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6 The Thermodynamics of Solutions
6.1
Ideal Solutions
A solution is 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 with ideal
solutions, which are defined to be solutions in which the chemical potential of each
component is given for all compositions by the formula
µ i (T , P) µ ∗
i (T , P) + RT ln(x i )
(6.1-1)
where µ ∗
i (T , P) is the chemical potential of the pure component i when it is at the same
temperature, T , and pressure, P, as the solution and where x i is the mole fraction of
component i in the solution.
Raoult’s Law
Raoult’s law is an empirical law nearly obeyed by some solutions:
P i P ∗
i x i (Raoult’s law)
(6.1-2)
where P i is the partial vapor pressure of substance i, defined as the partial pressure of
substance i in the vapor phase that is at equilibrium with the solution. The equilibrium
vapor pressure of the pure substance i at the temperature and pressure of the solution
is denoted by P ∗
i , and the mole fraction of substance i in the solution is denoted by
x i . If the solution is liquid and if substance i is a solid at equilibrium at the temperature 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 component i has the same value in the
solution and in the vapor:
µ
∗
i (T , P) + RT ln(x i ) µ
◦(g)
i
+ RT ln
P i
P ◦
(6.1-3)
where we now label the chemical potential of the gas in its standard state as µ
◦(g)
i . We
will have additional standard states, so we specify which one we are using.
Equations (6.1-1) and (6.1-3) contain µ ∗
i (T , P), the chemical potential of pure substance i at temperature T and pressure P. 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 that µ ∗
i
is independent of pressure.
E X A M P L E 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
6 The Thermodynamics of Solutions
6.1
Ideal Solutions
A solution is 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 with ideal
solutions, which are defined to be solutions in which the chemical potential of each
component is given for all compositions by the formula
µ i (T , P) µ ∗
i (T , P) + RT ln(x i )
(6.1-1)
where µ ∗
i (T , P) is the chemical potential of the pure component i when it is at the same
temperature, T , and pressure, P, as the solution and where x i is the mole fraction of
component i in the solution.
Raoult’s Law
Raoult’s law is an empirical law nearly obeyed by some solutions:
P i P ∗
i x i (Raoult’s law)
(6.1-2)
where P i is the partial vapor pressure of substance i, defined as the partial pressure of
substance i in the vapor phase that is at equilibrium with the solution. The equilibrium
vapor pressure of the pure substance i at the temperature and pressure of the solution
is denoted by P ∗
i , and the mole fraction of substance i in the solution is denoted by
x i . If the solution is liquid and if substance i is a solid at equilibrium at the temperature 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 component i has the same value in the
solution and in the vapor:
µ
∗
i (T , P) + RT ln(x i ) µ
◦(g)
i
+ RT ln
P i
P ◦
(6.1-3)
where we now label the chemical potential of the gas in its standard state as µ
◦(g)
i . We
will have additional standard states, so we specify which one we are using.
Equations (6.1-1) and (6.1-3) contain µ ∗
i (T , P), the chemical potential of pure substance i at temperature T and pressure P. 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 that µ ∗
i
is independent of pressure.
E X A M P L E 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