488
9 Chemical Equilibrium
with c being the concentration in moles per unit volume (normally cubic decimetres,
dm 3 ). The equilibrium constant K P (T ) for reaction (9.2.2) becomes
K P ≡
P
ν C
C P
ν D
D
P
ν A
A P
ν B
B
=
c
ν C
C c
ν D
D
c
ν A
A c
ν B
B
RT
P ◦
ν C +ν D −ν A −ν B
.
(9.2.16)
In this expression, the concentrations must be related to some standard concentration
c ◦ (which plays the same role for concentrations that P ◦ plays for pressures). Thus,
K P (T ) will be related to K c (T ) by
K P (T ) = K c (T )
c ◦ RT
P ◦
ν C +ν D −ν A −ν B
,
(9.2.17a)
with K c (T ) given as
K c (T ) =
(c C /c ◦ ) ν C (c D /c ◦ ) ν D
(c A /c ◦ ) ν A c B /c ◦ ) ν B
.
(9.2.17b)
Both K P and K c are unitless: the choice for P ◦ or c ◦ determines which units are
to be associated with the universal gas constant R. For example, P ◦ is typically
chosen to be 1 bar, while the most common choice for c ◦ is 1 mol dm −3 (1 dm
3
≡
1 litre): for these choices of P ◦ and c ◦ , the factor c ◦ RT /P ◦ becomes RT in unit
dm
3 bar mol
−1 , so that R must be given as R 0.083145 dm
3 bar mol
−1 K −1 . It is
not uncommon to find the relation (9.2.17a) expressed as
K P (T ) = K c (T )(RT )
ν C +ν D −ν A −ν B
(9.2.17c)
with it being implicitly understood that c ◦ = 1 dm
3 mol
−1 and P ◦ = 1 bar. For an
ideal gas, the volume dependence of a molecular partition function is contained
entirely in the translational component, so that z(T , V )/V is independent of volume
and depends only upon temperature.
At constant T and V , the total differential for the Helmholtz energy may be
written as
dA = μ A dn A + μ B dn B + μ C dn C + μ D dn D .
From the perspective of the extent of reaction, in which the numbers of moles of
reactants and products can be written as n A = n
(0)
A − ν A ξ , n B = n
(0)
B − ν B ξ ,
n C = n
(0)
C + ν C ξ , and n D = n
(0)
D + ν D ξ , we find that for chemical equilibrium, for
which dA = 0, we have
dA = [ν C μ C + ν D μ D − ν A μ A − ν B μ B ]dξ = 0 ,
or
ν C μ C + ν D μ D − ν A μ A − ν B μ B = 0 .
(9.2.18)
9 Chemical Equilibrium
with c being the concentration in moles per unit volume (normally cubic decimetres,
dm 3 ). The equilibrium constant K P (T ) for reaction (9.2.2) becomes
K P ≡
P
ν C
C P
ν D
D
P
ν A
A P
ν B
B
=
c
ν C
C c
ν D
D
c
ν A
A c
ν B
B
RT
P ◦
ν C +ν D −ν A −ν B
.
(9.2.16)
In this expression, the concentrations must be related to some standard concentration
c ◦ (which plays the same role for concentrations that P ◦ plays for pressures). Thus,
K P (T ) will be related to K c (T ) by
K P (T ) = K c (T )
c ◦ RT
P ◦
ν C +ν D −ν A −ν B
,
(9.2.17a)
with K c (T ) given as
K c (T ) =
(c C /c ◦ ) ν C (c D /c ◦ ) ν D
(c A /c ◦ ) ν A c B /c ◦ ) ν B
.
(9.2.17b)
Both K P and K c are unitless: the choice for P ◦ or c ◦ determines which units are
to be associated with the universal gas constant R. For example, P ◦ is typically
chosen to be 1 bar, while the most common choice for c ◦ is 1 mol dm −3 (1 dm
3
≡
1 litre): for these choices of P ◦ and c ◦ , the factor c ◦ RT /P ◦ becomes RT in unit
dm
3 bar mol
−1 , so that R must be given as R 0.083145 dm
3 bar mol
−1 K −1 . It is
not uncommon to find the relation (9.2.17a) expressed as
K P (T ) = K c (T )(RT )
ν C +ν D −ν A −ν B
(9.2.17c)
with it being implicitly understood that c ◦ = 1 dm
3 mol
−1 and P ◦ = 1 bar. For an
ideal gas, the volume dependence of a molecular partition function is contained
entirely in the translational component, so that z(T , V )/V is independent of volume
and depends only upon temperature.
At constant T and V , the total differential for the Helmholtz energy may be
written as
dA = μ A dn A + μ B dn B + μ C dn C + μ D dn D .
From the perspective of the extent of reaction, in which the numbers of moles of
reactants and products can be written as n A = n
(0)
A − ν A ξ , n B = n
(0)
B − ν B ξ ,
n C = n
(0)
C + ν C ξ , and n D = n
(0)
D + ν D ξ , we find that for chemical equilibrium, for
which dA = 0, we have
dA = [ν C μ C + ν D μ D − ν A μ A − ν B μ B ]dξ = 0 ,
or
ν C μ C + ν D μ D − ν A μ A − ν B μ B = 0 .
(9.2.18)