# 17.6: The Gibbs-Duhem Relation

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In a mixture of several components kept at constant temperature and pressure, the chemical potential µ_{i} of a particular component (which, under conditions of constant T and *P*, is also its partial molar Gibbs function, *g _{i}*) 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 *N*_{1}, *N*_{2}, *N*_{3}, 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 µ_{i} is just the partial molar Gibbs function, *g*_{i}, 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.