Page 31 - Chemical equilibria Volume 4

We can generalize this expression for any characteristic function Γ ,
defined by: Physico-Chemical Transformations and Equilibria 7
q
Γ = U − TS − ∑ X Y where 2 q≤≤ p [1.17]
ii
i= 2
By differentiation of [1.17], we obtain:

q q
i ∑
dΓ = dU − T d S − S dT − ∑ X i dY − Y dX i [1.18]
i
i= 2 i= 2
Thus, when we consider relation [1.15]:
i ∑
d −
dΓ =− ST ∑ q X i dY + p Y dX − A dξ [1.19]
m
m
=
i= 2 m q+ 1
we obtain a new expression of the affinity which generalizes relation [1.16]:
⎛ ⎞ ∂Γ
A =− ⎜ ⎟ [1.20]
⎝ ∂ξ X ⎠ ml YT

In particular, for chemical systems with the variables pressure and
temperature, the characteristic function is the Gibbs energy G. We obtain:

⎛ G ⎞ ∂
A =− ⎜ ⎟ [1.21]
⎝ ∂ξ ⎠ , PT

At constant pressure and temperature, the affinity is the opposite of the
partial derivative of the Gibbs energy in relation to the extent.

NOTE 1.1.– Expression [1.15] shows that the affinity, which is an extensive
value, and the extent, which is an intensive value, are two conjugate values.

1.3.3. Affinity and chemical potentials

If we consider relation [1.3], we can write:

⎛ G ⎞ ∂ ⎛ G ⎞ ∂
⎜ ⎟ = ν i ⎜ ∑ ⎟ = ∑ ν μ i [1.22]
i
⎝ ξ ∂ ⎠ , PT i ⎝ n ∂ i ⎠ PT i
,, j n