Physical Chemistry Third Edition

148 3 The Second and Third Laws of Thermodynamics: Entropy

A general procedure for the calculation of an entropy change in a closed system

without phase change or chemical reaction is to carry out the line integral

∆S

∫

c

dqrev

T

over a curve in state space corresponding to a reversible path.

The statistical entropy is defined by

SstkBln(Ω)+S 0

wherekBis Boltzmann’s constant and whereΩis the thermodynamic probability,

equal to the number of mechanical states that might be occupied by the system given

its macroscopic state. The quantityS 0 is an arbitrary constant whose value we can

take to equal zero. We showed that for an isothermal expansion of a model system

the change in the statistical entropy and the change in the thermodynamic entropy are

equal.

The third law of thermodynamics asserts that if the entropies of all samples of pure

perfect crystalline elements are taken as zero, the entropies of all samples of pure perfect

crystalline compounds can also consistently be taken as zero. Entropies relative to the

entropy at zero temperature are called absolute entropies. The values of these absolute

entropies can be used to calculate entropy changes of chemical reactions.

A consequence of the second and third laws of thermodynamics is that no finite

number of operations, either reversible or irreversible, can bring an object to zero

temperature.

ADDITIONAL PROBLEMS

3.45 a.Sketch a diagram representing the Carnot cycle in a
state space withSon the horizontal axis andTon the
vertical axis.
b.Show that the work done on the surroundings in
one cycle is equal to the area enclosed in this
diagram.

3.46Assume that a substance A can react without any energy
change to an isomer C (as in a racemization
reaction).

a.Assume that 1.000 mol of substance A is placed in a

container that is adiabatically isolated from any

surroundings and is allowed to equilibrate. Write an

expression for the entropy change of the system as a

function ofx, the amount of A that has reacted.

b.Find the value ofxat which the entropy of the system

is at a maximum.

c.Find the value of the equilibrium constant for the

reaction.

3.47Supercooled (metastable) water vapor commonly occurs in

the atmosphere if dust particles are not present to begin

condensation to the liquid. Sometimes small particles, such

as tiny crystals of silver iodide, are released from airplanes

in an attempt to begin condensation. This process is called

“cloud seeding.” At a certain location, water vapor at 25◦C

has a metastable partial pressure of 32.0 torr. The equilib-

rium value at this temperature is 23.756 torr. Consider the

air that is present to be the surroundings, and assume it to

remain at equilibrium at 25◦C. A tiny particle is added to

begin condensation. Calculate∆S,∆H, and∆Ssurrper

mole of water that condenses. State any assumptions.

3.48Without doing any detailed calculations, specify for each

process whether each of the following quantities is posi-

tive, negative, or equal to zero:q,w,∆U,∆S, and∆Ssurr.

a.The system consisting of two large objects and a small

bar between them is allowed to come to equilibrium

from an initial state in which the two objects are at

different temperatures. Assume that the objects have

fixed volume.