Physical Chemistry Third Edition

3.4 Statistical Entropy 133

d.Calculate the entropy changes of the system and the

surroundings if the initial and final states are the same

as in parts a and b, but if the gas is expanded

irreversibly and isothermally against an external

pressure of 1.000 atm and then heated irreversibly with

the surroundings remaining essentially at equilibrium

at 400 K.

3.22 Find∆H,∆S, andqfor the reversible heating of
0.500 mol of benzene from 25◦Cto100◦C at a constant
pressure of 1.000 atm. The normal boiling temperature of
benzene is 80.1◦C and the enthalpy change of vaporization
is 30.8 kJ mol−^1.

3.23 a.A sample of 2.000 mol of neon is expanded reversibly
and adiabatically from a volume of 10.00 L and a
temperature of 500.0 K to a volume of 25.00 L. Find
the final temperature,q,w,∆U,∆S, and∆Sunivfor the
process. State any assumptions or approximations.
b. The same sample is restored to its original state and is
first expanded adiabatically and irreversibly at a
constant external pressure of 1.000 atm to a volume of
25.00 L, then cooled reversibly to the same final
temperature as in part a at a constant volume of
25.00 L. Find the final temperature for the irreversible
step, and findq,w,∆U, and∆Sfor this entire

two-step process. What can you say about∆Sunivfor

each step of this two-step process?

3.24 1.000 mol of carbon tetrafluoride is vaporized at the normal

boiling point of− 128 ◦C and 1.000 atm (bothPandT

held constant). The molar volume of the liquid is

44.9 cm mol−^1. The enthalpy change of vaporization is

12.62 kJ mol−^1. Findq,w,∆S, and∆Hfor this process.

3.25A sample of 2.000 mol of a monatomic ideal gas is

expanded and heated. Its initial temperature is 300.0 K and

its final temperature is 400.0 K. Its initial volume is

20.00 L and its final volume is 40.00 L. Calculate∆S.

Does the choice of path between the initial and final states

affect the result?

3.26 The molar heat capacity of water vapor is represented by

CP, m 30 .54JK−^1 mol−^1 +(0.01029 J K−^2 mol−^1 )T

whereTis the absolute temperature.

a.Find∆Hfor heating 2.000 mol of water vapor at a

constant pressure of 1.000 atm from 100◦Cto500◦C.

b.Findq,w, and∆Ufor the process.

c.Find∆Sfor the process.

3.4 Statistical Entropy

Since molecules can occupy various states without changing the macroscopic state of

the system of which they are a part, it is apparent that many microstates of a macroscopic

system correspond to one macroscopic state. We denote the number of microstates

corresponding to a given macrostate byΩ. The quantityΩis sometimes called the

thermodynamic probabilityof the macrostate. The thermodynamic probability is a

measure of lack of information about the microstate of the system for a particular

macrostate. A large value corresponds to a small amount of information, and a value

of unity corresponds to knowledge that the system is in a specific microstate.

For a macrostate specified by values ofU,V, andn, Boltzmann defined astatistical

entropy:

SstkBln(Ω)+S 0 (definition of statistical entropy) (3.4-1)

wherekBis Boltzmann’s constant, equal toR/NAv. The constantS 0 is a constant that we

usually set equal to zero. The task of this section is to show that the statistical entropy

can be equivalent to the thermodynamic entropy for a model system that is called a

lattice gas. This system contains a numberNof noninteracting point-mass molecules

confined in a container with volumeV. We assume that the particles obey classical

mechanics. The lattice gas is in an equilibrium macrostate specified by values ofU,V

andN. The microstate of the lattice gas is specified by the positions and velocities of

all of the particles.