194
4 The Thermodynamics of Real Systems
0.2
0
21.4
21.2
21.0
20.8
20.6
20.4
Tangent point
at x 1 5 0.500 5 x´ 1
x 1
20.2
0
0.4
0.8
0.6
1.0
Value of
V 1 2 V 1 * at
x 1 5 0.500
_
_
Experimental
curve
Value of V 2 2 V 2 * at x 1 5 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.
P R O B L E M S
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
variables x, y, and z. Find the degree of each homogeneous
function. All letters except f , x, y, and z denote 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 + b ln(y/z) + c tan(x 3 y −3 )
c. f (x, y, z) ax 2 + b cos(x 2 y −2 ) + cz 2
4.46 Determine which (if any) of the following functions are
homogeneous with respect to all three independent
variables x, y, and z. Find the degree of each homogeneous
function. All letters except f , x, y, and z denote 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 ) + c exp(yz −1 )
b. f (x, y, z) ax 4 + bx 2 yz sin(x/z)
c. f (x, y, z) a cos(x/y) + b sin(x/z)
4.47 A mixture of 0.500 mol of ideal gas 1 and 1.500 mol
of ideal gas 2 is produced at a constant temperature