Gibbs-Konovalov theorem

We shall now look at a theorem developed by Gibbs and Konovalov. For the moment, we shall simply accept this theorem as true. We shall demonstrate it later when we examine equilibria between phases.  

Theorem 2.4.- In a system with two external intensive variables - pressure and temperature - at equilibrium, for any shift at constant temperature (or respectively at constant pressure), the pressure (or respectively the temperature) reaches an extremum for an indifferent state, and vice versa. If amongst the values of the pressure (or respectively the temperature) which keep the system at equilibrium at constant temperature (or respectively constant pressure), there is an extremum value, then the state corresponding to that value is indifferent. 

Note 2.3 - Obviously, this theorem only makes sense if the Gibbs variance is at least equal to 2. 

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GIBBS-KONOVALOV THEOREMS. Consider a binary system containing two phases (e.g.. liquid and vapor). Both components can pass from one phase lo another. The Gibbs Konovalov theorems refer to the properties of the phase diagrams of such systems (see also Azeotropic System). The lirst theorem is At constant pressure, the temperature of coexistence passes through tin extreme value (maximum, minimum or inflexion with a horizontal value), if the comfutsirlon of the two phases is the same. Conversely, al a point at winch the temperature passes through an extreme value, the phases have the same composition. The second theorem is similar. It refers lo the coexistence pressure at constant temperature. 

We note here that the value of this derivative is zero when the compositions of the two phases are equal in accordance with one of the Gibbs-Konovalov theorems. The second derivative which we choose to evaluate is (dT/dP)x.r We choose to work again with in order to introduce x[ as a variable. Therefore, we first eliminate d/q from Equation (10.124) and (10.125) to obtain... 

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