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# 17.6: The Gibbs-Duhem Relation

In a mixture of several components kept at constant temperature and pressure, the chemical potential µi of a particular component (which, under conditions of constant T and P, is also its partial molar Gibbs function, gi) depends on how many moles of each species i are present. The Gibbs-Duhem relation tells us how the chemical potentials of the various components vary with composition. Thus:

We have seen that, if we keep the pressure and temperature constant, and we increase the number of moles of the components by N1, N2, N3, the increase in the Gibbs function is

$d G=\sum \mu_{i} d N_{i}.$

We also pointed out in section 17.5 that, provided the temperature and pressure are constant, the chemical potential µi is just the partial molar Gibbs function, gi, so that the total Gibbs function is

$G=\sum g_{i} N_{i}=\sum \mu_{i} N_{i},$

the sum being taken over all components. On differentiation of equation 17.7.2 we obtain

$d G=\sum \mu_{i} d N_{i}+\sum N_{i} d \mu_{i}.$

Thus for any process that takes place at constant temperature and pressure, comparison of equations 17.6.1 and 17.6.3 shows that

$\sum N_{i} d \mu_{i}=0,$

which is the Gibbs-Duhem relation. It tells you how the chemical potentials change with the chemical composition of a phase.