Page 77 - Chemical equilibria Volume 4

and:
∂ 2 Γ Properties of States of Physico-Chemical Equilibrium 53
> 0 [2.68b]
ξ ∂ 2
Thus, the curve Γ(ξ) exhibits a minimum.

NOTE 2.6.– It may be that the first derivative (and even some of the
subsequent derivatives) of the affinity in relation to the extent will have a
value of 0, in which case we must continue the expansion [2.64] until we
find the first nonzero derivative. We then show that the system cannot be
stable if the first nonzero derivative is of an even order. The stability
condition becomes:

∂ 2n+ 1 A S < 0 [2.69a]
∂ξ 2n+ 1

or:
∂ 2n+ 1 Γ > 0 [2.69b]
ξ ∂ 2n+ 1

with all the derivatives of order less than 2n+1 having a value of 0.

2.6.4. Conditions of bilateral stability expressed in terms of
chemical potentials

If we express the stability condition [2.67] using expressions [1.23]
and [1.42], we obtain:

1 ∂ μν ⎛ i ⎜ ∑∑ i − ν ⎞ k 2 nn < 0 [2.70]
2 i k ∂ n k ⎝ n i n k ⎠ ⎟ ik

This inequality thus leads to the condition:
∂ μ
i < 0 for i ≠ k [2.71]
n ∂ k

Thus, the condition of bilateral stability can be expressed very simply in
terms of chemical potentials.