Page 195 - Physical Chemistry

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Chapter 6 having the following properties: (1) The equation of state PV n RT [Eq. (1.22)] is
tot
Reaction Equilibrium in Ideal Gas obeyed for all temperatures, pressures, and compositions, where n is the total num-
Mixtures tot
ber of moles of gas. (2) If the mixture is separated from pure gas i (where i is any one
of the mixture’s components) by a thermally conducting rigid membrane permeable to
gas i only (Fig. 6.2), then at equilibrium the partial pressure P x P [Eq. (1.23)] of
Ideal i i
gas mixture gas i in the mixture is equal to the pressure of the pure-gas-i system.
at T and P This definition makes sense from a molecular viewpoint. Since there are no inter-
i
with P i x P molecular interactions either in the pure ideal gases or in the ideal gas mixture, we
expect the mixture to obey the same equation of state obeyed by each pure gas, and
i
condition (1) holds. If two samples of pure ideal gas i at the same T were separated by
a membrane permeable to i, equilibrium (equal rates of passage of i through the mem-
i brane from each side) would be reached with equal pressures of i on each side.
Pure gas i Because there are no intermolecular interactions, the presence of other gases on one
at P* and T

i side of the membrane has no effect on the net rate of passage of i through the mem-
brane, and condition (2) holds.
The standard state of component i of an ideal gas mixture at temperature T is
At equilibrium, P* = P .
i i
defined to be pure ideal gas i at T and pressure P° 1 bar.
Figure 6.2 In Fig. 6.2, let m be the chemical potential of gas i in the mixture, and let m* i
i
be the chemical potential of the pure gas in equilibrium with the mixture through the
An ideal gas mixture separated membrane. An asterisk denotes a thermodynamic property of a pure substance. The
from pure gas i by a membrane condition for phase equilibrium between the mixture and pure i is m m*(Sec. 4.7).
permeable to i only. i i
The mixture is at temperature T and pressure P, and has mole fractions x , x , ...,
1 2
x , .... The pure gas i is at temperature T and pressure P*. But from condition (2) of
i
i
the definition of an ideal gas mixture, P*at equilibrium equals the partial pressure P
i i
x P of i in the mixture. Therefore the phase-equilibrium condition m m* becomes
i i i
m 1T, P, x , x , . . .2 m*1T, x P2 m* 1T, P 2 ideal gas mixture (6.3)
2
i
1
i
i
i
i
Equation (6.3) states that the chemical potential m of component i of an ideal gas mix-
i
ture at T and P equals the chemical potential m*of pure gas i at T and P (its partial
i i
pressure in the mixture). This result makes sense; since intermolecular interactions are
absent, the presence of other gases in the mixture has no effect on m .
i
From Eq. (6.2), the chemical potential of pure gas i at pressure P is m*(T, P )
i i i
m°(T ) RT ln (P /P°), and Eq. (6.3) becomes
i i
m m°1T2 RT ln 1P >P°2 ideal gas mixture, P° 1 bar (6.4)*
i
i
i
Equation (6.4) is the fundamental thermodynamic equation for an ideal gas mixture.
In (6.4), m is the chemical potential of component i in an ideal gas mixture, P is the
i i
partial pressure of gas i in the mixture, and m°(T) [ G° (T)] is the chemical poten-
i m,i
tial of pure ideal gas i at the standard pressure of 1 bar and at the same temperature T
as the mixture. Since the standard state of a component of an ideal gas mixture was
defined to be pure ideal gas i at 1 bar and T, m° is the standard-state chemical poten-
i
tial of i in the mixture. m° depends only on T because the pressure is fixed at 1 bar for
i
the standard state.
Equation (6.4) shows that the graph in Fig. 6.1 applies to a component of an ideal
gas mixture if m and m° are replaced by m and m°, and P is replaced by P .
i i i
Equation (6.4) can be used to derive the thermodynamic properties of an ideal gas
mixture. The result (Prob. 9.20) is that each of U, H, S, G, and C for an ideal gas mix-
P
ture is the sum of the corresponding thermodynamic functions for the pure gases
calculated for each pure gas occupying a volume equal to the mixture’s volume at a
pressure equal to its partial pressure in the mixture and at a temperature equal to its
temperature in the mixture. These results make sense from the molecular picture in
which each gas has no interaction with the other gases in the mixture.