Physical Chemistry Third Edition

4.6 Euler’s Theorem and the Gibbs–Duhem Relation 195

of 350.0 K and a constant pressure of

1.000 bar.

a.Find∆Smix.

b.Find∆Gmix.

c.Find the value ofμ−μ◦for each gas.

d.A small change in composition is made at constant

temperature and pressure such thatμ 1 changes by

0.100 J mol−^1. Find the corresponding change inμ 2.

Approximate your answer as though the changes were

infinitesimal.

4.48 a.From the data of Problem 4.41, make a graph of

∆Vm,mixas a function of ethanol mole fraction and

determine the partial molar volume of each substance

for the solution with ethanol mole fraction equal to

0.300, using the method of intercepts.

b.From the values of the partial molar volumes of water

and ethanol in the solution of part a, find the volume of

a solution containing 0.600 mol of ethanol and

1.400 mol of water. Compare this value with the value

obtained by interpolating in the list of density values

given in Problem 4.41.

Summary of the Chapter

In this chapter we have obtained two types of mathematical tools for the application

of thermodynamics to real systems: (1) criteria for spontaneous processes in terms of

properties of a system; (2) useful formulas.

The second law of thermodynamics provides the general criterion for spontaneous

processes: No process can decrease the entropy of the universe. For a closed simple

system at constant pressure and temperature the Gibbs energyGcannot increase, and

for a closed simple system at constant temperature and volume the Helmholtz energy

Acannot increase.

Several fundamental relations were obtained for closed simple systems. The first

relations were for the differentials of the different energy-related state variables for

closed simple systems. For example,

dG−SdT+VdP

The Maxwell relations were derived from these differentials. For example

(

∂S

∂P

)

T,n

−

(

∂V

∂T

)

P,n

The Maxwell relations were used to obtain a number of useful formulas, including the

thermodynamic equation of state:

(

∂U

∂V

)

T,n

T

(

∂P

∂T

)

V,n

−P

The molar Gibbs energy of a pure substance was expressed as the molar Gibbs energy

in the standard state,G◦m, plus another term. For an ideal gas

GmG◦m+RTln (P/P◦)

and for an incompressible solid or liquid

GmG◦m+Vm(P−P◦)

The description of open multicomponent systems was based on the Gibbs equation,

dG−TdS+VdP+

∑c

i 1

μidni