Physical Chemistry Third Edition

238 6 The Thermodynamics of Solutions

6.1 Ideal Solutions

Asolutionis a homogeneous mixture of two or more components (substances whose

amounts can be independently varied). We ordinarily apply the name only to solid and

liquid mixtures, although a gaseous mixture is also homogeneous. We begin withideal

solutions, which are defined to be solutions in which the chemical potential of each

component is given for all compositions by the formula

μi(T,P)μ∗i(T,P)+RTln(xi)

(6.1-1)

whereμ∗i(T,P) is the chemical potential of the pure componentiwhen it is at the same

temperature,T, and pressure,P, as the solution and wherexiis the mole fraction of

componentiin the solution.

Raoult’s Law

Raoult’s lawis an empirical law nearly obeyed by some solutions:

PiPi∗xi (Raoult’s law) (6.1-2)

wherePiis thepartial vapor pressureof substancei, defined as the partial pressure of
substanceiin the vapor phase that is at equilibrium with the solution. The equilibrium
vapor pressure of the pure substanceiat the temperature and pressure of the solution
is denoted byPi∗, and the mole fraction of substanceiin the solution is denoted by
xi. If the solution is liquid and if substanceiis a solid at equilibrium at the temper-
ature of the solution,P∗i must represent the vapor pressure of the supercooled liquid
substance.
Raoult’s law is named for Francois
Marie Raoult, 1830–1901, a French
chemist who was one of the founders of
physical chemistry.

We now show that a component of an ideal solution obeys Raoult’s law if the

solution is at equilibrium with an ideal gas mixture. From the fundamental fact of

phase equilibrium the chemical potential of componentihas the same value in the

solution and in the vapor:

μ∗i(T,P)+RTln(xi)μ

◦(g)

i +RTln

(

Pi

P◦

)

(6.1-3)

where we now label the chemical potential of the gas in its standard state asμ◦i(g).We

will have additional standard states, so we specify which one we are using.

Equations (6.1-1) and (6.1-3) containμ∗i(T,P), the chemical potential of pure sub-

stanceiat temperatureTand pressureP. The chemical potential of a pure substance

is equal to its molar Gibbs energy. It was shown in Chapter 4 that the molar Gibbs

energy of a pure solid or liquid is nearly pressure-independent. We will assume thatμ∗i

is independent of pressure.

EXAMPLE 6.1

The density of ethanol (substance 1) is equal to 0.7885 g cm−^3 at 19.0◦C. Its equilibrium

vapor pressure at 19.0◦C is equal to 40.0 torr. Find the change in its chemical potential at

19.0◦C if the total pressure is changed from 40.0 torr to 1.000 atm. Compare the value of this