EQUILIBRIUMEQUILIBRIUM

Introduction to Thermodynamics of MaterialsDavid R Gaskell

Taylor & Francis, New York (2003).

Recommended websitehttp://hyperphysics.phy-astr.gsu.edu/hbase/heacon.html#heacon

Thermodynamics and an Introduction to ThermostaticsHerbert B Callen

John Wiley and Sons, New York (2006).

MATERIALS SCIENCEMATERIALS SCIENCE&&

ENGINEERING ENGINEERING

Anandh Subramaniam & Kantesh Balani

Materials Science and Engineering (MSE)

Indian Institute of Technology, Kanpur- 208016

Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh

AN INTRODUCTORY E-BOOKAN INTRODUCTORY E-BOOK

Part of

http://home.iitk.ac.in/~anandh/E-book.htmhttp://home.iitk.ac.in/~anandh/E-book.htm

A Learner’s GuideA Learner’s GuideA Learner’s GuideA Learner’s Guide

Physical ChemistryIra N Levine

Tata McGraw Hill Education Pvt. Ltd., New York (2002).

Kinds of equilibrium. Stable, Metastable, Unstable & Neutral equilibrium states. Thermodynamic variables and potentials.

What will you learn in this chapter?

In this text only some aspects will be dealt with, readers may consult standard texts on thermodynamics/thermostatics for a other aspects/detailed account.

The fields of Thermodynamics and Kinetics are vast oceans and Chapter 2 will introduce the bare essentials required to understand the remaining chapters.

Please read Basics of Thermodynamics before reading this chapterPlease read Basics of Thermodynamics before reading this chapter

Let us start by performing the following (thought) experiment:Heat a rod of Al from room temperature to 500C As expected the rod will expand (A → B in figure below).

The expansion occurs because of two reasons: 1 Vibration of atoms (leading to an increase in average spacing between atoms→ the usual reason)(A → M in figure below).

2 Increase in the concentration of vacancies* (a vacancy is created when a Al atom goes to the surface and for every 4 vacancies created the volume equal to 1 unit cell is added). (M → B in figure below).

The 2nd reason is of subtler origin and must be surprising to many readers. Additionally, it is a smaller effect in terms of its contribution to the overall increase in length of the specimen (see solved example link below- it is about 1 in 10000 effect).

Motivation

* It costs energy for the system to put vacancies (broken bonds, distortion to the lattice) then why does the system tolerate vacancies?

1

These slides are intended to set the stage for understanding the ‘purpose’ and ‘power’ of thermodynamics and its ‘quantities’. Gibbs Free Energy (G) and Entropy (S) will be in special focus.

Click here for solved exampleClick here for solved example

Metal expands on heating due to 2 different

physical reasons!

Now let us perform another (thought) experiment to put in perspective the previous experiment:Heat a elastomer (cut rubber band) which has been stretched by a small weight by about 20C (room temperature + 20C) the stretched rubber band will contract!

The 2nd reason for the expansion of the Al rod is closely related to the contraction of the stretched rubber band! occurs because of thermodynamic reasons (quantities like Gibbs Free Energy (G) and Entropy (S)), which we shall learn in this chapter.

In the case of the heating of the Al rod- “how the vacancies form” is an issue of kinetics. Kinetics will be dealt with in the topic of kinetics and chapter on Diffusion.

A ‘stretched’ elastomer contracts on heating!

Let us next consider the melting of a pure metal at its melting point (MP) (at constant T and P) → by supplying heat to the sample of metal (so that the metal sample is only partly molten). At the MP the liquid metal is in equilibrium with the solid metal.

The liquid has higher potential energy as well as higher kinetic energy (at the molecular level)

than the solid. Then why does the liquid co-exist with the solid? The answer to this question lies in the fact that internal energy is not the measure of

stability of the system (under the circumstances). We will learn in this chapter that it is the Gibbs Free Energy (G). The molten metal has

higher energy (internal energy and enthalpy), but also higher Entropy. So the melting is driven by an increase in Entropy of the system. The molten metal and the crystalline solid metal have the same G → hence they co-exist in equilibrium.

2Motivation

Similarly we can consider the vaporization of water at 100C and 1 atm pressure. Not only that steam has higher internal energy, but also that work has to be done against

atmospheric pressure to accommodate the higher volume of steam!

Equilibrium refers to a state wherein there is a balance of ‘forces’*(as we shall see equilibrium points have zero slope in a energy-parameter plot).

Stability relates to perturbations (usually small perturbations** about an equilibrium state) (as we shall see stable relates to the curvature at the equilibrium points).

Stability and Equilibrium

* Force has been used here in a generalized sense (as an agent which can cause changes)** Perturbation is usually a small ‘force/displacement’ imposed in a short span of time.

Mechanical equilibrium Thermal equilibrium Reaction Equilibrium Phase Equilibrium

Kinds of Equilibrium

Material Equilibrium

Material equilibrium

Phase equilibrium

Reaction equilibrium

Equilibrium with respect to conversion of set of chemical species to another

Equilibrium with respect to transport of matter between phases of a system (without conversion of one to another)

Number of moles of each substance (in the closed system) remains constant with time

Note: nothing happens/changes in a system under equilibrium (at least macroscopically)

The fraction of reactants and products remains constant with time

The fraction of various phases remains constant with time

Let us start with a simple mechanical system (it is easier to visualize mechanical systems) → a rectangular block (Figure in next slide) (under an uniform gravitational potential).

The potential energy (PE) of the system depends on the height of the centre of gravity (CG).

The system has higher PE when it rests on face-A, than when it rests on face-B. The PE of the system increases when one tilts it from C1 → C2 configuration. In configurations such as C1,C2 & C3 the system will be in equilibrium (i.e. will not

change its configuration if there are no perturbations). In configuration C2 the system has the highest energy (point B) and any small

perturbations to the system will take it downhill in energy → Unstable state. Configuration C3 has the lowest energy (point C) and the system will return to this state

if there are small perturbations → the Stable state. Configuration C1 also lies in an ‘energy well’ (like point C) and small perturbations will

tend to bring back the system to state C1. However this state is not the ‘global energy minimum and hence is called a Metastable state.

Additionally, one can visualize a state of neutral equilibrium, like a ball on a plane (wherein the system is in a constant energy state with respect to configurations).

Equilibrium in a Mechanical System

Mechanical Equilibrium of a Rectangular Block

CentreOf Gravity

Pot

enti

al E

nerg

y =

f(he

ight

of C

G)

Metastable state

Unstable

Stable

Configuration

Lowest CG of all possiblestates

A

B

C1 C3C2

Ball on a planeNeutral Equilibrium

We start by considering the mechanical equilibrium of a block- this is to get a first feel- additional concepts will be required when dealing with condensed matter systems.

A

B

C

Equilibrium implies balance of (generalized) forces. Thermodynamic material equilibrium involves reaction and phase equilibria. Under equilibrium conditions nothing changes at the macroscopic scale, but this does not

mean than nothing is happening at the microscopic scale. E.g. consider water filled to half the volume of a closed vessel. Under equilibrium, the vapour pressure of water vapour is fixed (i.e. the amount of H2O in vapour and liquid forms

is fixed). However, at the microscopic scale, vapour molecules are constantly getting into the liquid phase and statistically an equal number of molecules are getting back into the vapour phase (i.e. there is dynamical equilibrium rather than a static one at the microscale). Similar situation also holds good for reaction equilibrium as well.

Points to be noted: A system can exist in many states (as seen even for a simple mechanical system: block on a plane)

These states could be stable, metastable or unstable (or even neutral)

Using the relevant (thermodynamic) potential* the stability of the system can be characterized

(In the case of the block it is the potential energy, measured by the height of the CG for the case of the block on the plane)

System will ‘evolve’ towards the stable state provided ‘sufficient activation’ is provided

(in the current example the system will go from C1 to C3 by ‘sufficient jolting/shaking’ of the plane)

However, this aspect is not within the domain of thermodynamics.

* The kind of ‘potential’ we are talking about depends on the conditions. As we shall see soon, at constant temperature and pressure the relevant TD potential is Gibbs free energy.

Q & A What is meant by equilibrium?

Kinds of Stability (Equilibrium)

Three kinds of equilibrium (with respect to energy) Global minimum → STABLE STATE Local minimum → METASTABLE STATE Maximum → UNSTABLE STATE Constant energy → Neutral State/Equilibrium

Kind of equilibrium can be understood by making perturbations to the system For the mechanical system (block) this corresponds to tilting the block If the system changes its state after small perturbations then the system

→ is in an unstable state If the system returns to its original state after a small perturbation (tilt) then the system

→ is in a stable or metastable state (lies in an energy minimum) If the system returns to its original position after small perturbations but does not do so for

large perturbations then the system → is in a metastable state (not in the global energy minimum)

If there is no change in energy for any kind of perturbation then the system → is in a state of neutral equilibrium (e.g. the case of the ball on a plane)

In a 2D system where perturbations are possible in more than one direction (i.e. the energy landscape is a surface), perturbations in one direction may be stable and in another direction it may be unstable (like on a surface with negative Gaussian Curvature)

Also next slide

If the system tends to return to the original state after the perturbation Stable/Metastable state

In Metastable state the system goes to a new state if the amplitude of the perturbation is large.

If the system goes to a different state on perturbation (the perturbation will tend to take the system to a new state ‘very different’ from the original state) Unstable state

If the system goes to the ‘new perturbed state’ without a change in energy (the perturbation will tend to take the system to a new state ‘close to’ the original state, as imposed by the perturbation)

Neutral state

In Materials Science we are mainly interested with condensed matter systems (solids and liquids) (also sometimes with gases)

The state of such a system is determined by ‘Potentials’ analogous to the potential energy of the block (which is determined by the centre of gravity (CG) of the block).These potentials are the Thermodynamic Potentials (A thermodynamic potential is a Scalar Potential to represent the thermodynamic state of the system).

The relevant potential depends on the ‘parameters’ which are being held constant and the parameters which are allowed to change. More technically these are the State/Thermodynamic Variables (A state variable is a precisely measurable physical property which characterizes the state of the system- It does not matter as to how the system reached that state). Pressure (P), Volume (V), Temperature (T), Entropy (S) are examples of state variables.

There are 4 important potentials (in some sense of equal stature). These are: Internal Energy (U or E), Enthalpy (H), Gibbs Free Energy (G), Helmholtz Free Energy (A or F). Of these internal energy can be conceived as a fundamental quantity, while enthalpy a quantity keep track of the available heat. G and A arise due to the fact that the entropy of the universe increases.****

Condensed Matter systems

Intensive and Extensive Properties

Intensive properties are those which are independent of the size of the system P, T

Extensive Properties are dependent on the quantity of material V, E, H, S, G

Thermodynamic potentials and the relation between them

There are 4 important potentials (in some sense of equal stature). These are: Internal Energy, Enthalpy, Gibbs Free Energy, Helmholtz Free Energy

The relation between these potentials and the state variables is as below.

U (or E) F (or A)

= U TS

H= U + PV

G= U + PV TS

TS

+PV

In terms of the stimulus and response*:

P → V (Pressure difference drives volume changes)

T → S (Temperature difference drives entropy changes)

→ Ni (Chemical potential difference drives mass transfer) (subscript i stands for the ith

species)

The relevant thermodynamic potential which characterizes the stability of the system is

dependent on the state variables which are held constant (as in the table below).

The most conditions are one of Constant P and Constant T

→ Hence G (Gibbs Free Energy) is the most relevant thermodynamic potential.

However, we should remember that depending on the state variables being held constant

any one of the four potentials (U, H, F, G) could be the relevant potential. (i.e. all of the

potentials have the same ‘stature’).

The relevant thermodynamic potential determining the stability of the system

Constant V Constant P

Constant S U H

Constant T F G

dH TdS VdP

dU TdS PdV

dG SdT VdP

dA SdT PdV

* The main response.

The total energy of a body is the sum of its kinetic energy (K), its potential energy (V) and internal energy (U).

Let us consider a container of gas moving at a height ‘h’ from the ground at a velocity ‘v’. The total energy of this system is (E) = K + V + U = (½ mv2)+(mgh)+U.

K & V are macroscopic quantities. U is related to the internal molecular structure (and the details therein).

U consists of: molecular translational energy, molecular rotational energy, molecular vibrational energy, energy due to electronic states, interaction energy between molecules, relativistic rest- mass energy (mc2) arising from electrons and nucleons.

In most cases in thermodynamics K and V are ignored.

Energy & Internal Energy

Gas

h

v

Internal Energy (U or E) = Kinetic Energy (KE) + Potential Energy (PE).(The above KE and PE are at the molecular level).

The origin of Kinetic Energy → Translations, Rotations, Vibrations.

The origin of Potential Energy → Bonding between atoms (interactions in the solid).

Internal energy is the total of the energies of molecules (or atoms or ions..) and their interactions.

The increase in internal energy on heating from 0 to T Kelvin is given by the equation below; where CV is the specific heat at constant volume and E0 is the internal energy of

the system at 0K.

T

V dTCEE0

0

Internal Energy (U)1

Enthalpy (H) = Internal Energy + PV (work done by the system). Measure of the heat content of the system. At constant pressure the heat absorbed or evolved is given by H. Transformation / reaction will lead to change of enthalpy of system.

Enthalpy (H) H = U + PV2

Enthalpy is also known as the ‘heat content’ of the system. Why do we need a quantity like enthalpy?

Suppose we burn a fuel (say petrol) in open air all the energy of the combustion is not available to us, as some energy is consumed in ‘making space’ for the products of the combustion. I.e. work is done by the system against the surrounding (atmosphere at 1atm pressure), to accommodate the products of combustion (and this work is not available to us for ‘useful’ purposes). If the combustion were to be carried out in a rigid (closed) container, then no work of expansion would have to be accounted for.

So, if internal energy (U) is the quantity of energy in a system (and say U is being made available during an experiment), then H is a measure of the part available to us (some gets ‘wasted’ in accommodating the products of the reaction→ i.e. out of the U only H is available to us). This is true only if the ‘entropy’ is constant (else we will have to get the amount available from the free energies (G or A)).

Gaseous state is considered as the reference state with no interactions. H is to constant pressure process as U is to a constant volume process. H = U + (PV). For condensed phases under ‘usual’ pressures the (PV) term is

much smaller than the U term H ~ U. The increase in enthalpy on heating from 0 to T Kelvin is given by the equation below;

where CP is the specific heat at constant pressure and H0 is the internal energy of the system at 0K (H0 represents energy released when atoms are brought together from the gaseous state to form a solid at zero Kelvin).

Enthalpy is usually measured by setting H = 0 for a pure element in its stable state at 298 K (RT).

T

PdTCHH0

0

298

T

PH C dT

Entropy (S)

Entropy is perhaps one of the most profound and subtle concepts of nature. Entropy can be understood looking at a Macroscopic picture (interpretation) or a

Microscopic picture (interpretation) (next slide).

Though these are different approaches to understand entropy– the result is the ‘same’ entropy.

In the Macroscopic view we ‘work’ at the system level and worry about observable average quantities. In the Microscopic view we go into all the ‘details’ about the system.

The entropy of an isolated system will increase in a spontaneous process (cannot spontaneously decrease)→ II law.

Since, the above can be applied to the entire universe (a large isolated system!**)→ the entropy of the Universe will always increase (here we are not using the word ‘spontaneous’ as the Universe cannot be

‘coaxed’ into doing a non-spontaneous processes!). The microscopic interpretation (view) is the Statistical Physics/Mechanics picture, which

is valid for large systems (i.e. systems with a large collections of atoms, molecules etc.). “Entropy is time’s arrow” → time increases in the direction of increasing entropy*. For a system the Internal energy (U) measures the quantity of the energy, while Entropy

(S) can be thought of a measure of the quality of that energy (lower value of entropy will imply a better quality→ as the entropy of the universe always* increases).

*The universe is in a expanding phase now. If it were to stop expanding and start contracting → entropy of the universe is expected to decrease in the contracting phase.

** We are ignoring ‘wormholes’, which connect our universe to other parallel universes for now (or take those universes part of ours)!!

What is entropy a measure of? (We will take this up later).

For an isolated system thermodynamic equilibrium is reached, when the entropy of the system reaches its maximum.

There are two equivalent descriptions of entropy: macroscopic (thermal entropy, Clausius view, classical thermodynamical view) microscopic (Boltzmann view, statistical physics definition).

These are equivalent under most circumstances (some deviations can occur at 0K). In the Macroscopic interpretation of entropy, one ‘sits’ on the system boundary and

computes Q/T. The details of the system or its response due to Q is not considered. In the Microscopic interpretation of entropy, one is ‘concerned’ about all the internal

microstates, in which the system can exist.

Entropy

Thermal$ Configurational + other**

lnkSconfig

Zero or +ve

Boltzmann constant (1.38 1023 J/K)

No. of different configurations the system

thermal

QS

T

Due to Rudolf Clausius Macroscopic view ‘Classical Thermodynamic’ definition Usually what is experimentally measured

Due to Ludwig Boltzmann Microscopic view Statistical Physics Definition Theoretical understanding

The entropy change of a system at temperature T absorbing* an infinitesimal amount of heat Q in a reversible way, is:

** For a system at Constant Energy, many microscopic states can give a macroscopic state of identical energy. These microscopic states could originate from various sources like Configurational (positional & orientational), Electronic, Vibrational and Rotational (etc.) states.

In many cases the configurational term may be the predominant one considered.

Configurational Vibrational Rotational Electronic NuclearS S S S S S

Two views and not two types*

These two descriptions are equivalent. Sometimes 0K there are some degenerate states → in

which case the configurational entropy is non zero (which can be confirmed by experiments, wherein one measures the ‘macroscopic’ entropy). This is called residual entropy.

These two descriptions are equivalent. Sometimes 0K there are some degenerate states → in

which case the configurational entropy is non zero (which can be confirmed by experiments, wherein one measures the ‘macroscopic’ entropy). This is called residual entropy.

* If heat is desorbed then entropy change will be negative.

$ Though other implications do exist in literature.

Warning! Busy slide → take time to understand

if heat input is at lower T (as compared to higher T) then for the same heat input (Q), there will be a higher increase in entropy.

lnkSconfig

We had noted that the Boltzmann constant arises because we try to make a ‘unit’ Kelvin equal to a unit C.

Suppose we measured temperature in terms of ‘’, then entropy would be a pure number.

A note on the equation

How do I get the total entropy? Can I add the thermal entropy and configurational entropy?

Funda Check

We have stated that there are two interpretations of (viewpoints) entropy– macroscopic (Clausius entropy or thermal entropy, based on classical thermodynamics) and microscopic (Boltzmann entropy, based on statistical thermodynamics).

The total entropy cannot be obtained by adding the two interpretations of entropy. Any one way of determining the entropy is fine (i.e. you should not do double counting).

In one case (macroscopic viewpoint) you can sit at the system boundary and compute Q/T (and need not worry about the details of the system). In the other case (microscopic viewpoint) you have to know the microscopic states (corresponding to a macrostate) and use the Boltzmann equation (S=kln).

Configurational entropy is one contribution to the microscopic interpretation (other degeneracies with respect to electronic, vibrational, rotational, etc. states have to be included).

Continued...Continued...

Say you want to measure the mass of a person (let the person called Sukriti). This can be done in two ways: (i) Sukriti stands on a weighing machine and this gives her mass (MSukriti) or (ii) she counts the

number of atoms of various types in her body (NO, NC, NFe, etc.) and then she multiplies these numbers

with their respective atomic masses and sums the results to get the total mass of her body (NOmO +

NCmC + NFemFe +..... = MSukriti).

By both ways we get the mass of Sukriti (MSukriti)

In method (i) in which we sit at the boundary of the system (called Sukriti) and need not know the microscopic details of the person, while in method (ii) we have all the microscopic (atomic composition) details and use this to compute the mass of Sukriti. Clearly we should not add the two methods of obtaining masses (to get the ‘total’ mass of Sukriti).

Mass of Sukriti

Weigh in a balance Do ‘atomic counting’

Two ways of determining the mass

25 kg

O, N, Fe,...

How do I understand the microscopically degenerate states?Funda Check

Let us consider a constant energy system a classroom of 50 students with a total energy of 100 U* (i.e. an average of 2 U per student). This energy can be distributed amongst the students in many ways.

Way-1: all the 50 students jump with 2U.Way-2: 25 students jump with 4U and the remaining are quiet.Way-3: 25 students are jumping with 2U and 25 are rotating with 2U.Etc.

So there are many ways that the 100U energy can be distributed amongst the students.

This represents the microstates in energy.

* U is some unit of energy!

Entropy (S)

One way of simply stating the ‘concept behind entropy’ is→ A system will, more often than not, be found in ‘states; with higher probability.

(This is nothing but the statement of the obvious!)

However, the implications of the above are ‘profound’. This can be best understood by considering the mixing of two ideal gases (or in the ‘toy model’ below as the mixing of 6 circles- 3 red and 3 blue, on 6 fixed lattice sites).

Assuming that red and blue circles can move about randomly on the fixed sites and also assuming that the probability of the occurrence of each state is identical (i.e. no state is preferred over any other state); there are 20 possible configurations as shown in the next slide.

As seen (from the figure in the next slide) the majority of the states (18/20) are ‘mixed states’ and only two are the ‘totally’ unmixed ones.

Hence, purely from a probabilistic point of view, mixed states occur more often than the unmixed ones.

This implies, if we start with a unmixed configuration as in the figure below and the system can access all possible states with equal probability → the system will go from a unmixed state (of low entropy) to a mixed state (of higher entropy).

Unmixed state

Two boxes separated by a barrier initially

63

6 6.5.420

3 3 3.2.1C

A simple understanding of entropy

A B

Unmixed state

Unmixed state

Mixed states with ‘various’ degrees of mixing

In the case of two gases initially separated by a barrier, which are allowed to mix on the removal of the barrier: the number of mixed states are very large compared to the unmixed states. Hence, if all configurational states are accessible, the system will more likely found in the mixed state. I.e. the system will go from a unmixed state to a mixed state (worded differently the system will go from ‘order’ to ‘disorder).

On the other hand it is unlikely (improbable) that the system will go from mixed state to a unmixed state. (Though this is not impossible → i.e. a mixed system can spontaneously get ‘unmix’ itself!!)

* We assume that all states have equal probability of occurring and are all accessible

Note: the profoundness of the concept of entropy comes from its connection to free-energy (via T)

18 mixed states2 unmixed states

In literature entropy is considered to be a measure of: the degree of ‘randomness’ in a system the ‘uncertainty’ about the system the ‘degeneracy’ of the system.

The terms randomness, uncertainty, etc. have other connotations in other contexts, which can lead to confusion to the reader. To avoid this we simply state that entropy is a measure based on the multiplicity of the system (of microstates of a system, corresponding to a macrostate).

If a system can exist in multiple states (corresponding to a particular macrostate), then this leads to an increased entropy. For a gas (with non-interacting atoms) some of these states could be ordered as below (State-A); but more likely these will be some ‘random disordered state’ (State-B). Note that the number of ‘disordered’ states is overwhelmingly large.

What is entropy a measure of?

Gas atoms in a box arriving at a regular configuration by chance.

Gas atoms in a box usual (disordered) state we expect.

State-A State-

B

In the microscopic (statistical mechanics) interpretation of Entropy, we take into account the multiplicity of microstates which give rise to a macrostate

For example for a constant energy system (with a constant number of particles/species and volume), the total energy of the system may be obtained due to a multiplicity of microstates originating from various sources like Configurational, Electronic, Vibrational and Rotational states.

If we consider only configurational entropy for now- the multiplicity of states can arise from various configurations of the atomic species. E.g. If we are talking about a pure crystal of A with just one B atom, this B atom could be in any one of the lattice positions (all of identical energy) giving rise to a multiplicity in the microstates. (for now we ignore the surface states)

Note again that for statistical mechanics to be valid we have to deal with large systems (for illustration purposes we often draw small systems!)

Calculating Configurational Entropy

3 configurations of equal energyOf course there are many

more such configurations-

3 are drawn for illustration

lnkSconfig

Zero or +ve

Boltzmann constant (1.38 1023 J/K)

No. of different configurations of equal energy for a constant energy system

Then the configurational entropy is given by

( & )mixed state pure elements A BS S S

Let us consider the entropy change due to mixing of two pure crystalline elements A & B (a simple case for illustration of the concept of entropy).

The unmixed state is two pure elements held separately. The mixed state (for now assuming that the enthalpy of mixing is negative- i.e. the elements want to mix) represents an atomic level mixing of the two elements. The mixed state is a disordered solid solution.

Let the total number of lattice sites (all assumed to be equivalent) be N. The Entropy of the unmixed state is zero (as in pure crystalline elements atoms are

indistinguishable and hence represent one state). Spure A = Spure B = k ln(1) = 0.

In the mixed state the entropy of the system increases (Smixed state) as multiple configurations

of the mixed state are possible. The number of permutations possible in the mixed system is . This boils down to

chosing ‘n’ (=nA) sites from N and putting ‘A’ atoms into these sites.

Entropy change due to mixing of two pure elements

Zero

!ln ln

( )! ( )!A B

NS k k

n n

An useful formula for evaluating ln(factorials) is the Stirling’s approximation:

( !) ln( )Ln r r r r

!ln ln

( )! !A A

NS k k

N n n

!

( )! ( )!A

Nn

A A

NC

N n n

!~ 2n

nn n

e

~ asymptotically equal, e = 2.718…!r r

At the melting point of a material when heat is supplied (Q) to the material it does not lead to an increase in the temperature. Instead, the absorbed heat leads to melting i.e. the energy goes into breaking of bonds in the solid and consequently a transformation in the state of the material (solid → liquid). The entire process of melting takes place at a constant temperature (Tm). The heat absorbed is called the Latent Heat of Fusion (Hfusion).

Suppose we take a mole of Al atoms in crystalline form and melt the same, then the change in entropy can be calculated as below.

In the solid state the atoms are fixed on a lattice (of course with vibrations!) and this represents a ‘low entropy’ state. On melting the entropy of the system increases as the atoms are free to move around and many configurations are possible. From this point of view often Entropy is considered as a measure of disorder (however, it must be clear that the phrase ‘measure of disorder’ is used with an understanding of the context).

Entropy change during melting

Data: Enthalpy of fusion (Hf) = 10.67 kJ/mole, Melting Point (Tm) = 933.4 K (660.25C)

310.67 1011.43 / /

933.4f

melting Alm

HQS J K mole

T T

SolvedExample

What is the entropy of mixing 0.5 mole of A with 0.5 mole of B on a mole of lattice sites (N0 sites)?

Data: 231.38 10 /k J K

The initial configurational entropy of solid A and Solid B (unmixed) is zero. This is because the atoms of solid A (and similarly solid B) are indistinguishable and hence these A and B atoms in a lattice of N0/2 sites represent just one configurational state, which implies that the entropy is zero.

final initial finalS S S S = 0

0 0

ln

2 2

Nk

N N

=

2

00ln ln

2

Nk N

0 0 00 0 0ln 2 ln

2 2 2

N N Nk N N N

00 0 0ln ln

2

Nk N N N

00 0ln ln

2

Nk N N

0 ln 2k N

23 23(1.38 10 )(6.023 10 ) ln 2 5.76 / /J mole K

Using: lnkSconfig ( !) ln( )Ln r r r r !

( )! ( )!A

Nn

A A

NC

N n n

We will take up a brief discussion about ‘free’ energies before considering G and A.

We all know that there is ‘no free lunch’ so what is meant by ‘free’ energy? It is the energy which is available to do work.From the second law we know that all the available energy cannot be converted into work some ‘has to be lost’ as heat (to the surrounding/sink).You cannot keep all your earnings some has to be paid as tax (and we are not even including ‘hafta’ here!!

We have noted that all the internal energy change may not be available as heat for us (some may be used up to do work to accommodate the products of say a combustion process) and we introduced the concept of enthalpy for this.

Needless to say, only a spontaneous process can be used for extracting work out of a system. (For a driven process we have to supply work!).

In a spontaneous process the entropy of the universe has to increase (2nd law of TD rephrased).

So: (Ssystem+Ssurrounding) > 0. (E.g. If heat enters the system from the surrounding then Ssystem will be positive, while Ssurrounding will be negative).

To decide the direction of ‘spontaneity of a process’ statement is enough → but then we have to do two separate computations (one for the system and one for the surrounding).

If we could come up with a quantity, which can help us do calculations only on the system this will be very useful. Further if the quantity is a state function, then we do not have to

‘worry’ about the reversibility of the heat transfer (remember: S = Qreverisble/T).

‘Free’ Energies

If entropy is constant in a process, then the direction (condition) for spontaneity is obvious → it is the direction in which the internal energy (U) or enthalpy (H) decreases. It is the ‘entropy tax’ which is making things difficult.

As pointed out before we will like to make computations of spontaneity purely looking at system based variables.

Under two conditions it is possible write the condition for spontaneity of a process, purely based on the system variables. If the process is carried out under constant T, V. If the process is carried out under constant T, P.

‘Free’ Energies cotd…

0 UniverseS

0 SystemG

0 SystemA

As we shall see in the coming slides:

Constant T, V

Constant T, P

At constant T,V (let Q enter the system)

Condition of spontaneity:

At constant V there can be no expansion work and if there is no other form of work (magnetic,

electrical etc.) then this energy will lead to an increase in the internal energy of the system (U).

Note that we do not have to write Ureversible as U is a state function (depends only on the

initial and final states).

Suppose we introduce a symbol A for UTS.

For a spontaneous process at constant T, V:

A is called the Helmholtz Free Energy → it is a thermodynamic potential (and a state function as U and S are state functions). Hence, if we know the initial and final values of A → we can conclude if the process is spontaneous (at constant T,V).

‘Free’ Energies cotd…

0revQS

T 0revQ

ST

T

Heat flow direction

T

Rigid wallsV constant

0U

ST

0T S U 0U T S or equivalently

Q

A U TS A U (T S S T ) Zero at constant T

0A U (T S )

At constant T,P (let Q enter the system):Condition of spontaneity:

At constant P, if there is only expansion work (no other form of work: magnetic, electrical etc.) then this energy will lead to an increase in the internal energy of the system (H).

Note that we do not have to write Hreversible as H is a state function (depends only on the

initial and final states).

Suppose we introduce a symbol G for HTS.

For a spontaneous process at constant T, H:

A is called the Gibbs Free Energy → it is a thermodynamic potential (and a state function as H and S are state functions). Hence, if we know the initial and final values of G → we can conclude if the process is

spontaneous (at constant T,P).

0revQS

T 0revQ

ST

0H

ST

0T S H 0H T S or equivalently

G H TS G H (T S S T ) Zero at constant T

0G H (T S )

THeat flow direction

T

Piston

Q

P

Helmholtz Free Energy (A or F) = E T.S S is the entropy of the system At constant V & T, for a process/reaction* to take place spontaneously the system has

to reduce its Helmholtz Free Energy. For a system to go from ‘state*’ 1 → 2 the change in F would be:

F2 F1 = F = (E2 E1) T (S2 S1) = E TS

This change of ‘state’ would take place spontaneously if F is Negative This implies that reactions which lead to an increase in the internal energy (E) are

allowed (at a ‘sufficiently high’ temperature) if there is a Entropic benefit for the process to occur(the concept of entropy will be dealt with in the context of Gibbs Free Energy)

Helmholtz Free Energy (A)

A = E T S

* used in the general sense

3

Gibbs Free Energy (G) = H T.S S is the entropy of the system

For a process/reaction* to take place spontaneously the system has to reduce its Gibbs Free Energy (at constant P & T). For a system to go from ‘state*’ 1 → 2 the change in G would be:

G2 G1 = G = (H2 H1) T (S2 S1) = H TS

This change of ‘state’ would take place spontaneously if G is Negative This implies that even Endothermic reactions are allowed (at a ‘sufficiently high’

temperature) if there is a Entropic benefit (weighed in with T) for the process to occur An example of the above is the presence of (‘equilibrium concentration’ of) vacancies

in a crystal (more about this later)

Many a times we are concerned with the relative stability of two phases at a given T and P. We asks questions such as at 1 atm. pressure & 50C, which of the phases is stable: ice, water or steam? (Answer can be found later in the slides).

Gibbs Free Energy (G) And the concept of Entropy

G = H T S

* used in the general sense

4

G = H T.S

As we have noted, G is the measure of stability at constant T,P (the usual conditions we work in condensed matter systems).

How does Gibbs free energy (G) vary with T? (illustration using the example of a pure metal)

Summary H increases with T (with positive curvature).

The Slope of the G vs T curve at 0K is zero. The curvature (G-T curve) is always negative. On ‘heating’ the H increases but G decreases

TSPVETSHG )(

( )

( )

dG dE PdV VdP TdS SdT

dq PdV PdV VdP TdS SdT

TdS VdP TdS SdT

P

dG VdP SdT

dG SdT

S

dT

dG

P

At T = 0 K, S = 0 J/K the slope of the G versus T curve at T = 0 is zero

2

2P

PP

Cd G dS

dT dT T

, P

P

dQWhere C

dT

As CP is always positive (i.e. when heat is supplied the

temperature of all materials increases), the RHS of equation is

negative

T (K)

C P

0

Pure Metal

T (K)

H

0

0

Pure Metal298 K

Slope

C P

Q & A

Revisit of the plots

H curve

G curve

Melting transition

Needless to say, there is no time involved here (and hence we cannot talk about heating or cooling). All we can comment is on the relative stability of the two phases involved (the solid and the liquid) at various temperatures. From this we can arrive at the stability regimes of the two phases (with T). (We often show this ‘as if’ some process is actually taking place– as shown with the red lines in the figure).

Since there are two phases involved, we need two H vs. T and two G vs T curves.

What happens during a phase transformation (say a solid to liquid transformation - melting)?

Liquid has higher H than solid at all T

Liquid has lower G than solid above Tm

Tm

Solid has lower G than liquid below Tm

Solid

Liquid

How long will the solid take to melt if heated above the melting point? Will the liquid freeze if cooled below Tm?

The above questions do not come under the preview of equilibrium thermodynamics. Some of these questions will be taken up for study in the chapter on phase transformations.

T (K)

H

G

Solid stable

Solid lines correspond to the solid and dotted to the liquid

Liquid stable

0 298

Solid

Liquid

Solid

Enthalpy of fusion

Liquid

T (K)

H

G

Solid stable

Solid lines correspond to the solid and dotted to the liquid

Liquid stable

0 298

Solid

Liquid

Solid

Enthalpy of fusion

Liquid

Another look

Why is water stable at 10C, does it not have higher ‘energy’ than ice?

Funda Check

True that ice has higher internal energy than water. But, internal energy (U) is not the measure of stability of a material system at constant T and P → it is the Gibbs free energy (G).

Hence, we have to compare the Gibbs free energy of water and ice at 10C (& 1 atm). The phase which has a lower G will be stable.

Ice has a lower enthalpy and a lower entropy, while water has a higher enthalpy with a higher entropy.

G = H TS.

Positive

icewater

Positive

icewatericewater SSTHHGG )()(

Additional point to note: below 0C (say at 5C) ice is stable, but we can cool water to this temperature (i.e. undercool) and hold it for a while (how long depends on other things).

Enthalpy cost not offset by entropy benefit, hence G > 0 (i.e. ice stable)Below 0C

Above 0CEnthalpy cost offset by entropy benefit, hence G < 0 (i.e. water stable)

Steam (with even higher entropy and enthalpy) becomes stable only at even higher temperatures (T > 100 C).