Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 2.2-8

THE CONDITIONS OF EQUILIBRIUM

Physics — Statistical Mechanics · multicomponent-energy-differential

Problem

Show that, in analogy with equation 2.25, for a system with rr components du=T ds−P dv+∑j=1r−1(μj−μr) dxjdu = T\,ds - P\,dv + \sum_{j=1}^{r-1}(\mu_j - \mu_r)\,dx_j where the xjx_j are the mole fractions (=Nj/N= N_j/N).
Intuitive explanation

The key insight is that chemical potentials μj\mu_j naturally appear when composition changes, but since mole fractions must sum to 1, only r−1r-1 of them are independent—the last one is determined by the constraint. By expressing the Gibbs free energy in terms of mole fractions instead of absolute mole numbers, you'll see how the differential picks up terms like (μj−μr)(\mu_j - \mu_r) rather than individual μj\mu_j values, reflecting that what matters physically is the relative chemical potential difference between components.

Intuitive explanation

The key insight is that chemical potentials μj\mu_j naturally appear when composition changes, but since mole fractions must sum to 1, only r−1r-1 of them are independent—the last one is determined by the constraint. By expressing the Gibbs free energy in terms of mole fractions instead of absolute mole numbers, you'll see how the differential picks up terms like (μj−μr)(\mu_j - \mu_r) rather than individual μj\mu_j values, reflecting that what matters physically is the relative chemical potential difference between components.

Related theory

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