Stability with regard to infinitesimal fluctuations

In general, the first derivative of the Gibbs energy is sufficient to determine the conditions of equilibrium. To examine the stability of a chemical equilibrium, such as the one described above, higher order derivatives of G are needed. We will see in the following that the Gibbs energy versus the potential variable must be upwards convex for a stable equilibrium. Unstable equilibria, on the other hand, are 

Let us assume the existence of a Taylor series for the Gibbs energy at the equilibrium point. This implies that the Gibbs energy and all its derivatives vary continuously at this point. The Taylor series is given as 

The equilibrium is unstable if this second derivative is negative. If 

Separates a stable region from an unstable region. 

If (a 3G / a 3 ) =0 0, it is possible to choose the sign of so that (-Ak s negative. Hence the equilibrium is unstable since small compositional fluctuations can have any sign. The stability criteria are summarized in Table 5.1. 

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