Thermodynamics and Chemistry

CHAPTER 9 MIXTURES

9.3 GASMIXTURES 240

A(g)

pDp^0

(A + B)(g)

pADp^0

pDpACpB

Figure 9.4 System with two gas phases, pure A and a mixture of A and B, separated

by a semipermeable membrane through which only A can pass. Both phases are ideal

gases at the same temperature.

gas. The standard chemical potentiali(g) of gaseousiis the chemical potential ofiin
this gas standard state, and is a function of temperature.
To derive an expression foriin an ideal gas mixture relative toi(g), we make an
assumption based on the following argument. Suppose we place pure A, an ideal gas, in
a rigid box at pressurep^0. We then slide a rigid membrane into the box so as to divide
the box into two compartments. The membrane is permeable to A; that is, molecules of
A pass freely through its pores. There is no reason to expect the membrane to affect the
pressures on either side,^6 which remain equal top^0. Finally, without changing the volume
of either compartment, we add a second gaseous substance, B, to one side of the membrane
to form an ideal gas mixture, as shown in Fig.9.4. The membrane is impermeable to B, so
the molecules of B stay in one compartment and cause a pressure increase there. Since the
mixture is an ideal gas, the molecules of A and B do not interact, and the addition of gas B
causes no change in the amounts of A on either side of the membrane. Thus, the pressure
of A in the pure phase and the partial pressure of A in the mixture are both equal top^0.
Our assumption, then, is that the partial pressurepAof gas A in an ideal gas mixture in
equilibrium with pure ideal gas A is equal to the pressure of the pure gas.
Because the system shown in Fig.9.4is in an equilibrium state, gas A must have the
same chemical potential in both phases. This is true even though the phases have different
pressures (see Sec.9.2.7). Since the chemical potential of the pure ideal gas is given by
D(g)CRTln.p=p/, and we assume thatpAin the mixture is equal topin the pure
gas, the chemical potential of A in the mixture is given by

ADA(g)CRTln

pA

p

(9.3.4)

In general, for each substanceiin an ideal gas mixture, we have the relation

iDi(g)CRTln

pi

p

(9.3.5)

(ideal gas mixture)

wherei(g) is the chemical potential ofiin the gas standard state at the same temperature
as the mixture.

Equation9.3.5shows that if the partial pressure of a constituent of an ideal gas mixture

is equal top, so that ln.pi=p/is zero, the chemical potential is equal to the standard

(^6) We assume the gas is not adsorbed to a significant extent on the surface of the membrane or in its pores.