Thermodynamics and Chemistry

CHAPTER 5 THERMODYNAMIC POTENTIALS

PROBLEMS 147

Problems

An underlined problem number or problem-part letter indicates that the numerical answer appears
in AppendixI.

5.1 Show that the enthalpy of a fixed amount of an ideal gas depends only on the temperature.

5.2 From concepts in this chapter, show that the heat capacitiesCVandCpof a fixed amount of

an ideal gas are functions only ofT.

5.3 During the reversible expansion of a fixed amount of an ideal gas, each increment of heat is

given by the expression∂qDCVdTC.nRT=V /dV(Eq.4.3.4).

(a)A necessary and sufficient condition for this expression to be an exact differential is that

the reciprocity relation must be satisfied for the independent variablesTandV(see Ap-

pendixF). Apply this test to show that the expression isnotan exact differential, and that

heat therefore is not a state function.

(b)By the same method, show that the entropy increment during the reversible expansion,

given by the expression dSD∂q=T, is an exact differential, so that entropy is a state

function.

5.4 This problem illustrates how an expression for one of the thermodynamic potentials as a func-

tion of its natural variables contains the information needed to obtain expressions for the other

thermodynamic potentials and many other state functions.

From statistical mechanical theory, a simple model for a hypothetical “hard-sphere” liquid

(spherical molecules of finite size without attractive intermolecular forces) gives the following

expression for the Helmholtz energy with its natural variablesT,V, andnas the independent

variables:

ADnRTln



cT3=2



V

n

b



nRTCna

Herea,b, andcare constants. Derive expressions for the following state functions of this

hypothetical liquid as functions ofT,V, andn.

(a)The entropy,S

(b)The pressure,p

(c)The chemical potential,

(d)The internal energy,U

(e)The enthalpy,H

(f)The Gibbs energy,G

(g)The heat capacity at constant volume,CV

(h)The heat capacity at constant pressure,Cp(hint: use the expression forpto solve forV

as a function ofT,p, andn; then useHDUCpV)

5.5 Figure5.2on the next page depicts a hypothetical liquid in equilibrium with its vapor. The

liquid and gas are confined in a cylinder by a piston. An electrical resistor is immersed in the

liquid. Thesystemis the contents of the cylinder to the left of the piston (the liquid, gas, and

resistor). The initial state of the system is described by

V 1 D0:2200m^3 T 1 D300:0K p 1 D2:50 105 Pa

A constant currentID0:5000A is passed for 1600 s through the resistor, which has electric

resistanceRelD50:00 . The piston moves slowly to the right against a constant external