Page 194 - Physical Chemistry

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Even if equilibrium is not reached, knowing the equilibrium constant is important Section 6.1
since this enables us to find the maximum possible yield of a desired product under Chemical Potentials in an
Ideal Gas Mixture
given conditions.
In aqueous solutions, reactions that involve ions are generally fast and equilibrium
is usually assumed; recall acid–base and complex-ion equilibrium calculations done in
general and analytical chemistry. Equilibrium analysis is important in environmental-
chemistry studies of the composition of water systems such as lakes and in dealing
with air pollution. Figure 6.5 shows that significant amounts of NO are present in
heated air at equilibrium. The formation of NO in automobile engines and in indus-
trial burning of coal and oil in power plants pollutes the atmosphere. (Figure 6.5 is not
quantitatively applicable to automobile engines because the combustion of the fuel
depletes the air of oxygen and because there is not enough time for equilibrium to be
reached, so NO formation must be analyzed kinetically. The equilibrium constant
determines the maximum amount of NO that can be formed.)

6.1 CHEMICAL POTENTIALS IN AN IDEAL GAS MIXTURE
Before dealing with m of a component of an ideal gas mixture, we find an expression
i
for m of a pure ideal gas.
Chemical Potential of a Pure Ideal Gas
The chemical potential is an intensive property, so m for a pure gas depends on T and
P only. Since reaction equilibrium is usually studied in systems held at constant tem-
perature while the amounts and partial pressures of the reacting gases vary, we are
most interested in the variation of m with pressure. The Gibbs equation for dG for a
fixed amount of substance is dG SdT VdP [Eq. (4.36)], and division by the
number of moles of the pure ideal gas gives dG dm S dT V dP, since the
m
m
m
chemical potential m of a pure substance equals G [Eq. (4.86)]. For constant T, this
m
equation becomes
dm V dP 1RT>P2 dP const. T, pure ideal gas
m
If the gas undergoes an isothermal change of state from pressure P to P , integration
2
1
of this equation gives
2 P 2 1
dm RT dP
P
1 P 1
m1T, P 2 m1T, P 2 RT ln 1P >P 2 pure ideal gas (6.1)
2
2
1
1
Let P be the standard pressure P° 1 bar. Then m(T, P ) equals m°(T), the gas’s
1 1
standard-state chemical potential at temperature T, and (6.1) becomes m(T, P )
2
m°(T) RT ln (P /P°). The subscript 2 is not needed, so the chemical potential m(T, P)
2 P P
of a pure ideal gas at T and P is
m m° 1T2 RT ln 1P>P°2 pure ideal gas, P° 1 bar (6.2)
Figure 6.1 plots m m°versus P at fixed T for a pure ideal gas. For a pure ideal
gas, m G H TS , and H is independent of pressure [Eq. (2.70)], so the
m m m m
pressure dependence of m in Fig. 6.1 is due to the change of S with P. In the zero-
m
pressure, infinite-volume limit, the entropy of an ideal gas becomes infinite, and m Figure 6.1
goes to q.
Variation of the chemical potential
m of a pure ideal gas with pressure
Chemical Potentials in an Ideal Gas Mixture
at constant temperature. m° is the
To find the chemical potentials in an ideal gas mixture, we give a fuller definition of standard-state chemical potential,
an ideal gas mixture than we previously gave. An ideal gas mixture is a gas mixture corresponding to P P° 1 bar.