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where S t is the total entropy of the system Combination of these expressions yields, upon rearrangement:
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(1461)
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Application of the first law provides:
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Combining this equation with Eq (1461) gives:
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Since this relation involves properties only, it must be satisfied for changes in state of any closed system of uniform T and P, without restriction to the conditions of mechanical and thermal reversibility assumed in its derivation The inequality applies to every incremental change of the system between nonequilibrium states, and it dictates the direction of change that
CHAPTER 14 Topics in Phase Equilibria
leads toward equilibriumThe equality holds for changes between equilibriumstates (reversible processes) Thus Eq (61) is just a special case of Eq (1462) Equation (1462) is so general that application to practical problems is difficult; restricted versions are much more useful For example, by inspection:
where the subscripts specify properties held constant Similarly, for processes that occur at constant Ut and Vt,
An isolated system is necessarily constrained to constant internal energy and volume, and for such a system it follows directly from the second law that the last equation is valid If a process is restricted to occur at constant T and P, then Eq (1462) may be written:
d(Ut
+ PV'
TSt)T,p5 0
From the definition of the Gibbs energy [Eq (63)],
Therefore,
(1463)
Of the possible specializationsof Eq (1462), this is the most useful, because T and P , which are easily measured and controlled, are more logical as constants than are other pairs of variables, such as Ut and V' Equation (1463) indicates that all irreversible processes occurring at constant T and P proceed in such a direction as to cause a decrease in the Gibbs energy of the system Therefore:
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:
(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
143 Equilibrium and Stability
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
from which xiGi < 0 (const T, P)
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