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law can be expressed in terms of a new thermodynamic variable called entropy.One statement of the second law is that in any physical process the total entropynever decreases.

For example, if we drop a red hot cannon ball into a bucket of cold water,we end up with a bucket of warm water and a warm cannon ball. However, ifwe start from a bucket of warm water and a warm cannon ball we never observethe cannon ball jumping out red hot leaving a bucket of cold water. Suchprocesses always go in one direction: heat flows from hotter to colder bodies.The explanation based on the second law is that the entropy of the bucket ofwarm water and the warm cannon ball is higher than the entropy of the bucketof cold water and the red hot cannon ball, and the second law requires that noprocess lead to a net decrease in entropy.

There are certain processes that are reversible, and for these the entropy

does not change. That is, reversible processes occur at constant entropy.The concept of entropy is a subtle one in classical thermodynamics. It has

a relatively simple interpretation in statistical mechanics: entropy is a measureof randomness, and the Universe evloves from a more ordered to a less orderedstate.

There are certain systems that superficially appear to violate the secondlaw of thermodynamics. One class involves self gravitating systems: interstellargas clouds collapse to form stars; also the galaxies must have formed from amuch more uniform distribution of matter in the early epoch of the Universe.In these systems, highly structured systems develop from essentially uniformlydistributed matter. Biological systems also evolve from simpler to more complexstructures. However, a more careful examination shows that in both cases the

total entropy increases as these systems evolve.

Physical quantities

The laws of thermodynamics force us to define certain physical variables and toquantify them. The basic thermodynamic variables are heat, temperature andentropy. There are other variable that we can use to describe a system. Supposefor example that the system is the air in this room. We can do measurementsto determine the volume of the system, the pressure that the air exerts, thedensity of the air, the chemical composition of the air, and so on. We arefamiliar with the concept of the temperature of the air, which we measure witha thermometer. However, the heat in the air and the entropy of the air arenot familiar. Nevertheless, all these are examples of physical quantities that

describe our system.A formal definition of a physical quantity is:

A physical quantity is defined by the sequence of operations used to determine

its value.

This sequence of operations will often include calculations as well as measure-ments.

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A physical law can now be defined:A physical law is a relation involving physical quantities that is found to be

satisfied by physical systems under specified conditions.

Some physical laws are very simple. For example, measuring a mass using a scaledefines a physical quantity, and measuring mass using a spring balance definesanother physical quantity. A simple physical law is that these two types of massare the same thing: this law implies that these are two different but equivalentways of measuring the same physical quantity (mass). A related physical law,with more profound consequences, is that inertial mass and gravitational massare equal.

In thermodynamics we are concerned with thermodynamic systems, with theair in this room being one illustrative example of a thermodynamic system. Asomewhat better example is the air inside a balloon. This system is characterized

by its volume, pressure, temperature and density. Another feature concerns thetransport of heat across the surface of the balloon. If no heat is allowed to crossthe surface (the balloon is made of thermally insulating material that allows noheat conduction), we say that the system is (thermally) isolated and the changesare then said to be adiabatic.

1.2 The laws of thermodynamics

After the first and second laws of thermodynamics had been recognized andthermodynamics was becoming a more formal subject, it was realized that therewas no consistent definition of temperature. A zeroth law of thermodynamicswas added to rectify this. Later still, a third law of thermodynamics was added

to summarize the observed properties of matter at extremely low temperatures.

Zeroth Law: If two systems are in thermal equilibrium with a third system,then they are in equilibrium with each other.The zeroth law allows one to order systems according to the direction heat flowswhen two systems are put into thermal contact. One system is said to be hotterthan another if heat flows from the former to the latter when they are put intothermal contact. This allows one to introduce a parameter, called an empiricaltemperature, which is the same for all bodies that are in thermal equilibriumwith each other. This is done by constructing one system, called a thermometer,that allows one to ascribe a number to the temperature, e.g., the height of acolored liquid in a glass tube.

First Law: Heat is a form of energy, and energy is conserved.The idea behind the first law is familiar to any thinking person in the modernworld. We pay for energy, in the form of electricity, gas, oil, etc., which we use forheating, and we know that the amount of heat that we get is proportional to theamount of energy that we pay for. Heat is said to be generated by doing work onthe system. The simplest example is the work done by a mechanical force: the

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work is then the force times the displacement. In simple thermodynamic systemsthis appears with the force being identified as the pressure (= force per unit area)and the displacement being identified as the change in volume (= displacementof a surface area times that area). Another example of work iinvolves electricity:the force is then identified as the EMF (= electrical potential difference) anddisplacement as the charge transferred (= electrical current time). Otherforce-displacement pairs that will appear in this course include surface tensionand surface ares, and the chemical potential and chemical concentration.

Second Law: There are several different ways of expressing the second law ofthermodynamics, and the following are three examples.Clausius principle: No cyclic process exists which has as its sole effect the trans-ference of heat from a colder body to a hotter body.

Kelvins principle: No cyclic process exists which produces no other effect thanthe extraction of heat from a body and its conversion into an equivalent amountof work.Caratheodorys principle: There are states of a system, differing infinitesimallyfrom a given state, which are unattainable from that state by any quasi-static

adiabatic process.The second law (especially in the Caratheodory form) allows one to order thestates according to the direction that the system is allowed to evolve. A pa-rameter, called an empirical entropy, is ascribed to each state, such that thisparameter never decreases spontaneously. In the absence of external agents do-ing work on the system, any change in the system is either reversible, whichinvolves no change in entropy, or irreversible, which involves an increase in en-tropy.

The bucket of warm water with the warm cannon ball has a higher entropythan the bucket of cold water and the red hot cannon ball, so that the evolutionis irreversible and can go in only one direction. To reverse the change we needto do work on the system: mechanical work to lift the cannon ball, and ther-modynamic work, using some form of refrigerator to decrease the temperatureof the bucket of water, and some other process to heat the cannon ball.

Third Law:The entropy of a system approaches a constant value as the temperature ap-

proaches absolute zero.The third law is relatively easy to understand from a statistical point of viewin which entropy is associated with disorder. As absolute zero is approached,all thermal motions cease, and any system must approach an ordered state in

which the particles do not move. Hence, the entropy of a system is definedonly to within an arbitrary constant, and only changes in entropy have physicalsignificance. The changes in entropy become negligibly small as absolute zerois approached.

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1.3 Thermodynamic variablesTo describe these ideas quantitatively we need to introduce thermodynamicvariables and state functions. For the present let the pressure, P, be the onlyforce that we consider. Then an infinitesimal change in the volume, dV, impliesan infinitesimal amount of work done on the system, dW = P dV. Note thesign: a positive change dV implies that the system expands a little, and hencethat the system does work against the external pressure force (P is alwayspositive), so that with dW defined as the work done on the system, we needa minus sign. Let dQ be the amount of heat added to the system. Let U bethe internal energy of the system. Then the first law requires conservation ofenergy, which is expressed through the relation dU = dQ + dW = dQ P dV.

The absolute temperature and the metrical entropy

The first law enables one to define an empirical temperature and the secondlaw allows one to define an empirical entropy. What this means is that thelaws allow us to order states with parameters t and s. Consider two states ofa system labeled 1 and 2, with parameters t1, s1 and t2, s2, respectively. Fort1 > t2 we say that state 1 is hotter than state 2. The observational fact is thatif the system is brought into thermal contact with some other system so thatheat can be added or taken away from our system, then heat is required to flowout the system to get from state 1 to state 2, and heat is required to flow intothe system to get from state 2 to state 1. For s1 > s2 it is possible for state2 to evolve into state 1 without any work being done on the system, but it isimpossible for state 1 to evolve into state 2.

The temperature that we are familiar with is the Celsius scale, which differsfrom the absolute temperature (Kelvin) scale, by 273.15 K. That is, absolutezero is 273.15C. The unit of the absolute temperature scale is the kelvin,denoted K. The existence of the absolute temperature scale involves a rathersubtle argument, which can be summarized as follows.

Consider the heat, Q, transferred to the system in some irreversible change(this is, found by integrating the elements dQ over a range of infinitesimalchanges). The zeroth and second laws imply that Q = Q(t, s) is some functionof t and s. For a reversible change we know that s does not change, and areversible change corresponds to no transfer of heat. Hence for a reversiblechange we have dQ = 0 and ds = 0. Now consider an irreversible change,ds = 0. In an irreversible change we have dQ = ds, where = (t, s) is some

function of the empirical temperature and the empirical entropy. The subtleargument implies (t, s) is of the form (t, s) = T(t)(s), that is, that (t, s)depends on t and s only in the form of a product of a function, T(t), of t anda function, (s), of s. The argument involves imagining splitting the systeminto two parts, labeled A and B say. (Imagine splitting the system consistingof the air in the room into two parts corresponding to the air in two halves of

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Appendix A: Rules of Partial Differentiation1). Let f(x,y,z) be any differentiable function. Then one has

df =

f

x

y,z

dx +

f

y

x,z

dy +

f

z

x,y

dz. (A.1)

The order of differentiation is reversible:

y

f

x

y,z

x,z

=

x

f

y

x,z

y,z

. (A.2)

2). Let there be a relation f(x,y,z) = 0 that can be solved for x = x(y, z),y = y(x, z) or z = z(x, y). Then the partial derivative of the first of thesefunctions with respect to y at fixed z is related to the partial derivative of thesecond with respect to x at fixed z by

x

y

z

y

x

z

= 1. (A.3)

The partial derivatives also satisfy the chain rulex

y

z

y

z

x

z

x

y

= 1. (A.4)

3). Let f(x, y) be a function of x and y with y = y(x, z) a function of x and athird variable, z. Then f = f

x, y(x, z)

may also be regarded as a function of

x and z. One has

df =

f

x

y

dx +

f

y

x

dy, (A.5)

df =

f

x

z

dx +

f

z

x

dz, (A.6)

dy =

y

x

z

dx +

y

z

x

dz. (A.7)

On inserting (A.7) into (A.5) and comparing the result with (A.6), one findsf

x

z

=

f

x

y

+

f

y

x

y

x

z

, (A.8)

f

z

x

=

f

y

x

y

z

x

. (A.9)

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Exercise Set 11.1). The coefficient of isothermal compressibility is

= 1

V

V

P

T

, (E1.1)

and the coefficient of thermal expansion is

=1

V

V

T

P

. (E1.2)

Show that for an ideal gas one has = 1/P, = 1/T.

1.2). Repeat the calculations in the previous problem, but for a van der Waalsgas. Show that one has

=v2(v b)2

RT v3 2a(v b)2, =

Rv2(v b)

RT v3 2a(v b)2. (E1.3)

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