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The equilibrium state of a closed system is that state for which the total Gibbs energy is a minimum with respect to all possible changes at the given T and P This criterion of equilibrium provides a general method for determinationof equilibrium states One writes an expression for Gt as a function of the numbers of moles (mole numbers) of the species in the several phases, and then finds the set of values for the mole numbers that minimizes G" subject to the constraints of mass conservation This procedure can be applied to problems of phase, chemical-reaction, or combined phase and chemical-reaction equilibrium; it is most useful for complex equilibrium problems, and is illustrated for chemical-reaction equilibrium in Sec 139 At the equilibrium state differential variations can occur in the system at constant T and P without producing any change in Gt This is the meaning of the equality in Eq (1463) Thus another form of this criterion of equilibrium is:

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(1464)

To apply this equation, one develops an expression for dGt as a function of the mole numbers of the species in the various phases, and sets it equal to zero The resulting equation along with those representing the conservation of mass provide working equations for the solution

1 0 0 + 1 2

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of equilibrium problems Equation (1464) leads directly to Eq (116) for phase equilibrium and it is applied to chemical-reaction equilibrium in Sec 133 Equation (1463) provides a criterion that must be satisfied by any single phase that is stable with respect to the alternative that it split into two phases It requires that the Gibbs energy of an equilibrium state be the minimum value with respect to all possible changes at the given T and P Thus, eg, when mixing of two liquids occurs at constant T and P , the total Gibbs energy must decrease, because the mixed state must be the one of lower Gibbs energy with respect to the unmixed state As a result: Gt r nG < x n i G i

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from which xiGi < 0 (const T, P)

According to the definition of Eq (1229), the quantity on the left is the Gibbs energy change of mixing Therefore,

1 0 0 1 2

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Thus, as noted in Sec 123, the Gibbs-energy change of mixing must always be negative, and a plot of AG vs xl for a binary system must appear as shown by one of the curves of Fig 1411 With respect to curve 1 , however, there is a further consideration If, when mixing occurs, a 1 system can achieve a lower value of the Gibbs energy by forming two phases than by forming a single phase, then the system splits into two phases This is in fact the situation represented between points a and B on curve I1 of Fig 1411, because the straight dashed line connecting points a and B represents the AG that would obtain for the range of states consisting of two phases of compositions x;Y and xf in various proportions Thus the solid curve shown between points a and p cannot represent stable phases with respect to phase splitting The equilibrium states between a and ,f consist of two phases These considerations lead to the following criterion of stability for a single-phase binary system:

At constant temperature and pressure, A G and its first and second derivatives must be continuous functions of XI, and the second derivative must everywhere be positive

Thus, d 2 AG >0 dx: (const T, P )

W H AT I T TA K E S T O B E # 1

2 0 0 + 1 2

(const T, P )

This requirement has a number of consequencesEquation (1230),rearranged and written for a binary system, becomes:

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