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 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 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.