Physical Chemistry Third Edition

188 4 The Thermodynamics of Real Systems

PROBLEMS

Section 4.5: Multicomponent Systems

4.41 Following are data on the density of ethanol-water
solutions at 20◦C. Calculate the volume of a solution
containing 0.700 mol of water and the appropriate amount
of ethanol for each of the data points except for the 0% and
100% data points. Determine the partial molar volume of
ethanol at an ethanol mole fraction of 0.300,
by either graphical or numerical means.

% ethanol by mass density/g mL−^1

00. 99823

46. 00 0. 9224

48. 00 0. 9177

50. 00 0. 9131

52. 00 0. 9084

54. 00 0. 9039

56. 00 0. 8995

58. 00 0. 8956

100. 00 0. 7893

4.42 The partial specific volume of a system is defined as
(∂V /∂wi)P,T,w′, wherewiis the mass of component
numberiand where the subscriptw′stands for keeping the
mass of every substance fixed except for substance number
i. All of the relations involving partial molar quantities can

be converted to relations for partial specific quantities by

consistently replacingnibywifor every substance and by

replacingxiby the mass fractionyi:

yiwi/wtotal

for every substance. Using the data of Problem 4.41, find

the partial specific volume of ethanol in the mixture with

mass fraction 0.500.

4.43 a.The pressure on a sample of liquid water is changed

from 1.000 bar to 100.000 bar at a constant temperature

of 298.15 K. Find the change in the chemical potential

of the water. State any assumptions.

b.The same change in pressure is carried out at the same

temperature on an ideal gas. Find the change in its

chemical potential.

4.44Assume that the volume of a two-component solution at

constant temperature and pressure is given by

Vn 1 Vm1∗ +n 2 Vm2∗ +na 20 x 12 +na 11 x 1 x 2 +na 02 x^22

whereVm1∗ andVm2∗ are the molar volumes of the pure

substances,n 1 is the amount of substance 1,n 2 is the

amount of substance 2,nn 1 +n 2 ,x 1 andx 1 are the

mole fractions, and theacoefficients are constants. Obtain

an expression for the partial molar volume of each

substance and write an expression forn 1 V 1 +n 2 V 2.

4.6 Euler’s Theorem and the Gibbs–Duhem Relation

Euler’s theorem is a mathematical theorem that applies to homogeneous functions.

A function of several independent variables,f(n 1 ,n 2 ,n 3 ,...,nc), is said to behomo-

geneous of degreekif

f(an 1 ,an 2 ,an 3 ,...,anc)akf(n 1 ,n 2 ,n 3 ,...,nc) (4.6-1)

whereais a positive constant. For example, if every independent variable is doubled,

the new value of the function is equal to the old value times 2k.IfTandPare held fixed,

any extensive quantity is homogeneous of degree 1 in the amounts of the components,

n 1 ,n 2 ,...,nc, and any intensive quantity is homogeneous of degree 0 in the amounts

of the components. For example, if the amount of every component is doubled at

constantTandP, the value of every extensive quantity doubles while the value of

every intensive quantity remains unchanged. Iffis a homogeneous function of degree