2.7 Extension to Multicomponent Systems
91
and called the (chemical) affinity. From Eq. (2.7.10b), it is clear that the affinity is
also related to the internal energy via
A(S, V , ξ) = −
∂U
∂ξ
S,V
,
and represents the isentropic-isochoric rate of change of U(S, V , ξ) with respect to
the degree of advancement of the chemical reaction.
Based upon expression (2.7.10b) for dU and the Legendre transforms H , A, and
G of U given by Eqs. (2.3.2), (2.3.5a), (2.3.6c), the total differentials dH , dA, and
dG, are related to the affinity via
dH = T dS + V dP − Adξ ,
(2.7.12a)
dA = −SdT − P dV − Adξ ,
(2.7.12b)
dG = −SdT + V dP − Adξ ,
(2.7.12c)
respectively, so that A is related to H , A, and G by
A = −
∂H
∂ξ
S,P
= −
∂A
∂ξ
T ,V
= −
∂G
∂ξ
T ,P
(2.7.13)
in terms of the relevant partial derivatives of H , A, and G.
As chemical reactions are most often carried out under conditions of constant
temperature and total pressure, Eq. (2.7.12c) will be the relevant expression for
discussing the thermodynamics associated with (inherently irreversible) chemical
reactions, such as that represented by Eq. (2.7.7). Hence, for reaction (2.7.7)
occurring at constant temperature, T , and total pressure, P , the affinity A will be
parameterized by the chosen values of T and P , and will be a function of the
extent of reaction, ξ . The change in the Gibbs energy, ((G) react ≡ ((G) T ,P , is
traditionally given as
((G) T ,P = ν C μ C (T , P C ) + ν D μ D (T , P D ) − ν A μ A (T , P A ) − ν B μ B (T , P B )
or, equivalently, in terms of the de Donder affinity as
((G) T ,P = −A(T , P ; ξ) ,
(2.7.14)
since the partial pressures P α are parameterized by P and are functions of ξ .
If reaction (2.7.7) is carried out at constant T , P , and starts out with n A (0) =
ν A 1 m , with 1 m representing one mole, n B (0) = ν B , n C (0) = n D (0) = 0, we may
write the numbers of moles of the chemical species present in the reaction mixture in
terms of the extent of reaction ξ as n A = ν A (1 m − ξ), n B = ν B (1 m − ξ), n C = ν C ξ ,
n D = ν D ξ , so that the mole fractions, x α , of the four participating species are given
as x α = n α (ξ )/n tot (ξ ) or, more explicitly, as
91
and called the (chemical) affinity. From Eq. (2.7.10b), it is clear that the affinity is
also related to the internal energy via
A(S, V , ξ) = −
∂U
∂ξ
S,V
,
and represents the isentropic-isochoric rate of change of U(S, V , ξ) with respect to
the degree of advancement of the chemical reaction.
Based upon expression (2.7.10b) for dU and the Legendre transforms H , A, and
G of U given by Eqs. (2.3.2), (2.3.5a), (2.3.6c), the total differentials dH , dA, and
dG, are related to the affinity via
dH = T dS + V dP − Adξ ,
(2.7.12a)
dA = −SdT − P dV − Adξ ,
(2.7.12b)
dG = −SdT + V dP − Adξ ,
(2.7.12c)
respectively, so that A is related to H , A, and G by
A = −
∂H
∂ξ
S,P
= −
∂A
∂ξ
T ,V
= −
∂G
∂ξ
T ,P
(2.7.13)
in terms of the relevant partial derivatives of H , A, and G.
As chemical reactions are most often carried out under conditions of constant
temperature and total pressure, Eq. (2.7.12c) will be the relevant expression for
discussing the thermodynamics associated with (inherently irreversible) chemical
reactions, such as that represented by Eq. (2.7.7). Hence, for reaction (2.7.7)
occurring at constant temperature, T , and total pressure, P , the affinity A will be
parameterized by the chosen values of T and P , and will be a function of the
extent of reaction, ξ . The change in the Gibbs energy, ((G) react ≡ ((G) T ,P , is
traditionally given as
((G) T ,P = ν C μ C (T , P C ) + ν D μ D (T , P D ) − ν A μ A (T , P A ) − ν B μ B (T , P B )
or, equivalently, in terms of the de Donder affinity as
((G) T ,P = −A(T , P ; ξ) ,
(2.7.14)
since the partial pressures P α are parameterized by P and are functions of ξ .
If reaction (2.7.7) is carried out at constant T , P , and starts out with n A (0) =
ν A 1 m , with 1 m representing one mole, n B (0) = ν B , n C (0) = n D (0) = 0, we may
write the numbers of moles of the chemical species present in the reaction mixture in
terms of the extent of reaction ξ as n A = ν A (1 m − ξ), n B = ν B (1 m − ξ), n C = ν C ξ ,
n D = ν D ξ , so that the mole fractions, x α , of the four participating species are given
as x α = n α (ξ )/n tot (ξ ) or, more explicitly, as