of a single polyatomic molecule

,

,

/21/2 3/2 3 5 or 6

/31

( / )

1

v i

v i

TnR

Ti

TV eq

e

. (7)

10 S. M. Blinder, Advanced Physical Chemistry: A Survey of Modern Theoretical Principles, New York: Macmillan, 1969 pp. 475–478.



where the rotational temperature R is defined in terms of the three principal moments of inertia AI , BI , and CI

for the molecule

2

1/32

( )8R A B C

B

hI I I

k

. (8)

As before, symmetry number is the number of different ways in which the molecule can achieve the same

orientation in space by rotation. Table 2 gives the symmetry numbers and other parameters of some representative

polyatomic molecules.

For a rigid polyatomic molecule with n atoms, the principal moments of inertia can be calculated from

2 2

1

[( ) ( ) ]n

A i i cm i cmi

I m y y z z

. (9)

2 2

1

[( ) ( ) ]n

B i i cm i cmi

I m x x z z

. (10)

2 2

1

[( ) ( ) ]n

C i i cm i cmi

I m x x y y

. (11)

where im stands for atomic mass, ( , , )i i i ir x y z for the atomic position in an arbitrary Cartesian coordinate, and

( , , )cm cm cm cmr x y z for the molecular center of mass.

1 1

/n n

cm i i ii i

r m r m

. (12)

The vibrational temperature in Eq.(7) is not directly related to the bond stretching but to the independent

vibrational modes of the atoms. For example, Figure 7 shows 4 vibrational modes for a linear molecule and 3

vibrational modes for a nonlinear molecule. Based on experimental infrared and Raman spectra, the four vibrational

temperatures for CO2 are 1890(I), 3360(II), 954(III) and 953 K (IV), and those for H2O are 5404 (I), 2295 (II) and

5262 K (III), respectively.



Figure 7 Vibration modes for a CO2 molecule (a) and for a H2O molecule (b)

From the single-molecular partition function, we can again derive the internal energy and heat capacity of

an ideal polyatomic gas:

3 6 or3 5

,

/1

1 1/ 3

2 1V

nn

V iIGT

i

U NT e

(13)

,

,

3 6 or 2 /3 5,

/ 21

3( 1)

V i

V i

nTn

V iVT

iB

C e

Nk T e

.

(14)

For an ideal gas of nonlinear polyatomic molecules, the entropy is

,

,

3 3 6/,3

/31

/1/( ) ln( ) 4 ln ln(1 )

1V i

V i

nTV iIG

B TiR

TTS Nk e

e

. (15)

A similar equation can be given for an ideal gas of linear polyatomic molecules

,

,

3 5/,3

/1

//( ) ln( ) 7 / 2 ln( / ) ln(1 ) .

1V i

V i

nTV iIG

B R Ti

TS Nk T e

e

(16)

We close this section by indicating that in principle the properties of an ideal gas can be predicted by

using the statistical‐mechanical equations but because of the complexity of the microscopic details of individual

molecules, the practical utility of the “classical” approach discussed above is rather limit. A faithful description of

the molecular properties requires sophisticated quantum‐mechanical considerations beyond the scope of this text.

(b) (a)

I.

II.

III.

IV.

O C O

O C O

O C O

O C O

I. II.

III.

O

H H

O

H H

O

H H



Table 1 Parameters for some polyatomic molecules

V , K

CO2 1890, 3360, 954, 953 2

N2O 1840, 3200, 850(2) 2

NH3 4800, 1360, 4880(2), 2330 (2)

3

CH4 4179, 2180(2), 4320(3), 1870(3)

12

H2O 2290, 5160, 5360 2

The numbers in the parentheses denote degeneracy. From D. A. McQuarrie, Statistical Mechanics,

University Science Books (2000) p137.



1.8Idealsolids

In an ideal solid, we assume that all atoms are placed in a perfect lattice and their local motions are

uncorrelated. A classical example is provided by a monatomic crystal at low temperature where the vibrating

atoms remain close to their equilibrium positions. Similar to the ideal‐gas model as a useful reference for real

fluids, an ideal‐solid model provides a convenient reference state for understanding properties of real crystals.

Relative to some ground‐state energy, the energy of an atom i in an ideal solid can be written as a sum of

the energies of harmonic oscillators in x, y and z dimensions:

, , , , , , , , ,

1 1 1( ) ( ) ( ) , 0,1, 2,

2 2 2i i x i x i y i y i z i z i x i y i zE n h n h n h n n n

(1)

where h is Planck’s constant, ,i xn , ,i yn and ,i zn represent quantum numbers, and , , ,, ,i x i y i z are characteristic

frequencies of the solid in 3‐dimensional space. Because the long‐wavelength vibrations give rise to sound in a

solid, the quantized modes of vibrations are called phonons11. Phonons play an important role in solid‐state

thermal and electrical conductivities.

Because the local motions of individual atoms are independent, we can conveniently evaluate the

thermodynamic properties of an ideal solid by using a canonical ensemble. For a solid with N atoms, the total

energy of the system is

3

1 1

1( ) 0,1, 2,

2

N N

i j j ji j

E E n h n

(2)

where subscript denotes a microstate specified by a set of quantum numbers 1,3{ }j j Nn for the 3N energy

levels of all harmonic oscillators12. Assuming that for all atoms the microstates and oscillations in the x‐, y‐ and z‐

directions are independent of each other, the canonical partition function Q can be calculated analytically

/ 213 3( )2

1 1 1

jj j

j

j

hN Nn hEh

nj j

eQ e e

e

(3)

11 The word phonon from Greek φονή (phonē) means voice. 12 A harmonic oscillator is a dynamic system where the restoring force from the equilibrium position is proportional to the displacement.



The Helmholtz energy is

3

/ 2 / 2

1

ln ln( )i i

Nh h

B Bi

F k T Q k T e e

. (4)

From Eq.(4), we obtain all thermodynamic properties by appropriate differentiations of F .

To evaluate the summation in Eq.(4), we need to find the frequencies of all harmonic oscillators. In

general, these frequencies depend on the structure of the atomic crystal and interactions between atoms on the

lattice. To the zeroth‐order approximation, assume that all the harmonic oscillators have the same vibrating

frequency . In this case, Eq.(4) simplifies to

/ 2 / 23 ln( )E ET TBF Nk T e e . (5)

Eq.(5) was first proposed by Einstein, where /E Bh k is called the Einstein temperature. For example, the

internal energy U and the constant‐volume heat capacity VC are

/

3 3

2 1E

E B B ET

N k NkFU

e

(6)

2 /

/ 2

13

( 1)

E

E

TE

v B TV

U eC k

N T T e

(7)

Eq.(7) indicates that /(3 )V BC k is a universal function of reduced temperature / ET . In other words, Einstein’s

theory suggests that the heat capacities of simple crystals obey a law of corresponding states.

In the limit of high temperature or small /E T , / 2 / 2( / ) /( 1) ~ 1E ET TE T e e . In that case, Eq.(7)

reduces to the Dulong‐Petit law that the molar heat capacity of all atomic crystals is 3R (about 25 J.mol‐1.K‐1). The

Dulong‐Petit law agrees reasonably well with experiment for a number of simple elements at ambient

temperatures where quantum effects are minimal. For example, at 298 K, the experimental heat capacity of Cu is

25.5 and that of Al is 24.4 J.mol‐1.K‐1.

In the low‐temperature limit, Einstein’s model predicts that the heat capacity of a monatomic solid

approaches

2/3 E TE

v BC k eT

( 0T ). (8)



Eq.(8) is not accurate when compared with experiments where vC ~3T as 0T . Nevertheless, Eq.(8) correctly

predicts that the heat capacity is a strong function of temperature at low temperatures and approaches a constant

at high temperatures. More importantly, Eq.98) correctly predicts that vC vanishes in the zero‐temperature limit.

Early in the 20th century, Einstein’s work on ideal solids provided strong support for the then new quantum theory.

Einstein assumed a very simple distribution of the frequencies, viz. all vibrating frequencies are the same.

A better approximation was proposed by Debye for the distribution of vibration frequencies in monatomic solids.

Debye’s theory is able to reproduce experimental observations at both high and low temperature limits.

Figure 8 Some possible standing waves (represented by different curves) in a one‐dimensional crystal lattice according to the Debye model. Here L is the total width of the lattice. The wavelength for each

standing wave satisfies 2 /n L n where 1,2,n a positive integer.

The Debye model is exact in the limit of low frequency or long wavelength where the atomic nature of the

solid is not important, and the crystal can be considered as a continuous elastic body. Debye’s theory assumes that

the harmonic oscillators are standing waves with frequencies independent of the atomic details of the solid. For N

atomic particles in a cube of side L , the wave vector ( k

) of any harmonic oscillator must satisfy

k nL

(9)

where ( , , )x y zn n n n , and , ,x y zn n n are positive integers. Figure 8 illustrates schematically some possible

frequencies for a one-dimensional crystal of length L .

L



Figure 9 The quantum numbers (nx,ny,nz) that satisfy

2 2 2 2( / )x y zn n n Lk are given by one‐eighth the

volume of a sphere of radius /Lk .

According to Eq.(9), the magnitude of a wave vector k

is given by

2

2 2 2 2( )x y zk n n nL

. (10)

The number of standing waves with the magnitude of wave vector less than k , ( )n k , is equal to the number of

possible combinations of , ,x y zn n n that satisfy

2

2 2 2 2( )x y zk n n nL

. (11)

As illustrated in Figure 9, all possible sets of quantum numbers that satisfy Eq.(11) are located within one-eighth of

a sphere with radius /kL . Therefore, the number of quantum states is equal to one-eighth of the volume of the

sphere

3

( )6

Lkn k

. (12)

The number of standing waves with the magnitude of wave vector between k and k dk is then given by

2

2( )

2

Vk dkdn k

(13)

where volume 3V L .

X1 Y1 Z1( )

n y

n x

n z

L k /



The frequency of a wave is related to the magnitude of wave vector k

through the relation

| | 2 / sk k c

(14)

where sc represents the velocity of sound. Therefore, the number of standing waves with frequency between and

d is given by

2

3

4( )

s

Vg d d

c

. (15)

Because each particle corresponds to one harmonic oscillator, integration of ( )g from zero to a maximum

frequency D equals the total number of standing waves

0

( ) 3D

g d N

. (16)


1/ 3

9

4D s

Nc

V

. (17)

This maximum frequency D is called the Debye frequency. In terms of D , the frequency distribution function

( )g becomes

23

90

( )

0

DD

D

Nd

g d

. (18)

With Debye’s assumption for the frequency distribution, the Helmholtz energy of a monatomic crystal

becomes

/ 2

0( ) ln

1

h

B h

eF k T g d

e

. (19)

Upon substituting Eq.(18) into Eq.(19)

/ 2

23 0

9ln

1

Dh

Bh

D

Nk T eF d

e

. (20)

From the Helmholtz energy, we can derive the internal energy

2/( )3 0

9 /( )

2 1

D

B

B Bh k T

D B

Nk T h k TF hU d

k T e

, (21)



and the constant-volume specific heat capacity

3 4/

20/ 9

( 1)

Dx

T

V B xD

T x eC k dx

e

, (22)

where /D D Bh k is called the Debye temperature. Similar to the Einstein theory, the Debye theory also predicts

a law of corresponding states for the heat capacity.

/ ET or / DT

Figure 10 Reduced heat capacity of a monatomic crystal according to the theories of Einstein and Debye.

E is the Einstein temperature and D is the Debye temperature.

Figure 10 compares predictions of the Debye theory and the Einstein theory for the heat capacity of a

simple solid. While both Debye’s theory and Einstein’s theory predict that the heat capacity converges to the

classical limit at high temperatures, they differ significantly at low temperatures. The Einstein theory predicts that

near 0T K the specific heat capacity is

2

/3 E TEV BC k e

T

, (23)

whereas the Debye theory predicts

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

Debye

Einstein

3V

B

C

k



3412

5V BD

TC k

. (24)

The low-temperature limit of the Debye theory is in good agreement with experiment. For Einstein’s theory, the

low-temperature heat capacity erroneously falls to zero more rapidly than 3T . Because the Debye theory considers a

crystal to be a continuous elastic body, in principle, the Debye temperature can be calculated in terms of the crystal’s

elastic constants. However, in practice, the Debye temperature is often obtained by fitting experimental VC data at

low temperature. Table 2 gives the Debye temperatures for some atomic crystals.

The Debye theory for atomic solids provides a highly idealized model for crystalline solids. When

compared with experiment, as shown qualitatively by Figure 11, at high frequencies there are serious errors in the

Debye theory, even though it agrees reasonably well with experiment in the low-frequency end. Fortunately, for

macroscopic thermodynamic quantities, such as heat capacity, errors at high frequencies are often not important.

Figure 11 A typical frequency distribution ( )g for a monatomic crystal. The solid curve represents

experimental results while the dashed curve represents Debye’s approximation. Here the maximum

Debye frequency is 0.8. Einstein’s model uses only a single frequency, i.e., ( )g is given by a Dirac delta

function.

The Debye theory of atomic crystals is remarkably successful for describing experimental observations of

heat capacity at low temperatures for pure crystalline solids including metals, alkali halides and diamond. At low

temperatures, the low-frequency part of the vibrational modes are well approximated by the continuous theory of

elasticity because at high temperatures, the heat capacity is determined by an average over the entire frequency

g(v)

v



range with the limit 3V BC k as / DT . As a result, to a good approximation, errors of the Debye theory for

the high-frequency contributions tend to cancel.

Table 2 The Debye temperatures (K) for some atomic crystals

Ag 225 Fe 470 Na 158 Si 645

Al 428 Ga 320 Nb 275 Sn 200

Ar 92 Gd 200 Ne 75 Sr 147

As 282 Ge 374 Ni 450 Ta 240

Au 165 Hf 252 Os 500 Te 153

Ba 110 Hg 71.9 Pb 105 Th 163

Be 1440 In 108 Pd 274 Ti 420

Bi 119 Ir 420 Pt 240 Tl 78.5

C 2230 K 91 Rb 56 U 207

Ca 230 Kr 72 Re 430 V 380

Cd 209 La 142 Rh 480 W 400

Co 445 Li 344 Rn 64 Xe 64

Cr 630 Lu 210 Ru 600 Y 280

Cs 38 Mg 400 Sb 211 Yb 120

Cu 343 Mn 410 Sc 360 Zn 327

Dy 210 Mo 450 Se 90 Zr 291

From C. Kittel, Introduction to Solid State Physics (7th Ed, 1995).

As the ideal gas models discussed in Sections 1.6 and 7, the theories of Einstein and Debye illustrate the

general procedure for applying statistical mechanics to calculation of macroscopic thermodynamic properties. The

partition function is evaluated using a microscopic model for the substance of interest, in this case, a simple

monatomic crystal. The model describes the substance in terms of molecular properties, in this case, characteristic

energies of vibration and their distribution. Once we have an expression for the partition function Q , we obtain

macroscopic thermodynamic properties through the key equation lnBF k T Q .



1.9Photonsandthermalradiation

Photons13 provide an example to illustrate that statistical thermodynamics is useful not only for systems

containing molecules but also for those containing a large number of virtually any elements or particles. The

difference between molecular systems and statistical systems of “particles” is only reflected by the way we

describe the energy or the dynamic behavior of individual particles that comprise the statistical‐mechanical

system.

Planckdistribution

Early in the twentieth century, Planck suggested that photons are distinct units of an electromagnetic

wave such as gamma‐ and X‐rays, visible light, infrared and radio waves. The energy of a photon is proportional to

the frequency of the electromagnetic wave

h (1)

where h is Planck’s constant, and is the frequency. Different from atoms or electrons, photons do not interact

with each other because they have neither electric charge nor rest mass14.

Consider a system of photons in thermal equilibrium with its container. The photons are

undistinguishable from each other but, because of emission and absorption from the container wall, the total

number of photons is not conserved. We seek to derive the distribution of photons among different energy states

at a given temperature. A microstate of the system is specified by the number of photons 1 2, ,n n … in different

energy states denoted, respectively, by 1 , 2 , … . The occupation number, i.e., the number of photons in each

energy state, may vary from zero (unoccupied) to infinity. Because photons are non‐interacting, the total energy

of the system is a sum of all energies for individual photons

1 1 2 2E n n (2)

At a given temperature, the canonical partition function,

13 Photon is derived from the Greek word phos that means light. 14 According to Einstein's special theory of relativity, the mass of an object increases with its velocity. When an object is at rest (relative to the observer), it has the usual mass that we call the “rest mass”.



exp( )Q E

, (3)

can be evaluated analytically

1 2

1 2

1 1 2 20 0

1 1 2 20 0 1

exp ( )

1exp( ) exp( )

1 i

n n

n n i

Q n n

n ne

(4)

With the help of Eq.(2), we find the average number of photons at an energy state i

,

( / )ln 1

1

v v

v v i

j i

E Ei i

i E Ei T

n e eQ

ne e e

(5)

Eq.(5) is the Planck distribution; it describes the thermal average number of photons in a single energy state or

mode of frequency.

Thermalradiation

At a finite temperature, any material emits or absorbs electromagnetic waves or radiation; some

materials emit (or absorb) radiation more efficiently than others, but regardless of the specific properties of a

material, it is the temperature of the material that plays a major role in thermal radiation. A black body provides

an idealized model of thermal radiation where the efficiency of absorption and emission is 100% for

electromagnetic waves of arbitrary frequency.

Consider photons in a container shown schematically in Figure 12. The distribution of the microstates

satisfies the Planck distribution (Eq.5). From the canonical partition function (Eq.4), we find the internal energy of

the system

1 1

ln

1 1i i

i ih

i i

hQU

e e

. (6)

Because the container emits and absorbs photons, the internal energy can be understood as the total energy of

the black‐body radiation.

.



Figure 12 A simple model of a black body is provided by a small hole on the side of a container. Radiation entering the hole from the outside is repeatedly reflected inside the container with essentially no chance of escape, i.e., the radiation is completely absorbed. Similarly, when the container is heated, the efficiency of emission through the small hole is also 100%.

To evaluate the summation in Eq.(4) or (6), we assume that the container is a cubic box with length L .

Imagine that each photon is a standing wave contained in the box. The photon wavelength is then related to the

box length by

/ 2L n

(7)

where n is a positive integer. Because the photon frequency is related to the speed of light c

2

c cn

L

, (8)

the summation of quantum states in Eq.(4) or (6) is thus equivalent to integrations in the continuum limit

3

23 30 0 0 0

0 0 0

(2 ) 22 2 4

x y z

x y zi n n n

L Vd d d d

c c

(9)

where 3V L is the system volume. The factor 2 comes from photon polarization, i.e., photons can be electrically

polarized in clockwise or counter‐clockwise directions.

Because the frequencies are very closely spaced, we replace the summation in Eq.(6) with integration of

the frequency according to Eq.(9)

Entering radiation Isothermal wall of the

container at temperature, T



2 5

3 3 40

8 8

1 15( )h

h V d VU

e c hc

. (10) 15

Recalling that1( )Bk T , Eq.(10) predicts that, as experimentally observed, the total energy of photons (or

energy of thermal radiation) is proportional to 4T .

Figure 13 The spectrum of black body radiation according to Eq.(12). As predicted by Wien’s law, the peak of emission occurs at a wavelength inversely proportional to temperature (in Kelvins), i.e.,

3max 2.9 10 /T (in meters).

According to Eq.(10), the internal energy per unit volume for photons with frequency between and

d is

3

3

8( )

1h

h du d

e c

(11)

Because /c and 2( / )d c d , Eq.(11) becomes

5 /

8 1( )

1hc

hcu

e

(12)

15 Because

3 4

0 1 15x

x dx

e

, we obtain Eq.(10) by replacing x with h .

0

1

2

3

4

0 1000 2000 3000

Rad

iation intensity (eV/m

4)

Wavelength, nm

5000

4000

3000

Temperature, K



The spectral energy density function ( )u has units of energy per unit wavelength per volume. As shown in Figure

13, ( )u exhibits a maximum at a wavelength approximately equal to

3max 2.9 10 /T . (13)

where max is meters and the temperature T is in Kelvins. Prior to Planck, Eq.(13) is known as Wien's displacement

law, first derived in 1893 by applying the laws of thermodynamics to electromagnetic radiation.

The internal energy given by Eq.(10) corresponds to the total energy of black‐body radiation in all

directions. At a single direction, the radiation energy per surface area and per unit time is

2

5 /

2 1( ) ( )

4 1hc

c hcI u

e

(14)

where 4 accounts for the solid angle (because the radiation is the same in all directions), and the speed of light

c is equivalent to the volume of photons propagated per unit time per unit area. Eq.(14) is known as Planck’s law

of black‐body radiation, in excellent agreement with experiment. Planck’s law provided strong evidence in favor of

the quantum theory that led to quantum mechanics about 25 years later.

From the internal energy of photons (Eq.10), we can also derive the Stefan-Boltzmann law for the total

amount of thermal radiation. As illustrated in Figure 14 for radiation from a unit area of a black body A , the

radiation energy per unit time is given by the energy density multiplied by the volume of photons propagated per

unit time in area A . The total amount of energy radiated per unit time per unit area from all directions is given by

integration with respect to the solid angle:

5

2 423 40 0

8sin cos

4 4 15( )

c U cI d d T

V hc

(15)

where 4282345 KW/m1067.5)15/(2 chk B is called the Stefan-Boltzmann constant. While was

determined by experiment before the work by Planck, Eq.(15) relates this macroscopic constant to the

microscopic properties of an idealized (black-body) radiator. Derivation of the Stefan-Boltzmann relation tells us

why total radiation intensity I is proportional to the fourth power of absolute temperature T as had been previously

observed experimentally.



Figure 14 From an area A at the black‐body surface in the direction specified by angles ( , the

radiation energy per unit time is given by the cylinder volume cosc dt A multiplied by radiation

energy intensity /U V . The total energy per unit area per unit time at all angles (0 2 ,

02

) is 2

2

0 0

cossin

U cd d

V

, leading the Stefan‐Boltzmann relation (Eq.15).

This brief discussion of radiation illustrates how statistical mechanics provides a powerful tool for

understanding a well-known natural phenomenon.

Direction of radiation specified by a solid angle

A

Direction normal to the unit area



1.10Isingmodel

To gain an understanding of how the microscopic details of individual particles determine macroscopic

properties of a thermodynamic system, it is useful to consider simple mathematical models. Once these models

are understood, it is then possible to apply corrections toward representing real systems. This procedure is often

used in science. For example, we use the well‐understood properties of an ideal gas as a basis for understanding

real gases; we apply corrections through a compressibility factor that perhaps we obtain from a corresponding‐

states correlation or from a realistic equation of state. Similarly, for vapor‐liquid equilibria of nonelectrolyte liquid

mixtures we start with an ideal mixture (Raoult’s law) and then apply corrections through activity coefficients.

For possible extension to real systems, we desire an idealized model that retains the essentials of some

aspect of nature without attention to details. One model that possesses these features is the Ising model, where

each element in the thermodynamic system has only two energy states. The Ising model provides a simple

representation of diverse phenomena in nature, in particular, phenomena related to phase transitions, including

the critical region.

Figure 15 The concept of spin in a magnetic material. Here the particle of interest is an electron.

When Wilhelm Lenz and his PhD student Ernst Ising introduced the Ising model in 1925, they were

concerned with a model for ferromagnetism, a property exhibited by some metals or alloys, such as the various

form of iron, steel, cobalt and nickel; for these materials, below a certain temperature, the atomic magnetic

moments tend to line up in a common direction. The concept of spin was used to represent an intrinsic angular

Spin of an electron around nucleus is clockwise

Spin of an electron around nucleus is counter-clockwise

Spin is aligned with magnetic field giving a lower energy, h

Spin is counter to magnetic field giving a higher energy, h

External magnetic field



momentum of particles such as electrons and protons as indicated in Figure 15. Intuitively, elementary‐particle

spin is a rotation about the axis of the particle; only two spin states are possible, one clockwise and one

counterclockwise. In the presence of an external magnetic field, the particle’s magnetic dipole tends to align with

the external field, giving a lower energy h . When the magnetic dipole is counter to the external field, it has a

higher energy h . As demonstrated later in this section, we can establish the canonical partition function Q for a

one‐ or two‐dimensional Ising model. Regrettably, the exact partition function for a three‐dimensional Ising model

remains unknown but it is possible to obtain a good three‐dimensional approximation from molecular simulations.

We now discuss the thermodynamic properties of Ising models and illustrate their application to three

phenomena: the charged‐uncharged transition of a weak polyelectrolyte, a monolayer of carbon monoxide on a

graphite surface, and binary liquid‐liquid equilibria near the consolute (critical) solution temperature. The next

section discusses another application of the Ising model: the helix‐coil transition of a biomacromolecule (e.g., DNA

or a polypeptide).

Ising chain

Figure 16 shows a schematic representation of the one‐dimensional Ising model, in effect, an Ising chain.

Each spin can only be in one of the two energy states represented by up and down arrows. For an Ising chain with

ferromagnetic interactions16, the energy between two neighboring spins is negative ( ) when they are in the

same direction (aligned); otherwise it is , as indicated in Figure 17. In the presence of an external magnetic

field, each spin has a negative energy h when it is up (aligned with the external field), and a positivity energy h

when it is down (otherwise).

Figure 16 The one‐dimensional Ising model is a linear chain of N spins. Here each arrow represents the direction of a spin. In a paramagnetic system, there is a positive coupling energy when two immediate

neighboring spins are in the same direction (aligned) and a negative energy when they are in opposite

16 If 0 , the interaction is ferromagnetic and if 0 , it is antiferromagnetic.

1 4 3 N - 2 N - 1 2 N



directions. In a ferromagnetic system, the coupling energies are negative when two immediate neighboring spins are aligned and positive otherwise.

For an Ising chain of N spins, each microstate is specified by the spin orientations (i.e., up or down), i.e.,

1 2, , ,...,i Nv s s s s (1)

where 1is means spin up, and 1is means spin down. In the presence of an external magnetic field, the

system energy is

1 2 1 2 2 3 1( ) ( )v N N NE s s s h s s s s s s . (2)

where is h stands for the energy of spin i due to the external field, and 1i is s stands for the interaction

energy between spin i and its nearest‐neighbor 1i .

Figure 17 Interactions of neighboring spins in a ferromagnetic system. If neighboring spins are aligned, the pair energy is . If neighboring spins are not aligned, the pair energy is .

The canonical partition function is defined as a summation of the Boltzmann factors for all microstates:

1

11 1

11

N N

i i iv i i

i

N h s s sE

v si

Q e e

. (3)

where 1

Bk T .

or

Pair energy:

or

Pair energy:



To a good approximation, the partition function of an Ising chain is given by (see Box 3)

NQ (4)

with

2 4cosh( ) sinh ( )e h h e

. (5)

From the partition function, we find the Helmholtz energy

ln lnB BF k T Q Nk T . (6)

All thermodynamic properties of the system can be obtained from differentiation of Eq.(6).

Box3PartitionfunctionforanIsingchain

In the limit of large N (or the chain is cyclic), the reduced total energy can be expressed as

N

i

N

i

N

iiiiiiiiv sssshssshE

1

1

1 1111 2/)( .

Because

1 1

( )

1 1 1( )1 1

exp[ ( / 2 / 2) ]i i i i

hT

i i i i ihs s s s

e eh s s s s s

e e

where (1,0)is for 1is and (0,1)is for 1is , the partition function can be written as

1 2 2 3 1{ }

( ) ( ) ( ) ( )v

i

E T T TN

v s

Q e s M s s M s s M s (*)

where ( )

( )

h

h

e eM

e e

. Because i

Ti i

s

s s I , the unit matrix, Eq. (*) can be simplified as

1

1 1 Tr( )N T N

s

Q s M s M

where Tr( )NM stands for the trace of matrix NM~

.

The standard way for evaluating a trace of a matrix is by diagonalization. The eigenvalues of matrix M~

can be found from

0)~

(Det IM (**)

where Det stands for determinant of a matrix. Solution of Eq.(**) for gives two eigenvalues

42 )(hsin)cosh( ehhe .

The trace of NM~

is then given by

QMTr NNN )~

( .

When N is large, and NN , the partition function, can be approximated by

NQ .

This approximate partition function is used in Eq.(4) of the main text.



Figure 18 Average magnetization of an Ising chain as a function of reduced external field h , and

reduced interaction energy .

The average magnetization is defined as the ensemble average of s

1 2

1Ns s s s

N . (7)

When 0s , the spins have the same probability of up and down, i.e., there is no net magnetism. If 0s ,

the system exhibits a net magnetism. From the partition function (Eq.3), the average magnetization is

2 4

ln1 ln sinh( )

sinh ( )

Q hs

N h h h e

(8)

where sinh is the hyperbolic sine function.

‐1

‐0.5

0

0.5

1

‐2 ‐1 0 1 2

<s>

h

=0.01

=1

=0.01



Figure 18 shows that the average magnetization s is a smooth function of the reduced external energy

h . For small h and low temperature, 2sinh ( ) exp[ 4 ]h and 1s . In the limit of zero temperature,

h is large for any non‐vanishing external field, and the system reaches magnetic saturation17.

ThermodynamicPropertiesofIsingChain

We now consider the thermodynamic properties of an Ising chain in the absence of an external field.

When h=0 (no external field), the Helmholtz energy of an Ising chain (Eq.6) simplifies to

/ ln[2cosh( )]F N . (9)18

From Eq.(9), we can derive the internal energy

/ ( / ) / tanh( )U N F N (10)

and the heat capacity

2

2

( )/

cosh ( )V BC U T Nk

. (11)

Figure 19 shows Helmholtz energy, internal energy, entropy, and heat capacity of an Ising chain. All these

thermodynamic properties are smooth functions of reciprocal reduced temperature . As temperature

approaches zero ( ), the spins are in the lowest energy state where they are perfectly aligned in one or

the other direction. In this case, the heat capacity vanishes because, if Bk T is too small compared to the coupling

energy , a small rise in temperature does not change the spin alignment. As temperature increases further, the

internal energy becomes less negative because fewer spins are aligned. In the limit of high temperature (small

), the entropy dominates the Helmholtz energy and the spins favor a random distribution. In this case, both

internal energy and heat capacity vanish. Whereas Helmholtz energy, entropy and internal energy are all

monotonic functions of , the heat capacity exhibits a maximum at approximately 1.2 .

17 At saturation, all spins are aligned, i.e., s 1 or ‐1. 18 This equation is valid only in the limit of large N. It can be shown that the exact partition function for an Ising chain of

arbitrary N at zero field is )(cosh2 1 NNQ .



Figure 19 Thermodynamic properties of an Ising chain in the absence of an external field as functions of

reduced temperature / Bk T . At low temperature (right end of the first diagram), F U , but at

high temperature, F TS .

In the absence of an external field ( 0h ), the average magnetization of an Ising chain is always zero. In

other words, without an external field, the spins in an Ising chain will not be spontaneously aligned. Spontaneous

magnetization is absent because in an Ising chain, the energy reduced due to alignment is insufficient to overcome

the entropy lost relative to the state where the spin orientations are random.

To illustrate the interplay of entropy and internal energy, consider an Ising chain where all spins are up, as

shown in Figure 20. In this case, the spins are in an ordered state, 1s and the total energy is ( 1)oE N

.

Figure 20 An ordered Ising chain refers to the state when all spins are aligned.

Now consider a set of disordered states, all with the same energy. One disordered state is shown in Figure 21.

Suppose that for all positions from 1 to l, the spin is up and for all subsequent positions, the spin is down.

‐3

‐2

‐1

0

0 1 2 3

F/N U/N

‐S/(NkB)

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5

Cv/(Nk B)

1 2 3 4 N



Figure 21 A disordered Ising chain where 1is for 1 to l, and 1is for 1i l to N .

There are N‐1 such states because l can vary from 2 to N. Regardless of l, the system energy is

2oE E . At temperature T, the Helmholtz energy difference between the two states shown in Figures 20

and 21 is approximately19 given by

2 ln( 1)BF k T N (12)

In the limit N , F is always negative for all 0T , indicating that disordered states are always more likely.

Therefore, an infinitely‐long one‐dimensional Ising model with only nearest‐neighbor interactions20 does not show

spontaneous magnetization.

Titrationofaweakpolyelectrolyte

We now apply the Ising‐chain model to the acid‐base equilibrium of a weak polyelectrolyte, such as a

polyamine. Figure 22 shows the chemical structure of a poly (ethylene imine) chain and a schematic representation

of an Ising model. In this example, a spin corresponds to one repeating unit of the poly(ethylene imine) chain. Let

in indicate the state of an ionizable group: 0in if an ionizable site (amine group) is neutral, and 1in if it is

protonated. A microstate of this system is specified by the protonation states of the amine groups of the entire

molecule, characterized by 1 2 3, , ...n n n . We want to find the fraction of protonated amine groups as a function of

pH.

19 Eq.(12) is approximate because energy difference U and entropy difference S are evaluated independently, with

approximations 2U and )1ln( NkS B .

20 For a chain of finite size, spontaneous magnetization ( s greater than zero) occurs approximately at

)1ln(/2/ NTkB . Order‐disorder transition of a finite Ising chain has been discussed by Altenberger A.R. and Dahler

J.S., Advances in Chemical Physics, vol.112, 337‐356 (2000).

1 2 3 ℓ ℓ+1 N



Figure 22 Representation of a weak polyelectrolyte, poly (ethylene imine), and the corresponding Ising model. Here, each spin corresponds to a CH2CHNH2 or CH2CHNH3

+ group. In positions 1 and 2, where the ionizable groups (NH2) are neutral, they are in the spin‐up state, while in position 3, where the NH2 is protonated, we have a spin‐down state.

For a particular microstate, the total energy can be approximated by the nearest‐neighbor interactions

among all protonated sites

'

v i jij

E n n (13)

where the primed sum extends to only the nearest‐neighbor pairs, and stands for magnitude of the interaction

energy between two neighboring spins. The canonical partition function of the system is given by

'

0,11

i j iij i

i

n n nN

ni

Q e

(14)

where stands for the protonation energy. Because applies to individual sites, it is equivalent to an

external one‐body potential in the Ising‐chain model. Different from the canonical partition function for an Ising

chain, however, we use in 0 or 1 instead of ‐1 or 1. That difference can be removed by changing of variables

2 1i is n . (15)

With Eq.(15), the canonical partition function can be rewritten as

11

exp[ ' / 4 ( ) / 2 constant]i

N

i j is ij ii

Q s s s

. (16)

where the constant has no effect on the properties of the system. The average degree of protonation in can be

found from a comparison with the equation for the average magnetization (Eq.8)

NH3+H2N H H H H2N

C C C C C C

HH H H H H

H

1 2 3



2

1 sinh[ ( ) / 2] 1

2 22 sinh [ ( ) / 2]

sn

e

(17)

To use Eq.(17), we need to find a relation between the energy of protonation and pH. Toward that end, we

consider the dissociation equilibrium of a single base group (with all neighboring bases remaining neutral)

aKBH B H (18)

where the dissociation constant Ka can be approximated by that of the monomeric group. The Henderson-

Hasselbalch equation relates the pH with apK

loga

BpH pK

BH

(19)

where the square brackets stand for activity. From Eq.(19), we can calculate the protonation energy

ln[ ] ln[ ] ln(10) ( )aBH B pK pH . (20)

Figure 23 Titration curves for a weakly ionizable polyelectrolyte and its corresponding monomer. The curve for the monomer is calculated according to the standard base‐acid equilibrium and that for the polyelectrolyte is from the one‐dimensional Ising model [Eq.(17)].

Figure 23 shows the degree of protonation, n , as a function of pH for a long ionizable polyelectrolyte

with 10apK and 2 . For comparison, Figure 23 also shows the degree of protonation as a function of pH

for a monomer with only one ionizable group. The titration curve for the polybase is different from that for its

Fra

ctio

n o

f p

roto

nat

ion

2 4 6 8 10 12 0

0.2

0.4

0.6

0.8

1

monomer =2 polyelectrolyte

pKa=10

pH



monomeric counterpart; the former proceeds in two steps due to the interaction between nearest neighbors,

while the latter proceeds in a single step. The prediction of the Ising model provides only qualitative agreement

with experiments; because Coulomb forces are long‐range, it is not sufficient for Eq.(13) to consider only nearest‐

neighbor interactions. To obtain quantitative agreement with experiment, we must also consider interactions

between protonated groups that are not nearest neighbors21. Nevertheless, this example illustrates how a simple

model (the one‐dimensional Ising chain) is useful for obtaining the essential features of a real phenomenon. For

accurate results, corrections must be applied.

Two‐dimensionalIsingmodel

In the two‐dimensional Ising model, each spin is in contact with four neighboring spins. At a given

microstate, specified by the orientations of all spins, the system energy is

, 1, , , 1( )v ij i j i j i j i ji j i j

E h s s s s s (21)

Eq.(21) is the two‐dimensional form of Eq.(2) for the one‐dimensional Ising chain. In Eq.(21), , 1i js stands for a

spin orientation, and subscripts i and j give the spin index (row and column). As in the one‐dimensional Ising

chain, h stands for the interaction energy between a spin and an external field, and is a coupling parameter

such that, when the interaction energy of two aligned neighboring spins is , the system is called ferromagnetic

but if it is , it is called paramagnetic.

21 Borkovec M. and Koper G.J.M., Ber. Bunsenges. Phys. Chem., 100(6), 764‐769 (1996).



Figure 24 In a two‐dimensional Ising model each spin interacts with its four immediate neighbors.

In the absence of an external field, the partition function for a two‐dimensional Ising model was first

derived by Lars Onsager in 1944. However, no analytical solution has as yet been found if there exists an external

field. Because the mathematical details of the derivation are complex, the derivation is not given here22. We

discuss only the thermodynamic properties derived from Onsager’s partition function for the ferromagnetic case.

22 A detailed derivation is presented in Equilibrium Statistical Physics, M. Plischke and B. Bergersen, World Scientific Publishing Co. Pte. Ltd.(1994).



Figure 25 The effect of temperature on magnetization for a two‐dimensional Ising model. In the absence

of an external field, the reduced critical temperature /B ck T of the phase transition is 2.269.

Using Onsager’s partition function, the average magnetization is given by

4 1/8[1 sinh(2 ) ]

0c

c

T Ts

T T

` (22)

where /B ck T 2.269 is the critical temperature or Curie temperature23. Unlike a one‐dimensional Ising chain,

the two‐dimensional Ising model exhibits spontaneous magnetization at reduced temperatures below cT . Figure

25 shows s as a function of temperature. When cT T , the majority of the spins are aligned ( s is greater

than zero) without an external field and the system is magnetized, i.e., in a ferromagnetic state. However, when

cT T , the spins are randomly oriented and the average magnetization is zero, corresponding to a paramagnetic

state. Because s reflects the status of “order” or alignment of spins in the Ising model, it is often referred to as

the order parameter.

23 The Curie temperature is the maximum temperature where the material is ferromagnetic. At any temperature above the Curie temperature, there is no spontaneous magnetization.

0

0.2

0.4

0.6

0.8

1

1 1.5 2 2.5

Magnetization

kBT/



From Onsager’s partition function, we obtain the Helmholtz energy, internal energy and heat capacity for

a systems of N spins at zero field

2 2

/ 2

0

1 1 sin1/ ln[2cosh(2 )] ln

2

q xF N dx

(23)

212/ coth(2 ) 1 2 tanh (2 ) 1

KU N

(24)

2 2 21 2

4/( ) coth(2 ) 1 tanh (2 ) / 2 2 tanh (2 ) 1V BC Nk K K

(25)

where

22sinh(2 ) / cosh (2 )q (26)

2 2

/ 2

1 0 2 2

sin

1 sin

q xK dx

q x

(27)

1/ 2/ 2 2 2

2 01 sinK q x dx

(28)

Figure 26 Thermodynamic properties of the two‐dimensional Ising model at zero external field. For the heat capacity, the solid line is from Eq.(25) and the dashed line is from the scaling law Eq.(30). The

important result shown here is the singularity in VC at the critical temperature / 2.269Bk T . The

reciprocal of 2.269 is 0.441.

0 0.2 0.4 0.6 0.82

1.5

1

0.5

0

-S/NkB

U/N

F/N

0.434 0.439 0.443 0.4481

3

5

7

Cv/NkB



Figure 26 presents thermodynamic properties of the two‐dimensional Ising model as functions of . At

first sight, the Helmholtz energy, internal energy and entropy profiles in a two‐dimensional Ising model (Figure 26)

appear to be similar to those for a one‐dimensional Ising chain (Figure 19). In both cases, the major contribution

to Helmholtz energy F is U at low temperature and TS at high temperature. However, closer inspection of

Figure 26 indicates that in the two‐dimensional case, the internal energy and entropy profiles show inflection

points. These inflection points indicate the occurrence of a singularity in heat capacity, as shown in Figure 26.

Singularities of some thermodynamic properties represent the most important characteristics of a phase

transition24.

Near the critical temperature, the Ising model indicates that the average magnetization and heat capacity

obey scaling laws25

1/ 8( ) ,c cs T T T T (29)

28/ ( ) ln 1 constantB

V cc

k TC N

T

. (30)

The critical exponent 1/8 and the logarithmic function characterize a phase transition in a two‐dimensional Ising

model.

To illustrate the scaling law for a two‐dimensional phase, Figure 27 presents the singularity of heat

capacity for a monolayer of carbon monoxide on graphite. Near the critical temperature, the effect of

temperature on heat capacity agrees well with the two‐dimensional Ising scaling law (Eq.30).

24 A phase transition is second order if the order parameter, here s , is continuous when the system changes from one phase

to another. The phase transition of the two‐dimensional Ising model at the critical point provides an example of a second‐order phase transition. A phase transition is first order if the order parameter varies discontinuously, as in the condensation of a

vapor to a liquid where the order parameter is the fluid density. Scaling law for s is shown in Eq.(29).

25 A scaling law describes how some quantity (e.g., s or vC ) varies with some independent variable as ( TTc ) or

( ) /c cT T T , the deviation of the system’s temperature from its critical temperature.



Figure 27 Heat capacity CV versus | | /c cT T T for N carbon monoxide molecules adsorbed on a

graphite surface at 75% coverage. The points are obtained from measurements and the solid lines are fitted to the two‐dimensional Ising scaling law, Eq.(30). (From H. Wiechert and S.A. Arlt, Phys. Rev. Lett., Vol. 73, 2090, 1993).

Three‐dimensionalIsingmodel

For many years, phase transition occupied a much‐studied position in statistical physics. Although

examples of phase transitions are abundant (e.g., the freezing of water or the condensation of a vapor to a liquid),

the fundamental physics of phase transition is hard to understand, especially near the critical point where two

coexisting fluid phases become identical26. The three‐dimensional Ising model provides a suitable model system

for describing some of the essential features of phase transitions. Regrettably, however, no one has been able to

derive exact analytical expressions for the thermodynamic properties of the three‐dimensional Ising model; it now

appears that such analytical expressions may never be found. However, numerical solutions from computer

simulations and from calculations based on renormalization‐group‐theory have provided some useful insights into

26 Solid‐liquid transitions do not have a critical point.

log

-3 -2 -1 0

Cv/

Nk B

0 .0

0.5

1.0

1.5

2.0

T<Tc

T>Tc



the phase transition of the three‐dimensional Ising model. In this section, and the next, we briefly discuss how

these insights help to interpret some experimental data.

As in the two‐dimensional Ising model, a three‐dimensional Ising model shows a phase transition at

temperatures below a critical temperature. The exact value of the critical temperature is sensitive to the details of

the model, including the type of lattice (e.g. face‐centered or body‐centered cubic lattice). However, near the

critical temperature, the scaling laws describing the structure and thermodynamic properties of the Ising model

are independent of those details. It has now been well established that, according to a three‐dimensional Ising

model, the heat capacity near the critical temperature diverges following the power law

/ | |V CC N T T . (31)

In addition, near the critical temperature, the order parameter s as defined in Eq.(7) also follows a power law

~ ( )cs T T . (32)

From simulations and from experiments the approximate critical exponents are 0.110 and 0.326 .

It is a striking feature of phase transitions that the critical exponents and depend only on the

spatial dimensionality of the system, independent of model details. As a result of this striking feature,

experimental results for the thermodynamic properties of one system near its critical point can provide results for

other systems with the same dimensionality. The commonality of critical exponents is called universality, which

holds for a variety of different systems. Systems with the same spatial and order dimensionalities are said to

belong to the same universality class. For example, vapor‐liquid equilibria of fluids and fluid mixtures belong to the

three‐dimensional Ising universality class because in these systems, the order parameter is scalar (a one‐

dimensional variable), and the spatial dimensionality is three. Close to the critical point, thermodynamic

properties of all these systems can be described by the same scaling laws27.

With a simple change in variables, the Ising model can be applied to describe liquid‐liquid or liquid‐vapor

phase transitions of monatomic fluids. To illustrate, consider a lattice model for a binary liquid mixture containing

27 However, the proportionality constant (not shown in Eqs.(29) and (30)) depends on the particular system of interest.



components A and B 28. In this lattice model, each lattice site is occupied by one and only one molecule of

species A or B ; the lattice is completely filled. To study the liquid‐liquid phase transition of the binary mixture,

we consider only nearest‐neighbor interactions, and the interaction energy between two identical molecules is

arbitrarily set equal to zero.29 In this system, liquid‐liquid phase transition is driven by the number of contacts

between different species and by the corresponding energy parameter AB . A microstate of the system is

specified by the arrangements of A and B molecules on the lattice. The system energy is given by

/ 2AB ABE N (33)

where ABN stands for the number of nearest‐neighbor A‐B pairs and the factor ½ prevents counting AB pairs

twice.

To transfer this lattice model for liquid‐liquid phase transition to an equivalent three‐dimensional Ising

model, we designate 0in when a lattice site is taken by A , and 1in when it is taken by B . With the

identification of molecules represented by in 0 and 1, and remembering that 0AA BB , the nearest‐

neighbor interaction energy ij between two adjacent sites i and j can be expressed as

( 2 ) / 2ij i j i j ABn n n n (34)

For any microstate, the system energy can be expressed as

( 2 ) / 2i j i j ABij

E n n n n (35)

where the summation extends only to nearest‐neighbor pairs.

For convenience, we make a variable change; let

2 1i is n and 2 1j js n (36)

Substitution of Eq.(36) into Eq.(35) gives

i j Tij

E s s N Z (37)

28 A similar description can be applied to the vapor‐liquid transition. In that case, the lattice is either occupied by a molecule or it is empty. 29 The results are not affected by this assumption.



where / 4AB , Z is the lattice coordination number (i.e., the number of nearest neighbors), and TN is the

total number of lattice sizes. The last term on the right side of Eq.(37) is a constant irrelevant to the

thermodynamic properties of the system. Except for the irrelevant constant, Eq.(37) is identical to the energy of

an Ising model (e.g., Eq.13) except here, a “spin up” corresponds to a lattice site that is occupied by a molecule of

species B, and a “spin down” corresponds to an occupancy of molecule A. As a result, the thermodynamic

properties and phase behavior derived from the Ising model are applicable to binary mixtures of liquids.

Figure 28 Coexistence compositions of a symmetric binary liquid mixture A+B in a simple cubic (SC), a body‐centered‐cubic (BCC), and a face‐centered‐cubic (FCC) lattice. The vertical axis gives the reduced temperature, and the horizontal axis is the mole fraction of component B. While the critical mole fraction is always 0.5, the reduced critical temperature depends on lattice geometry. The symbols are from Monte Carlo calculations and the lines are to guide the eye.

The phase transition of the lattice model for a binary liquid mixture has been investigated using Monte

Carlo calculations30. Figure 28 shows the phase coexistence curves for three lattices: simple cubic (SC), body‐

centered‐cubic (BCC) and face‐centered cubic (FCC). The shapes of the curves are similar, characterized by a nearly

flat section in the middle. However, the critical temperature depends on the lattice geometry. Near the critical

temperature, the curvatures of these coexistence lines are alike, indicating the universal scaling law of the three‐

30 Lambert SM, PhD thesis, The University of California at Berkeley, 1995.

k B

T/

AB

Mole fraction B

1.13

1.61

2.46



dimensional Ising universality class. The flatness of the vapor‐liquid coexistence shown in Figure 28 is in agreement

with experiment. The flat curvature cannot be achieved by classical solution theories or by classical equations of

state.



1.11Helix‐coiltransitionofpolypeptides

A protein is a polypeptide chain that often adopts a unique, well-defined 3-dimensional conformation

(native structure) essential for its biological functions. For small globular proteins, it is often hypothesized31 that

the native structure is determined only by the amino‐acid sequence along the linear backbone, that is, by its

primary structure. Because virtually all biological activities of a protein are determined by its structure,

understanding the relation between the 3‐dimensional structure of a protein and its amino‐acid sequence plays a

crucial role in protein engineering. In this section, we apply a canonical partition function to describe the helix‐coil

transition of a polypeptide. Such a description is useful for understanding the stability of protein structures and

may provide insight on protein folding.

Figure 29 Formation of a peptide bond between two amino‐acid molecules (such as those shown in Table 3) by the release of a water molecule. R1 and R2 denote amino‐acid residues.

When unfolded, proteins are linear polymers; more precisely, they are polypeptides of various amino acids linked

by peptide bonds32. As illustrated in Figure 29, a peptide bond is formed by the condensation of two amino acids,

that is, by the reaction of an amino group with a carboxyl group, eliminating water. The functional groups,

31 It is known as Anfinsen's dogma or the thermodynamic hypothesis in molecular biology. 32 While the number of possible different amino‐acid groups is very large, only twenty amino groups are observed in natural proteins.

O

H3+N

H R1

C

C

O

H C C

H

H

H

H

R2

O

O

N

H3+N

H R1

C

C H C

C

H R2

O

O

N

O

H

H2O

72 | T h e

represente

Table 3 Str

Aliphatics

Aromatics

Polar

Charged

m a t e r i a l

ed by R1 and R

ructure of stan

(

L‐I

L‐(

L‐A(

T

l i s s u b j e

R2, are amino‐

ndard amino ac

Glycine (Gly / G)

Isoleucine (Ile / I)

‐Histidine (His / H)

Asparagine (Asn / N)

L‐Lysine (Lys / K)

Thermodynam

e c t t o c o

‐acid residues.

cids, three‐lett

L‐Al(Ala

L‐Pr(Pro

L‐Pheny(Ph

L‐Glut(Gln

L‐Arg(Arg

mics for Molec

p y r i g h t

Table 3 show

er symbols, an

aninea / A)

rolineo / P)

ylalaninee / F)

taminen / Q)

ginineg / R)

cular Enginee

ws for the struc

nd one‐letter sy

L‐Valin(Val / V

L‐Cyste(Cys /

L‐Tyros(Tyr /

L‐Serin(Ser /

L‐Aspartic(Asp /

ering, Wu and

ctures of 20 st

ymbols.

neV)

ineC)

sineY)

neS)

c acidD)

d Prausnitz, 3

andard amino

L‐Leucine(Leu / L)

L‐Methionin(Met / M)

L‐Tryptopha(Trp / W)

L‐Threonin(Thr / T)

L‐Glutamic a(Glu / E)

/2010

acids.

ne

an

e

cid



While the peptide bonds are the same for all proteins, the amino‐acid residues vary from one protein to

another. In solution, because amino groups and acid groups are connected by peptide bonds, the backbone of a

polypeptide is not electrically charged (except at the ends), but an amino‐acid residue (i.e., branches or side chains

from the backbone) may or may not be charged, depending on its ionization properties and on the pH of the

solution. The net charge of a protein is the algebraic sum of all charges on the protein molecule. The isoelectric

point (pI) of a protein is the pH of the solution where the net charge is zero.

Figure 30 Hydrogen bonding between amino acids in an ‐helix chain. Because a hydrogen bond spans

over four amino acid residues (shown by open arrows) along the chain, hydrogen‐bond donors and acceptors are not satisfied near the ends (shown in black and gray arrows). Here i stands for amino‐acid index along the chain, R denotes an amino‐acid residue, and H, C, N and O are chemical elements. Cα is the carbon in each amino‐acid residue where the amino and carboxylate groups are attached.

An ‐helix33 is formed by hydrogen bonds between the carbonyl oxygen of amino‐acid residue i and the

amide hydrogen of residue i+3. Here, i and i+3 refer to the positions of amino‐acid residues in the protein

sequence. Figure 30 illustrates the interlacing hydrogen‐bond network in an ‐helix. Because a hydrogen bond

spans over four amino‐acid residues along the chain, the three carbonyl oxygens at the C‐terminus34 and the three

33 The nomenclatures α‐helix and β‐sheet were introduced by William T. Astbury, an English physicist and molecular biologist who made pioneering X‐ray diffraction studies of biological molecules. 34 The C‐terminus (or C‐terminal end) is the end of an amino‐acid chain terminated by a carboxyl group (‐COOH).

H N

Cα N H

Cα

O

O

O

H N

Cα

O

N H

Cα

O

H N

Cα

O

N H

Cα

O

H N

R

R

R

R

R

R

Unsatisfied hydrogen-bond acceptors at C-terminus

Unsatisfied hydrogen-bond donors at N-terminus

i i+1 i+2 i+3 i+4 i+5



amide hydrogen atoms at the N‐terminus35 do not form hydrogen bonds. In space, the polypeptide is arranged in

a helical conformation with about 3.6 residues per turn and a translocation of 0.15 nm per residue.

In an ‐helix, a hydrogen bond goes from the thi to the ( 3)thi residue; formation of the first hydrogen

bond requires ordering of three intervening residues. Such ordering leads to a large entropic cost36 to initiate the

formation of an ‐helix. However, after the first hydrogen bond is formed, formation of an additional hydrogen

bond at the end of a pre‐existing helix requires only one additional residue to be fixed in the correct orientation.

Consequently, for the formation of a second, third, etc. hydrogen bond, the entropic cost is much smaller

compared with that for formation of the first hydrogen bond. Because all of the hydrogen bonds and peptide

groups point in the same direction, the dipoles of the hydrogen‐bonding groups are in near‐perfect alignment. The

side chains of residues project outward from the ‐helix, minimizing steric interference.

The ‐helix/coil transition of a polypeptide refers to the equilibrium of the polymer chain between the

two coexisting states: in spring‐like helical structure or in a random‐coil structure. To construct a model for the ‐

helix/coil transition, we assume that each amino‐acid residue exists in one of two possible states: helix (h) or coil

(c). In a polypeptide of m residues, we have a maximum of 2m residues that can adopt the helical state

(because the two end residues are unconstrained), and a maximum of 4m hydrogen bonds that can be formed

along the backbone. To find the distribution of the residues between the two states, we use the coil state as a

reference and assume that its Helmholtz energy is zero. Because of hydrogen bonding, the Helmholtz energy of a

residue in the helix state depends on the status of its neighboring residues. It is denoted as su if it is the leading

residue in an ‐helix, or as au if it follows a pre‐existing ‐helix (i.e., the preceding residue is also in the helical

state). The Helmholtz energy difference between a coil state and a helical state can be understood as the

difference in Helmholtz energy when the two states of a residue are at equilibrium. Near the transition, this

Helmholtz energy is on the order of Bk T .

35 The N‐terminus (or N‐terminal end) refers to the end of a protein or polypeptide chain terminated by an amino group (‐NH2). 36 In general, nature wants to increase entropy. Therefore, if a process is accompanied by a decrease in entropy, that decrease is considered a “cost”. Sometimes “penalty” is used instead of “cost”.



A microstate relevant to the ‐helix/coil transition is specified when each amino‐acid residue of the

polypeptide chain is given helical or coil status. The problem of our interest is to determine how many residues in

a polypeptide chain are in helical structures at a fixed solution condition. To solve this problem, we consider a long

polypeptide where the end effects can be neglected. The canonical partition function of an m‐residue polypeptide

is given by

1 2 3 1

... expm

m

ir r r r i

Q u

, (1)

where ir represents the conformation of residue i ( ir h or c , where h stands for helical and c stands for

coil), and iu stands for the change of the Helmholtz energy of residue i from a coil state to a helix state; iu is

zero for a coil. For a helix, iu is su or au , depending on the conformation of the preceding residue.

Eq.(1) resembles the partition function of an Ising chain. With some algebra (see Box 4), this partition

function can be evaluated in the limit of large m.

Q m 1 , (2)

where

21 (1 ) (1 ) 4 / 2a a su u ue e e

. (3)

The average number of residues that are in a helical structure hn can be divided into the number of

“adding” helical residues an and the number of “starting” helical residues sn . From the partition function in

Eq.(2), an and sn can be calculated by differentiating lnQ with respect to au and su , respectively

2

1, ,

ln ( 1)

2 2 (1 ) 4

a a a

a s

a

u u u

a u ua T m u

Q m e e en

u e e

, (4)

2

1, ,

ln

(1 ) 4

s

a s

a

u

s u us T m u

Q m en

u e e

. (5)

The total number of residues in the helical structure is



sa

aasa

uu

uuuu

sahee

eemennn

4)1(

121

2 2

)(

1. (6)

The property of greatest interest in helix-coil transition is the average fraction of residues in the helical

state, or the helix fraction, denoted by . From Eq.(6), we have

ss

ss

m

nh

4)1(

121

2 21. (7)

where s e ua and e u us a( ) . Because the energy difference between a helical state and a coil state is au ,

parameter s is the Boltzmann factor for the distribution of a single residue between the coil and helical states.

Parameter reflects additional energy in initiating a helical residue. Eq.(7) indicates that for a long polypeptide, the

fraction of residues that are in a helical state is completely determined by two parameters, s and , both depending

on temperature.

Figure 31 Theoretical and experimental results of the helical fraction as a function of mT T for poly‐‐benzyl‐L‐glutamate chains of three different lengths, denoted by m. The points represent experimental

data; lines are calculated from the theory (Eq.7). mT =285K is the midpoint temperature of the transition

curve for m=1500 (Reproduced from Zimm B. H., Doty P. and Iso K., Proc. Natl. Acad. Sci., 45:1601‐1606, 1959).

0.0

0.2

0.4

0.6

0.8

1.0

‐15 ‐5 5 15 25 35

T‐Tm, K

m=1500

m=46

m=26



The analysis of helix‐coil transition has been tested using experimental data. Figure 31 compares

theoretical and experimental helical fractions for poly‐ ‐benzyl‐L‐glutamate chains with average length 1500, 46

or 26 residues. The solvent is 70:30 (by weight) dichloroacetic acid and ethylene dichloride. The long‐chain model

is used for the polypeptide with 1500 residues; the others are calculated using the short‐chain model (see Problem

17). In correlation of the experimental results, we assume that 42 10 independent of temperature.

Parameter s is expressed through energy parameter ae , which is related to ua through the Gibbs-Helmholtz

relation

( / )a au e , (8)

Let Tm be the temperature where ua 0 . Integration of Eq.(8) gives

ln( ) (1/ 1/ ) /a m a Bs u T T e k . (9)

Using 890 /ae cal mole obtained by best fitting, we see that agreement between calculated and experimental

data is reasonable, suggesting that the simple theory captures the essential physics of the helix/coil transition.



Box4Partitionfunctionforhelix/coiltransition

It is convenient to use the transfer-matrix method to evaluate the partition function for the -helix/coil

transition. Let Qmc represent the partition function of an m -residue polypeptide whose final residue is in a coil state,

and let Qmh represent the partition function when the final residue is in a helical state. In general, the partition

function of an m -residue polypeptide is given by

Q Q Qmc

mh .

This equation can be expressed as the inner product of two vectors,

1

1),(

1

0),(

0

1),( h

mcm

hm

cm

hm

cm QQQQQQQ .

The two unit vectors in the above equation can be understood as the coil and helical states of residues. When the

final residue is in a helical state, Qmh can be related to the partition functions of an ( )m1 -residue polypeptide

1 1s au uh c h

m m mQ Q e Q e

Similarly, cmQ can be written as

Q Q Qmc

mc

mh 1 1 .

Thus, the vector ( , )Q Qmc

mh can be calculated from a recursive matrix

( , ) ( , )~

Q Q Q Q Mmc

mh

mc

mh 1 1 ,

where

~

1

1

s

a

u

u

eM

e

is called the transfer matrix.

Because the first residue can only be in a coil state or in a starting helical state, the partition function of one residue is given by

1 1

1( , ) (1, 0)

1

s

a

uc h

u

eQ Q

e

Using this equation, and the recursive relation, we have the desired expression for the partition function of an m -residue polypeptide,

1 1

(1, 0)1 1

s

a

mu

u

eQ

e

.



Box4Partitionfunctionofhelix/coiltransition(continued)

The mth-power matrix in the above equation can be evaluated by diagonalization of M

~ using a similarity

transformation

2

11

~~

1

~

0

0

TMT ,

where T~

is the transformation matrix, T~

1 is the matrix inverse of T~

and 1 2, are the eigenvalues of the matrix ,

given by

21,2 (1 ) (1 ) 4 / 2a a su u ue e e

.

The transformation matrix T~

is

2 1

~

1 1

1 1T

and its inverse is

2

1

21

1

~ 1

1

1

11

T .

Diagonalization of M~

yields an analytical expression of the partition function

11

~ ~2

1 1 212 1

2 2 1 2

1 11 2 2 1 1 2

0 1(1, 0)

10

0 /( )(1 ,1 )

/( )0

(1 ) (1 ) /( )

m

m

m

m

m m

Q T T

(*)

Since 1 2 , and for a long polypeptide m 1, the partition function in Eq.(*) can be approximated by

Q m 1 .

The error of this approximation becomes vanishingly small for 1m .



1.12Grandcanonicalensemble

The previous sections are concerned with closed systems where the number of particles is fixed but

where energy can fluctuate around its mean value. Many thermodynamic systems of practical concern, however,

are in contact with an external environment that allows both mass and energy exchanges. Common examples

include molecules near a substrate as in heterogeneous separation and reaction processes in engineering and

biological systems. The most convenient way to define an open system is by temperature (T ), volume (V ), and

chemical potential i for every component i of interest. In thermodynamics, a chemical potential is an intensive

property that specifies the reversible work required to transfer molecules to the system from a reference state

(e.g., an ideal gas). In an open system, the chemical potential of each species is everywhere uniform.

A grand canonical ensemble refers to all accessible microstates of an open system at equilibrium. To find

the distribution of microstates and corresponding thermodynamic properties at equilibrium, we consider an m ‐

component open system at given T and V. To help define the chemical potential of individual species, we assume

that the system coexists with an “imaginary” bulk system, as shown schematically in Figure 32. The chemical

potential of each species in the bulk can be uniquely defined by its temperature, pressure and composition.

2

3

2

3

m

m

……

1

1

Figure 32 A schematic representation of an m ‐component open system in equilibrium with its environment. The vertical dashed line indicates that particles are allowed to transfer through this boundary.

Following a procedure similar to that for a closed system, the distribution of microstates in an open

system can be determined from the second law of thermodynamics. As in a canonical ensemble, the entropy is

uniquely defined by the distribution of microstates



lnB v vv

S k p p . (1)

For an open system at equilibrium, the entropy attains a maximum subject to the constraints of constant average

total energy, constant average total number of particles, and the normalization condition for the probability

distribution

v vv

U p E (2)

,i v i vv

N p N

1, 2, ,i m (3)

1vv

p (4)

where U stands for the internal energy, i.e., average total energy of the system, and vp is the probability that the

system is at microstate v.

As discussed before for isolated and closed systems, the conditional maximum can be found by using the

Lagrange-multiplier method. Differentiation of the entropy with respect to the microstate probability yields

,(ln 1) 0v v i i vi

p E N (5)

where , , and i are Lagrange multipliers arising, respectively, from the normalization condition (Eq.4), and

constraints for the internal energy (Eq.2) and average number of molecules (Eq.3). Rearrangement of Eq.(5) gives

,exp[ 1 ]v v i i vi

p E N (6)

Substitution of Eq.(6) into (1) yields

/ ln 1B v v i iv i

S k p p E N (7)

From the thermodynamic relations,

,1/ ( / )iV NT S U (8)

, ,( / )j ii i U V NT S N (9)

we find the Lagrange multipliers:

1/ Bk T (10)



i i (11)

Finally, the multiplier is determined from the normalization condition (Eq.4),

ln 1 (12)

where

exp[ ( )],

E Ni ii

(13)

is the grand partition function, and 1/ ( )k TB .

The final expression for the probability distribution is obtained by substitution of Eqs.(10)-(12) into Eq.(6)

,

1exp[ ( )]v v i i v

i

p E N (14)

and the entropy of an open system is obtained from Eq.(7)

/ / /i ii

S U T N T T (15)

where

lnBk T . (16)

The quantity defined in Eq.(16) is called the grand potential, a free‐energy that is equivalent to Helmholtz

energy for closed systems. As the Helmholtz energy is introduced in a canonical ensemble for the convenience of

applications, the grand potential is an extensive thermodynamic function that is convenient to use for open

systems. For example, we can readily express the internal energy and the average number of particles i in terms of

the grand potential:

1, 1,, ,( ln / ) ( / )

i m i mv v V Vv

U E p

(17)

, , ,( ln / ) ( / )i i v v i V i Vv

N N p (18)

Eq.(17) is equivalent to the familiar Gibbs‐Helmholtz equation in classical thermodynamics. The grand partition

function also gives the mean‐square deviation for the number of particles i and the mean square‐deviation for the

total energy

22 2

, ,/

j ii i i i i V

N N N N

(19)



1,

22 2

,/

i mVE E E U

. (20)

Because is an intensive variable, Eq.(19) suggests that in an open system, the relative deviation of the number

of particles i, 2 /i iN N , scales as 1/ iN . For a typical macroscopic system, the average number of

particles of species i is on the order of 23~ 10iN . As a result, the fluctuation of the number of particles is

extremely small (except at the critical point). Similarly, Eq.(18) suggests that the relative deviation of the total

energy is extremely small for a typical macroscopic system.

In summary, this section introduces the grand partition function (Eq.13) for describing the probability

distribution of microstates in an open system (Eq.14). In addition, we introduce grand potential as a

thermodynamic free energy convenient for applications of statistical mechanics to open systems. Because in an

open system, the temperature, chemical potential, internal energy and average number of particles i are fixed, it

follows from Eq.(16) that the grand potential is a minimum for an open system at equilibrium. This is another

equivalent statement of the second law of thermodynamics.



1.13Gasadsorption

Gas‐solid physical adsorption equilibria are required for designing separation and purification processes.

The equilibrium between the adsorbate (adsorbing phase) and the adsorbent (the solid surface, also called the

substrate) is expressed by an adsorption isotherm that describes the amount of adsorption per unit area of

adsorbent as a function of pressure37 at constant temperature. In this section, we illustrate how the grand

partition function can be used to establish a few useful gas‐adsorption isotherms.

Langmuiradsorptionisotherm

Figure 33 Schematical representation of the Langmuir adsorption model. Each surface site can adsorb one and only one gas molecule. In simple Langmuir adsorption, the solid surface is homogenous: all sites are equivalent, i.e., the interaction energy between a surface site and a gas molecule is the same for all sites. There is no interaction between adsorbed gas molecules.

Consider the adsorption isotherm of a pure gas in equilibrium with a homogeneous solid surface at low

pressures. Figure 33 gives a representation of gas adsorption on a flat surface. We assume that the solid surface

(substrate) can be represented by a two‐dimensional lattice that contains distinguishable but identical interaction

sites. There is only one layer of adsorbed molecules (monolayer adsorption). Each interaction site can adsorb only

one gas molecule. For simplicity, we assume that the surface sites are independent, i.e., the occupation status of

one site is independent from that of another site, and that the gas molecules on the surface do not interact with

each other. Because the gas molecules on the solid surface represent a non‐interacting open system, it is

appropriate to describe the system with a grand canonical ensemble. In this case, a microstate of the system is

specified by occupation states of the lattice sites, that is, by how many of the surface sites are occupied. The grand

partition function is given by

37 When the adsorbing component is in a gaseous mixture, we use the component’s partial pressure or more precisely, its fugacity.

. . . . . .

gas molecule surface site



0,1

( )

1

(1 )s

i i s

i

Nn n Nin in

ni

q e q e

(1)

where sN is the total number of surface sites, 0,1in denotes the occupancy status of site i (If 0in , the site

is empty. If 1in , the site is occupied.) The interaction energy between a gas molecule and a surface site is given

by ; is the chemical potential of the gas, and inq denotes contributions from the internal degrees of freedom

of a gas molecule on the surface38. For a pure gas, chemical potential depends on the temperature and

pressure of the bulk gas phase in equilibrium with the surface. The exponent sN in Eq.(1) results from the

assumption that all adsorption sites are independent. From the grand partition function, we obtain the average

number of gas molecules adsorbed on the solid surface

ln

1in

sin

q eN N

q e

(2)

At low pressure, the chemical potential of a pure gas can be approximated by that of an ideal gas

0 ln P (3)

where 0 is the chemical potential of an ideal gas at system temperature and unit pressure. Substitution of Eq.(3)

into Eq.(2) yields the fraction of surface sites occupied by gas molecules

1s

N bP

N bP

(4)

where 0exp( )inb q is a parameter that depends only on temperature. Eq.(4) is the Langmuir

adsorption isotherm.

Using much simpler arguments than those used here, Langmuir derived Eq.(4) to describe the adsorption

of gases on metal surfaces. Although the Langmuir adsorption isotherm can be derived from a variety of ways, the

statistical thermodynamic derivation clarifies the underlying physics and the necessary molecular details required

38 Internal degrees of freedom refer to rotation and vibration of atoms within each gas molecule and to quantum states of electrons. For simple gas molecules, to a good approximation, an internal degree of freedom is not affected by the proximity of

other molecules. In deriving the Langmuir adsorption isotherm we assume that inq for adsorbed molecules is equal to that for

gas‐phase molecules.



to calculate parameter b . More important, the statistical thermodynamic derivation provides a basis for

systematically improving the Langmuir isotherm by relaxing some of its severe simplifying assumptions, as shown

later.

Figure 34 Langmuir isotherms for adsorption of nitrogen, ethylene, and nitrous oxide on a steamed‐

charcoal surface at 20 oC. Here, P is gas pressure and is the fraction of adsorption sites occupied by

an adsorbed gas molecule. Parameters b for the three gases are, respectively, 0.2, 0.8, 1.8 atm‐1.

At low pressure, a variety of experimental data (e.g., adsorption of ethyl chloride on charcoal, nitrogen on

TiO2, and O2 or CO on silica) are well represented by the Langmuir adsorption isotherm. To illustrate, Figure 34

shows that the Langmuir adsorption isotherm represents well the experimental results for the adsorption of

nitrogen, ethylene, and nitrous oxide on a steamed‐charcoal surface at 20oC. When parameter b is obtained by

careful measurements of the amount of adsorption, it is then possible to estimate the total surface area of the

adsorbent, provided that we have a reasonable estimate of the projected area of each gas molecule sitting on an

adsorption site. This method can be applied to characterize porous media (e.g., catalyst or catalyst supports).

However, because of its severe simplifying assumptions, the Langmuir isotherm provides little more than an

empirical tool for representing adsorption data. For physical (as opposed to chemical) adsorption, a much better

0

20

40

60

0 20 40 60 80

P, atm

P, atm

Nitrogen

Ethylene

Nitrous oxide



adsorption isotherm is provided by allowing adsorbing molecules to form more than one adsorbed layer leading to

the BET isotherm.

Brunauer‐Emmett‐Teller(BET)adsorptionisotherm

Figure 35 Schematic view of multilayer adsorption illustrating surface coverage by several adsorbed layers. Each square stands for a lattice site that can accommodate a single gas molecule.

In deriving the Langmuir isotherm, we assume that the gas molecules form a monolayer on the solid

surface. This monolayer assumption can be relaxed to allow multilayer adsorption. In the multilayer model (Figure

35), a surface site may adsorb a vertical column of gas molecules: the gas molecule closest to the surface has an

adsorption energy 0 , determined by the interaction between the gas molecule and the surface; all successive gas

molecules in the vertical column have an adsorption energy 1 , determined primarily by the interactions between

two vertical gas molecules. As in the Langmuir model, we assume that all surface sites are identical and

independent and that gas molecules within the same layer do not interact with each other. In other words, we

consider vertical adsorbed gas‐adsorbed gas interactions but we neglect horizontal adsorbed gas‐adsorbed gas

interactions.

The microstate in this model is specified by the gas occupation numbers for each surface site. In this case,

the grand partition function is

0 0 1 0 1( ) 2 2 32 3

1 0

(1 )S

i i s

i

Nn n Nin in in in

i n

q e q e q e q e

(5)

2nd layer

Lattice site Gas molecule

Surface

3rd layer

1st layer . . .

. . .

. . .



where 1( 1)i o in n for 1in and 0 for 0in . As in Eq.(1), inq is the internal partition function of the

molecule. For convenience, we define 0e Pe 39, 00 inq q e and

11 inq q e . The average number

of gas molecules adsorbed on the solid surface can be obtained from the grand partition function:

2

01 10 2

0 1 1 1 0 1

1 2 3( )ln

1 [1 ( ) ] (1 )(1 )s S

qq qN N q N

q q q q q q

(6)

Thus, the average number of adsorbed molecules per surface site is given by

0 0

1 0 1 1 0 1(1 )(1 ) (1 )(1 )S

N q k P

N q q q k P k P k P

(7)

where

0o

ok q e (8)

1 1ok q e (9)

Eq.(7) is the Brunauer‐Emmett‐Teller (BET) adsorption isotherm; at a fixed temperature, it contains two adjustable

parameters, 0k and 1k . At temperatures near or below the gas critical temperature and at advanced pressures,

adsorption on a solid surface is more likely to follow the BET isotherm than the Langmuir isotherm. When

1 1k P (low pressure), and 0 1 (adsorption in the first layer is strongly favored relative to adsorption of

higher layers), the BET equation reduces to the Langmuir form (Eq.4), because in that event, 0 0/(1 )k P k P .

39 See Eq.(3).



Figure 36 Nitrogen adsorption on nonporous silica at 77 K as given by the linearized form of the BET isotherm. Experimental data are from D. H. Everett et al.(1974). Here is the fraction of surface sites

covered by nitrogen molecules, P is pressure and Ps is the vapor pressure of nitrogen.

To fit experimental data, the BET equation has one parameter more than the Langmuir equation.

However, parameter 1k in the BET equation can be identified as the reciprocal of the saturation vapor pressure of

the gas because as pressure approaches saturation, the bulk liquid begins to condense on the adsorbing surface,

leading to . In other words, as we must have 1 1sk P where sP is the saturation pressure. As a

result, the BET isotherm (Eq.7) can be rewritten as

0

0

/

(1 / / )(1 / )S

S S S

k P P

P P k P P P P

(10)

Rearrangement of Eq.(10) gives

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4

SP

P

)( PP

P

S



0 01/ (1 1/ ) /( ) S

S

Pk k P P

P P

(11)

The linear equation (Eq.11) provides a convenient justification of whether experimental data follow the BET

adsorption isotherm. Figure 36 shows a successful application of the BET equation to adsorption of nitrogen on

nonporous silica at 77K. Here nitrogen is well below its critical temperature (126.2 K).

In deriving the BET adsorption isotherm, we assume a uniform surface and no horizontal interactions

between adsorbed molecules. These assumptions are not valid for typical gas‐adsorption systems. Nevertheless,

compared with experimental results, the BET equation gives good representation of experimental data provided

that, empirically, the range of reduced pressure is restricted to 0.05 / 0.35sP P . At reduced pressures below

0.05, the BET equation underestimates adsorption because of surface heterogeneity; gas molecules have a

relatively strong affinity for the few high‐energy surface sites during the initial stages of adsorption. At reduced

pressures above 0.35, the equation overestimates adsorption because the adsorption energy falls as the number

of adsorption layers further increases.

Gasadsorptioninporousmaterials

The grand partition function can also be used to develop useful adsorption isotherms for gas adsorption in

porous materials (e.g., zeolites). To illustrate, consider a low-pressure gas in equilibrium with a porous medium

that contains Nc pores or cages where each cage can contain up to m molecules. Figure 37 presents a schematic

picture of cages and resident molecules. We assume that all cages are independent, i.e., interactions between

molecules in neighboring cages can be neglected. The grand partition function of gas molecules in Nc independent

cages is

Nc (12)

where stands for the grand partition function for each cage.



Figure 37 Schematic diagram for gas adsorpton in a porous medium. In this picture, the solid contains four cages (pores). Spheres represents gas molecules in the gas.

To find , we assume that there is an “average” interaction energy between a gas molecule and the cage,

represented by . Suppose that a cage of volume v may contain a maximum of m molecules. The grand partition

function for the gas molecules in the cage is

sm

ss

sssinin vsv

s

eqv

eq])1([

!1 0

233

, (13)

where h Mk TB/ 2 represents the thermal wave length40, M is the mass of a gas molecule, qin stands for the

contributions to the partition function from the internal degrees of freedom of a gas molecule inside the cage, and ov

is the volume of a gas molecule. Approximately, 0)1( vsv stands for the volume accessible to the ths molecule

in a cage that is already occupied by 1s molecules. The maximum number of molecules in the cage, m , is

determined by the largest integer less than 0/ vv . The first term on the right side of Eq.(13) corresponds to the

empty cage; the second term corresponds to a single molecule in the cage, and the summation corresponds to

contributions due to multi-occupied cages.

40 In Eq.(13), the quantity v / 3 can be understood intuitively as the number of microstates of a gas molecule in a space of

volume v . Imagine that the microstate of a gas molecule is fixed by a small volume of 3. The total number of microstates

for a single gas molecule in a cubic cage of volume v is given by v / 3. When there are two gas molecules in the cage, the

total volume accessible to each molecule is v v 0 . In that event, the number of microstates for each gas molecule is

( ) /v v 03 . A factorial (2!) accounts for the fact that two gas molecules are indistinguishable. Similar analysis applies to a

cage containing s gas molecules where ms 1 .



The average number of gas molecules adsorbed in the porous medium can be derived from the grand

partition function. For convenience, we define a temperature-dependent constant

K q vein 0 3/ (14)

At low pressure, the chemical potential of a gas can be approximated by 0 ln P [Eq.(3)]. The number of

gas molecules per cage is then given by

N

N N

KP KP s v v s

KP KP s v v sc c

s

s

m

s

s

m

11 1 1

1 1 1

02

02

ln{ [ ( ) / ]} / ( )!

{ [ ( ) / ]} / !

(15)

Two model parameters appear in Eq.(15): a temperature-dependent parameter K and a characteristic

molecular volume of the cage v0 . These parameters are normally treated as adjustable parameters by fitting to

experimental results. Figure 38 shows an application of Eq.(15) to describing single-gas adsorption (O2, N2 , CO) in

zeolites over a wide range of temperature and low pressure. Table 4 presents pertinent parameters. While 0v is

independent of temperature, K is a function of temperature (Eq.14). Figure 38 shows that the simple statistical-

mechanical model provides an excellent correlation for the gas adsorption data. The model has been successfully

extended to predict equilibrium sorption of binary gas mixtures41.

41 Principles of adsorption and adsorption processes. Ruthven D.M., New York: Wiley, 1984.



Figure 38 Equilibrium isotherms for adsorption of O2, CO and CH4 in 5A zeolite. Curves are from Eq.(15) with parameters in Table 4 (from Ruthven 1984).

Table 4 Molecular parameters in Eq.(15) for correlating single-gas adsorption data on a 5A Zeolite (after Ruthven, 1984).

Gas KT , K (Torr) 0v (Å3)

O2 145 0.31 46 201 0.007 46 CO 145 50 59 CH4 195 - 64.5 212 0.083 64.5 230 0.033 64.5 253 0.011 64.5

1 10 100 1000 0

1

2

3

4

5

6

7

8

9

10

11

12

CO - 145 K

O2 - 145 K

CH4 - 195 K

CH4 - 212 K

CH4 - 230 K

CH4 - 253 K

O2 - 201 K

cNN /

torr,P



1.14Isobaric‐isothermalsystems

Most isothermal chemical and physical phenomena occur at constant pressure, not constant volume. It is

therefore useful to construct an ensemble for an isobaric‐isothermal system where the total number of particles is

fixed but where both the total energy and volume fluctuate around their mean values at equilibrium. The

assembly of all accessible microstates of a closed system at constant temperature and pressure is called an

isobaric‐isothermal ensemble. In this ensemble, pressure, temperature and number of particles are the

independent fixed variables.

Figure 39 A system containing m different species at constant temperature and pressure.

To derive the distribution of microstates in an isobaric‐isothermal ensemble, we consider an m‐

component system at equilibrium, as illustrated in Figure 39. We follow a procedure similar to that used for other

ensembles. The second law of thermodynamics requires that, at equilibrium, the system entropy attains a

maximum, subject to three constraints: constant average total energy, constant average volume and normalization

of microstate probability42. These constrains are described by

42 At equilibrium, there is no transfer of net energy between the system and its surroundings. As a result, at constant pressure,

0pdV . The constraint of constant pressure in an isobaric‐isothermal ensemble is therefore equivalent to that of an

equilibrium system at constant average volume.

T, P, N1, N2, …, Nm



i

E p E (1)

v vi

V p V (2)

1vv

p (3)

where, as before, P is the probability that the system is in microstate ; E is the total energy of the system at

this microstate, and V is the total volume.

Application of the Lagrange‐multiplier method to find the constrained maximum of entropy yields

1 2 3(ln 1) 0Bk p E V (4)

where 1 2 3, , are Lagrange‐multiplier constants that can be determined from classical‐thermodynamic

identities: 1 ( / )VT S E and / ( / ) EP T S V .

Following some rearrangements, we find that the probability that an isobaric‐isothermal system is in

microstate with energy E and volume V is given by

exp( ) /p E PV Y (5)

where

exp( )Y E PV

. (6)

Y is the isobaric‐isothermal partition function.

From the probability distribution of microstates (Eq.4), we can write the entropy of an isobaric‐isothermal

system:

ln / / lnB BS k p p E T P V T k Y

. (7)

The classical thermodynamic Gibbs energy is defined by G U TS PV . From this definition,

/ / /S E T P V T G T . (8)

A comparison of Eqs.(7) and (8) gives

lnBG k T Y (9)



Similar to Helmholtz energy and grand potential, the Gibbs energy is a free energy convenient for closed systems

at constant temperature and pressure.

The average energy and average volume of an isobaric‐isothermal ensemble can be determined from the

probability distribution [Eq.(5)] and Eqs.(1) and (2). Following some algebraic rearrangement, these average

quantities are related to the Gibbs energy by

1, 1,, ,( ln / ) ( / )

i m i mN P N PE Y G

(10)

1, 1,, ,( ln / ( )) ( / )

i m i mN NV Y P G P

(11)

Because the partial derivative on the right side of Eq.(10) is equivalent to

1 1 1, , , , , ,

/ / /i i iN m P N m P N m

G G P G P

, (12)

the enthalpy of the system H is given by

1, ,( / )

i mN PH E P V G

(13)

Eq.(13) is the familiar Gibbs‐Helmholtz equation in classical thermodynamics.

We now consider an application of the isothermal‐isobaric ensemble for practical systems. Point defects

in solids refer to localized anomalies of atomic dimensions that occur in an otherwise perfect crystalline lattice.

These anomalies include vacancies, misplaced atoms or impurity atoms that are deliberately introduced (dopants)

to control the electronic properties of a semiconductor, or that are inadvertently incorporated (contaminants)

during material growth or processing.

Figure 40 shows some common point defects of a solid crystal. Because point defects can have a strong

influence on the electronic properties of a crystal, they are of much interest in the electronics industry.



Figure 40 Various point defects in a crystal. As shown by the particle at the bottom, a Schottky defect (vacancy) is created when a bulk atom is removed to the surface due to thermal fluctuations.

We apply the isobaric‐isothermal ensemble to a particular kind of defect that arises when an atom is

removed from a bulk lattice site to the surface of the crystal as a result of thermal fluctuations. The resulting

vacancy is known as a Schottky defect, or vacancy, that is common in crystals of alkali halides. We want to find the

equilibrium vacancy concentration of a crystal at a given temperature and pressure.

Consider a crystal lattice containing N atoms. Let v be the volume of each unit cell and let 0 be the

energy related to the formation of a Schottky vacancy, i.e., the energy required to move an atom from its bulk

lattice site to a site on the crystal surface. We assume that the Schottky vacancies do not interact with each other,

i.e., formation of vacancy sites are independent. In this case, a microstate of the system is specified by the number

and arrangements of vacancies in the lattice and by their arrangement in the system. At a given temperature and

pressure, the distribution of microstates is determined by the isobaric‐isothermal partition function

0,

0,0, 0 0,

0 0,

( )!exp( ) exp ( )

! !n

N nY E PV n Pv N n

N n

. (14)

vacancy

interstitial

misplaced

dopant impurity

Schottky



In Eq.(14), 0,n denotes the number of Schottky vacancies in a microstate . The proportionality factor preceding

the exponential takes into account the number of ways (degeneracy) we can arrange 0,n vacancies at 0,N n

lattice sites.

Using the mathematical identity

1

0

( )! 1,

! ! (1 )k

Nk

N kx

N k x

(15)

where 0exp( )x Pv , we now evaluate the partition function

1

exp( )

[1 exp( )]No

NPvY

Pv

(16)

and the average number of vacancies in the crystal

0

20

, ,

ln ( 1)exp( )

[1 exp( )]o o N

N P

N Pvn P n

Pv

(17)

A silicon crystal has a zincblend structure whose lattice constant is 0.543 nm at room temperature. The

volume of each vacancy v = 0.01 nm3. For this crystal, the energy of formation of a Schottky defect 0 = 2.6 eV. At

atmospheric pressure, the Pv term is much smaller than 0 . Because 1N , the denominator in Eq.(17) is

essentially unity. It follows that at low or moderate pressure, the fraction of Schottky vacancies in a silicon crystal

can be approximated by

0 0/ exp( )n N (18)

At room temperature, 1 40 BeV k T . Eq.(18) indicates that the fraction of Schottky vacancies in the

crystal lattice is on the order of 10‐44, a very small number. However, high temperature (e.g., at 1000K), the

fraction of Schottky vacancies is about 10‐13 for the same 0 . At high‐pressure conditions, it is necessary to

consider the effect of Pv in Eq.(17) because this term may become comparable to vacancy formation energy 0 .

For example, at 1000K and 1000 bar, the fraction of Schottky vacancies in silicon crystal is 10‐5, which may have

significant effect on the properties of the crystal.



The result predicted from Eq.(18) is somewhat expected. Why, then, do we need a statistical‐mechanical

derivation for such a simple equation? We need it, first, to provide theoretical support for what we might have

intuitively assumed and second, to tell us how the fraction of Schottky effects is influenced by temperature and

pressure.



1.15Gashydrates

A gas hydrate is a crystalline solid where a “guest” molecule (e.g., methane) is surrounded by a cage of

“host” water molecules. Large deposits of hydrates are located in Canada, the Gulf of Mexico, and Siberia under

the seabed and under permafrost. The amount of natural gas trapped in hydrates is estimated to be more than

twice the amount of carbon in all other known fossil fuels on earth. Hydrates represent a huge potential resource

for future energy needs43.

When hydrates form in natural‐gas lines, they prevent fluid flow. Whether or not hydrates form depends

on gas composition, temperature, pressure and water content. Because transport of natural gas is vital in the U.S.

(and elsewhere), natural gas companies go to much trouble and expense to prevent hydrate formation.

Gas hydrates are also called clathrates; the Greek root of clathrate means claw. Figure 41 shows three

crystal structures of gas‐clathrate hydrates. These structures were obtained from x‐ray‐diffraction data. Structure

I hydrates contain 46 water molecules per eight guest molecules. The water molecules form two small

dodecahedral cavities (cages) and six large tetradecahedral cavities. These cavities can hold only small guest

molecules such as methane and ethane, with molecular diameters not exceeding 0.52 nm. Structure II hydrates

consist of 136 water molecules per 24 gas molecules. The water molecules form sixteen small dodecahedral

cavities and eight large hexakaidecahedral cavities. They may contain guest molecules with molecular dimensions

from 0.59 to 0.69 nm, such as propane and isobutane. Finally, structure H hydrates have cavities for both small

and large occupants. They contain 34 water molecules per six gas molecules. The large cavities of structure H

hydrates can hold larger molecules like isopentane and ethylcyclohexane.

43 www.hydrate.org provides excellent information on gas hydrates.

101 | T h e

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e m a t e r i a

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T

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In off‐shore wells or in cold climates (e.g., Alaska), moist natural gas may form hydrates in transfer lines.

Hydrate formation must be prevented lest solid hydrate crystals plug the transfer lines. A reliable molecular‐

thermodynamic model is useful for predicting gas‐solid equilibria where the gas is a natural gas (mostly methane

plus small amounts of light hydrocarbons and perhaps nitrogen, carbon dioxide and hydrogen sulfide) and the solid

is a hydrate.

To find the amounts of various gases in a hydrate, we consider a clathrate hydrate in equilibrium with an

m‐component gas mixture. We assume that the hydrate consists of n types of cages and that each cage can

contain at most one guest molecule. The guest molecules within the hydrate cannot diffuse from one cage to

another and there is no interaction between guest molecules that reside in different cages. With these

assumptions, the hydrate phase is equivalent to a set of independent open subsystems where each subsystem is a

cage in equilibrium with an m‐component gas mixture. Therefore, it is appropriate to apply the grand canonical

ensemble for gas adsorption in a porous medium.

We want to find the amount and composition of natural‐gas molecules in hydrates at a given

temperature, pressure, humidity and gas composition. For each subsystem (cage), with microstates specified by

the occupation numbers of the gas molecules, the grand canonical partition function for a cage i is

,1

1 j

m

i i jj

q e

(1)

where m is the number of components, subscripts i and j denote the type of cage and type of gas molecule,

respectively; ,i jq is the partition function of a j molecule in a type i cage; and, following usual notation, and

are chemical potential and 1/( )Bk T , respectively. The first term (unity) on the right side of Eq.(1) corresponds

to the microstate where the cage is empty, and ,j

i jq e corresponds to the microstate where the cage i is

occupied by a gas molecule of type j . For a cage in equilibrium with an m‐component gas mixture such that the

cage can contain at most one gas molecule, there are m+1 possible situations including an empty cage or a cage

containing one of the m different gas molecules. Figure 42 illustrates the number of ways that a hydrate cage can

be in equilibrium with a two‐component gas mixture.



Figure 42 Number of ways for a hydrate cage (shown as a pentagon) can be filled with one gas molecule from a binary mixture. The cage contains at most one gas molecule: (a) the cage is unoccupied; (b) the cage contains a gas molecule of type 1; (c) the cage contains a gas molecule of type 2. Thus, the number of ways is three.

Because all subsystems (cages) are independent of each other, the partition function of the hydrate is

given by the product of the partition functions of individual cavities44,

,11

1i W

j

k Nn m

i jji

q e

(2)

where WN is the total number of water (ice) molecules and ik is the number of cages of type i per water

molecule. From the partition function, we can derive desired macroscopic thermodynamic properties.

The average number of guest molecules of type j in the hydrate phase, jN , can be determined by

differentiating ln with respect to j :

,

1,

1

ln

1

j

j

Ni W i j

j mij

i jj

k N q eN

q e

(3)

where jN is the number of guest molecules of type j in all types of cages.

Because jN , an extensive property of the system, is a linear, homogeneous function45 of the total

number of cages, we can derive the number of molecules of type j in cages of type i from Eq.(3).

44 This assumption is equivalent to saying that the grand potential of the hydrate is given by the sum of the those of the individual cavities.

Type 1: Type 2: Gas Phase

Hydrate Phase

Gas Molecule

(b) (c) (a)



,

,

,1

1

j

m

i W i ji j M

i mm

k N q eN

q e

. (4)

It follows that, ,i jx , the fraction of cages of type i occupied by gas molecules of type j , is given by

, ,

,

,1

1

j

m

i j i ji j M

i Wi m

m

N q ex

k Nq e

(5)

Near atmospheric pressure, the chemical potential of a gas molecule can be approximated by that of an

ideal gas46,

0 lnj j jP (6)

where superscript 0 denotes the standard state, defined as pure gas j at system temperature and unit pressure,

and jP is the partial pressure of j in the gas mixture. Substituting Eq.(6) into Eq.(5) gives the fraction of cages i

that are occupied by a gas molecule of type j

,

,

,1

1

i j ji j M

i m mm

C Px

C P

(7)

where

0

, ,j

i j i jC q e (8)

is a model parameter depending on microscopic interactions between the cage and its guest gas molecule.

Summation with respect to index j on both sides of Eq.(7) yields

1

, ,1 1

1 1m m

i j j i jj j

C P x

. (9)

Eq.(8) is the key equation for correlating natural‐gas/hydrate equilibria. This equation is similar to the

Langmuir isotherm equation for multi‐component systems. Parameter ijC can be determined from a potential

45 A linear homogeneous function satisfies ( ) ( )f x f x

where is an arbitrary scalar parameter and x represents a set

of independent variables. 46 If the gas is nonideal, the partial pressure in Eq.(7) should be replaced by the corresponding fugacity.



function that describes the interaction between a guest molecule and a host molecule, that is, between a gas

molecule and a hydrate‐forming water molecule (Box 5). However, in practical applications, ijC is usually

obtained by fitting Eq.(8) to some experimental data for a hydrate in equilibrium with a pure gas. Once all ijC are

fixed, Eq.(8) provides a useful method for predicting gas sorption and hydrate composition at various operating

conditions for mixed gases. For example, Figure 43 shows theoretical predictions for hydrate formation for

mixtures of propane and methane. At a given gas composition, the gas is in equilibrium with liquid water at

temperatures above 0°C. As the temperature falls, hydrates form and three phases are in equilibrium. With

further reduction of temperature, the water phase disappears. Figure 43 also shows the dramatic effect of gas

composition on hydrate formation pressure.

Temperature (K)

270 272 274 276 278 280 282 284 286 288

Pre

ssu

re (

kPa)

100

1000

10000

Mol% of Propane

0

1.0

2.6

4.8

11.7

28.8

63.8

100

Figure 43 Predicted (lines) and measured (points) phase boundaries for mixtures of methane and propane in a system containing liquid water with the model parameters from pure‐gas hydrate data (from Sloan E. D., Energy & Fuels, 12, 191‐196, 1998).



Because the grand partition function depends on the total number of cages or, equivalently, the total

number of water molecules in all cages, the derivative of ln with respect to WN represents an excess chemical

potential of water, relative to its hydrate‐free state, due to formation of the hydrate. The excess chemical

potential reflects a propensity of water molecules to form a gas hydrate. From Eq.(2), we find the reduced excess

chemical potential of water

,1 1

lnln(1 )j

n MexW i i j

i jW

k q eN

(10)

Using Eqs.(8) and (9), we can rearrange the above equation

,1 1

ln 1n m

exW i i j j

i j

k C P

. (11)

Because ik and ,i jC are positive, Eq.(11) indicates that the occupation of hydrate cavities by guest molecules

reduces the chemical potential of water, making the hydrate phase thermodynamically more stable. This result

explains why hydrate is often formed in the presence natural gas.



Box5CijandIntermolecularInteractions

Parameter ijC in Eq.(8) can be calculated from intermolecular interaction between a gas molecule and the

hydrate cage. The connection provides a theoretical basis to find the effect of temperature on gas sorption or hydrate composition.

Consider the canonical partition function of a single gas molecule inside a hydrate cage. For simplicity, we assume that both cage and gas molecule are spherical so that the interaction potential between the gas molecule

and the cage can be represented by an isotropic function ( )r , where r is the distance of the guest molecule’s

center from the cage center. The canonical partition function is related to that of a free gas molecule 0q by

( )roqq dre

v

where v denotes the internal volume of the hydrate cage, and the integral is performed with respect to the cage

volume. The proportionality constant 0 /q v can be calculated from the ideal‐gas law and the standard chemical

potential of the gas

00

0 /q v P e

where 0P and 0 are, respectively, the pressure and chemical potential at the standard state. Substitution of the

above two equations into Eq.(9) gives

0

0 ( ), ,

j ri j i jC q e P dre

.

With a suitable potential for interaction between the cage and a gas molecule, this equation allows us to predict

parameter ijC and its temperature dependence.



1.16Partitionfunctionsforclassicalsystems

In macroscopic systems the dynamic properties of individual molecules are often described by Newtonian

mechanics instead of quantum mechanics. In that case, a microstate is represented by two types of continuous

dynamic variables: momenta and positions of individual particles that provide a classical representation of the

molecules. As a result, summation of microstates in a partition function must be replaced by integration in the phase

space, i.e., a multi-dimensional continuous space defining the momenta and positions of classical particles. The

connection between quantum and classical descriptions of molecular systems can be established by considering an

ideal gas of atomic molecules.

Consider an ideal-gas system containing N identical monatomic molecules in volume V . As discussed in

Section 1.4, the canonical partition function is defined by a summation over all accessible microstates. Because

molecules are non-interacting in an ideal gas, the canonical partition function is related to q , the partition

function of individual molecules

!

NIG q

N (1)

where !N accounts for the indistinguishablilty of the ideal-gas molecules. For a single atomic molecule, the

canonical partition function is

0 3

Vq q

(2)

where 0q is the intrinsic partition function related to the motions of electrons and nuclei, 2 / 2h m stands

for the thermal wavelength, h is the Planck constant, 1/( )Bk T , and m is the molecular mass. For a system

free of chemical reactions, 0q depends only on temperature47.

In classical mechanics, a microstate of the ideal-gas system is specified by the positions and momenta of

individual molecules. The 6N–dimensional hyperspace is called phase space ( , )N Nr p

, where

47 While in principle thermodynamics remains applicable in quantum mechanics to determine the distributions of electrons and nuclei that underlie the bond connectivity, we confine our interest to systems free of chemical reactions such that the electronic properties are not of concern.



1 2{ , , , }NNp p p p denotes the momenta of all molecules, and 1 2{ , , , }N

Nr r r r stands for the positions of

all molecular centers. Because there is no intermolecular interaction in an ideal-gas system, the total energy depends

only on the momenta of individual molecules:

2

1

( , )2

NN N i

i

pE r p

m

. (3)

Correspondingly, the canonical partition function is evaluated by integration over the phase space

3 /22 3

2

1

1 2exp exp( / 2 )

! 2 ! !

NN NNNIG N N i

Ni

p V V mdp dr p m dp

N C m N C N C

, (4)

where !N accounts for indistinguishablility, and constant C is introduced to make the partition function is

dimensionless. This constant can be fixed by a comparison of the partition function from quantum mechanics (Eq.2):

30/C h q (5)

Eq.(5) suggests that in evaluation of a partition function for a classical system, the summation over microstates

should be replaced by integration over the phase space

03

NN N

N

qdp dr

h

. (6)

The above procedure can be similarly extended to polyatomic systems and other ensembles. Table 4

summarizes the expressions of the partition functions and the phase-space probability densities in isolated,

canonical, grand canonical and isothermal-isobaric ensembles of one-component classical systems. Here =( Np

,

Nr

) stands for a particular point in phase space and N Nd dp dr , and 0q is omitted because in most cases it

does not contribute to the thermodynamic properties of the system.



Table 4 Partition functions and probability densities for a one‐component classical fluid in different ensembles

Ensemble Partition Function Probability Density

Micro canonical 3, .

1( , , )

! NE V constW N V E dv

N h 3

1 1( )

! Np v

W N h

Canonical ( )

3( , , )

!

E v

N

eQ N V T dv

N h

( )

3

1 1( )

!E v

Np v e

Q N h

Grand canonical [ ( ) ]

3( , , )

!

E v N

NN

eV T dv

N h

[ ( ) ]

3

1 1( )

!E v N

Np v e

N h

Isobaric-isothermal [ ( ) ( )]

3( , , )

!

E v PV v

NN

eY N P T dv

N h

[ ( ) ( )]

3

1 1( )

!E v PV v

Np v e

Y N h



Summary

Thermodynamics concerns macroscopic systems, i.e., those containing a large number of microscopic

elements such as atoms, molecules, electrons, or photons. While at equilibrium a thermodynamic system can be

specified by a few macroscopic variables such as temperature, pressure, volume or composition, because of the very

large number of elements (or particles), the microscopic details can only be specified by statistical means. To link

the microscopic details, called microstates, to macroscopic thermodynamic properties, we use statistical

thermodynamics. Statistical thermodynamics provides a formal mathematical framework to describe the distribution

of microstates in a thermodynamic system.

For a typical thermodynamic system, the number of microstates is extensively large. As a result, the

distribution of microstates is exactly solvable only for a few highly idealized systems. For more realistic systems, we

need to use a molecular model that describes how molecules interact with each other (intermolecular forces) and

where molecules are located in space (structure). As illustrated by the examples in this chapter, using simple

molecular models, we can establish quantitative connections between macroscopic properties of realistic systems

and microscopic details.

Conceptschecklist

Ensemble: a large number of systems with identical macroscopic thermodynamic prescriptions

Ergodic: equivalence of time average and ensemble average

Microstate: a detailed specification (e.g., positions and momenta) of all microscopic elements

Constraints: conditions imposed on the distribution of microstates

Partition function: normalization factor in the statistical description of microstates

Internal energy: energy of microscopic elements

Entropy: freedom of microscopic elements

Ideal gas: a gas free of intermolecular interactions



Problems

1. Discuss whether the following macroscopic variables are necessary and sufficient to define an equilibrium

system:

a) a pure liquid of known mass at given temperature and density.

b) a pure liquid of known mass at given temperature and pressure.

c) a pure liquid of known mass at given density and pressure.

d) a two-phase system at given temperature, pressure and compositions of one phase.

e) a pure liquid at given total energy, volume and mass.

f) a pure liquid at given total energy, volume and pressure.

g) a solid at given temperature, pressure and composition.

h) a pure vapor at given total entropy, temperature and volume.

2. According to Newtonian dynamics, the kinetic energy of a gas molecule is given by 2( ) / 2v mv

where m stands for the molecular mass and v for velocity. Derive the average kinetic energy using the Boltzmann

distribution law. What’s the physical significance of temperature?

3. An ethane molecule may be in one of the two conformation states as shown in the following

Staggered conformation Eclipsed conformation

Because of the steric effects, the energy at the staggered conformation is 12.5 kJ/mole lower than that at the eclipsed

conformation.

a) Confining attention to the two conformers only, calculate the canonical partition function for 10 independent

ethane molecules at 298 K.

b) Find the relative populations of the two conformers in an ideal gas of ethane molecules at 298 K.

113 | T h e

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a) Derive t

b) Calculat

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d) Why is t

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6. Based on the Maxwell velocity distribution, derive that the flux of gas molecules, i.e., the number of gas

molecules passing through a small hole per unit area and unit time, is

2Bk T

Jm

where is the number density of gas molecules, and m is molecular mass.

7. Effusion was used for separation of 235U from 238U isotopes in the development of the atomic bomb. To use the

effusion method, the uranium isotopes are first converted to uranium hexafluoride (UF6) gas, which is forced to

diffuse through porous barriers repeatedly, each time becoming a little more enriched in the slightly lighter 235U

isotope. The 235U to 238U ratio in a typical uranium ore is 1:140. Based on the equation derived from Problem 6,

estimate how many separation stages are required to obtain 90% 235U.

8. Consider a liquid in equilibrium with its vapor. The rate of condensation (per unit area) can be estimated from the

number of the gas molecules impinging on the solid or liquid surface multiplied by an efficiency factor

2Bk T

Jm

At equilibrium, the rate of gas molecule condensation is the same as the rate of evaporation. Assume 0.5 ,

estimate how long it takes to dry up a pond of 100 m2 in surface area and 10 m in depth open to 50OC dry air.

9. Calculate the entropy, Helmholtz energy, chemical potential, and heat capacity of argon gas at 25oC and 1 atm by

using the equations for a monatomic ideal gas. Compare your result with the entropy for argon obtained from

calorimetric experiments (155 JK-1mol-1).

10. A biomacromolecule such as a polypeptide or a DNA chain may form a helical structure, i.e., organization of the

monomers along the polymer backbone. To the zeroth-order approximation, we may assume that each monomer can

be in one of two states different in length but similar in energy. Imagine that we have a very long idealized linear



biomacromolecule as shown above that consists of N identical segments and each segment can exist in length 0l

or

0l l . The total length of the polymer is 1

N

ii

L l

where il is the length of segment i.

a) Derive an expression for entropy ln ( , )BS k W N L

b) Based on the result from (a), derive an expression for the tension ,U N

Sf T

L

as a function of temperature

T, polymer length L, and number of segments N.

11. NMR measurement is based on the population of proton spins in a magnetic field. The nuclear of a hydrogen

atom, a proton, has a magnetic moment of 241005.5 J Tesla-1. In a magnetic field, the proton has two energy

states corresponding to spin up and spin down. The energy difference between these two states can be expressed as

B 79.2

where B is the magnitude of the magnetic field. For a 300 MHz NMR instrument, 7B Telsa.

a) Compute the average energy per proton and the relative distribution of protons at 298T K;

b) Describe how the relative distribution varies with temperature.

12. The energy of a gas molecule in the earth’s gravitational field is a function of the altitude z

mgzz )(

where m is the molecular mass, g is the gravitational constant. Assume that the atmosphere is at equilibrium and

at a constant temperature of 298 K, estimate the atmospheric pressure at 100 km above the sea level.

13. Consider a system that has N distinguishable non-interacting particles with two energy levels for each particle:

a ground state with energy zero, and an excited state with energy 00 . Find the average energy per particle and

the constant volume heat capacity.



14. The degeneracy in a canonical ensemble can be understood as the number of microstates that have the identical

energy. Suppose that a system has energy levels at 0 , 300 Bk , and 600 Bk with degeneracies 1, 3, and 5,

respectively. Here the energy has the units of Joule and Bk is the Boltzmann constant.

a) Calculate the Canonical partition function, the relative populations of energy levels, and the average energy at

298T K;

b) At what temperature is the population of the energy level at 600 Bk equal to that at 300 Bk ?

15. Based on Einstein’s model of ideal solids and the ideal-gas law for the vapor phase, derive an equation to predict

the vapor pressure of an ideal solid.

16. The molar heat capacity of a monatomic solid at 300 K is 2R where R is the ideal-gas constant. Use the Einstein

model to calculate the frequency of vibration.

17. The end effects are non-negligible for the helical-coil transition of a short polypeptide chains. But in this case,

we may assume that there is only one helical region for a polypeptide with m residues and that the maximum

number of hydrogen bonds is 4m . Show that the partition function for a short chain is given by

Q m i si

i

m

1 31

4

( ) ,

where ( )m i 3 takes into account the number of ways to arrange an i -residue helical region in an m-residue

polypeptide chain. From the partition function, derive the helix fraction

1

4

3

4 1 3

1

4

1

4( )

ln

ln

( )

( )[ / ( ) ]m

d Q

d s

i m i s

m m i s

i

i

m

i

i

m

and plot the function for different values of s and .

18. For a one-component system, we may reformulate the mean-square deviations of the number of particles in

terms of the isothermal compressibility,



1

TTP

,

where /N V is the average number density. At constant T , the Gibbs-Duhem equation for a one-

component fluid is

T T

P

.

For a one-component fluid, pressure and chemical potential, as well as their derivatives with respect to density,

depend only on temperature and density. Show that the isothermal compressibility can be expressed as

2

2 2, ,,

1T

T V T VT V

NN

N N

19. Demonstrate that both heat capacity and isothermal compressibility must be non-negative for an open system at

equilibrium. Discuss the physical significance of these quantities to thermodynamic stability.

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