Physical Chemistry Third Edition

194 4 The Thermodynamics of Real Systems

0 0.2

2 1.4

2 1.2

2 1.0

2 0.8

2 0.6

2 0.4

Tangent point

atx 15 0.500 5 x ́ 1

x 1

2 0.2

0

0.4 0.6 0.8 1.0

Value of

V 12 V 1 * at

x 15 0.500

_ _

Experimental

curve

Value of V 22 V 2 * at x 15 0.500

_ _

DV

m,mix

/cm

3 mol

21

Figure 4.3 The Change in the Mean Molar Volume on Mixing for an Ethanol-Water

Solution as a Function of Mole Fraction of Ethanol.

PROBLEMS

Section 4.6: Euler’s Theorem and the Gibbs–Duhem
Relation

4.45 Determine which (if any) of the following functions are
homogeneous with respect to all three independent
variablesx,y, andz. Find the degree of each homogeneous
function. All letters exceptf,x,y, andzdenote constants.
For the expressions that are homogeneous, verify that they
conform to Euler’s theorem.
a.f(x,y,z)ax^2 +bx^3 y−^1 +cy^2 +dyz
b.f(x,y,z)ax^2 y−^2 +bln(y/z)+ctan(x^3 y−^3 )
c.f(x,y,z)ax^2 +bcos(x^2 y−^2 )+cz^2

4.46 Determine which (if any) of the following functions are

homogeneous with respect to all three independent

variablesx,y, andz. Find the degree of each homogeneous

function. All letters exceptf,x,y, andzdenote constants.

For the expressions that are homogeneous, verify that they

conform to Euler’s theorem.

a.f(x,y,z)az^3 +bx^3 cos(x^4 y−^4 )+cexp(yz−^1 )

b.f(x,y,z)ax^4 +bx^2 yzsin(x/z)

c.f(x,y,z)acos(x/y)+bsin(x/z)

4.47A mixture of 0.500 mol of ideal gas 1 and 1.500 mol

of ideal gas 2 is produced at a constant temperature