The Thermodynamics of Mixing

Mixing is a fascinating phenomenon in thermodynamics, particularly when considering the Gibbs energy of a mixture at constant temperature and pressure. This energy, denoted as G, tends to decrease during spontaneous mixing processes, offering insights into the composition changes that occur.

Imagine two ideal gases, initially separated in different containers, with amounts n_A and n_B, respectively, both at a temperature T and pressure p. The chemical potentials of these gases have their 'pure' values, determined by the equation μ = G_m, where G_m(p) = G°_m + RT ln(p/p°).

Simplifying the notation by replacing p/p° with p, the total Gibbs energy of the separated gases is G_i. When these gases spontaneously mix, leading to partial pressures p_A and p_B, with p_A+p_B =p, the total Gibbs energy becomes G_f. The difference G_f − G_i represents the Gibbs energy of mixing, denoted as ΔG_mix.

To express ΔG_mix in terms of mole fractions (x_Jn), where n is the total amount of A and B and p_J/p = x_J for each component, gives the Gibbs energy of mixing for ideal gases. As mole fractions are never greater than 1, the logarithms in this equation are negative, confirming that ΔG_mix<0. This implies that ideal gases mix spontaneously in all proportions.

Further exploration of the system reveals that (∂G/∂T)_p = −S. It follows that, for a mixture of ideal gases at the same pressure, the entropy of mixing ( ΔS_mix) is −(∂ ΔG_mix/∂T)_p. Since lnx<0, ΔS_mix>0 for all compositions. The entropy of mixing increases for ideal gases at constant temperature and pressure, affirming their spontaneous mixing in all proportions.

Under constant pressure and temperature conditions, the enthalpy of mixing ( ΔH_mix) can be calculated from ΔG = ΔH−TΔS. Notably, the enthalpy of mixing is zero, indicating no interactions between the molecules in the gaseous mixture. The driving force for mixing arises from the increase in system entropy, as the entropy of the surroundings remains unchanged.

In summary, the thermodynamics of mixing, as demonstrated through the Gibbs energy, entropy, and enthalpy changes, elucidates the spontaneous nature of ideal gas mixing in various proportions.