The Thermodynamics of Mixing

The thermodynamics of mixing describes the decrease in Gibbs energy during spontaneous mixing of ideal gases at constant temperature and pressure.

Consider two ideal gases with amounts n_A and n_B at the same temperature and pressure. Their chemical potentials follow from the molar Gibbs energy of an ideal gas.

The total Gibbs energy changes from G_i before mixing to G_f after mixing, with partial pressures p_A and p_B.

The Gibbs energy of mixing is obtained from G_f minus G_i and simplified using the relation between partial pressure and mole fraction of any gas J.

For an ideal gas, the partial derivative of G with respect to T at constant p equals minus the entropy. Applying this to the Gibbs energy of mixing gives the entropy change of mixing.

Since mole fractions are less than one, the logarithmic terms are negative, so mixing is spontaneous, with negative Gibbs energy of mixing and positive entropy of mixing.

From ΔG = ΔH − TΔS, the enthalpy of mixing is zero. Because unlike and like interactions are similar, mixing does not change enthalpy.