Non-reactive Mixtures

What a non-reactive mixture means

The species remain chemically distinct and their amounts change only by mixing, separation or transport, not chemical reaction. Composition is an independent part of the thermodynamic state. An ideal-gas mixture is a useful model when intermolecular effects are small; it should not be applied automatically to dense gases or liquid mixtures.

Mole and mass fractions

For species i, mole fraction y_i = n_i/n_total and mass fraction w_i = m_i/m_total. Each set sums to one, but y_i and w_i are generally different.

Mean molar mass M_mix = sum(y_i M_i). w_i = y_i M_i/M_mix. y_i = (w_i/M_i)/sum(w_j/M_j).

Use compatible units for molar mass and the universal gas constant. The mass-specific gas constant is R_mix = R_u/M_mix.

Ideal-gas mixture relations

The equation of state is PV = n_total R_u T = m_total R_mix T, with absolute P and T. Dalton's relation gives partial pressure p_i = y_i P. Partial pressure is the pressure species i would exert if it alone occupied the mixture volume at the same temperature.

For an ideal-gas mixture, molar internal energy and enthalpy are composition-weighted sums: u_bar_mix = sum(y_i u_bar_i(T)) and h_bar_mix = sum(y_i h_bar_i(T)). On a mass basis, use mass fractions and mass-specific properties. Heat capacities depend on temperature and composition; gamma_mix is c_p,mix/c_v,mix, not generally a simple average of the species gammas.

Worked composition example

An ideal-gas mixture contains 0.7 mol N₂ and 0.3 mol O₂ at total pressure 200 kPa. For this exercise use rounded molar masses M_N2 = 0.028 kg/mol and M_O2 = 0.032 kg/mol, with R_u = 8.314 J/(mol K).

Total amount = 1 mol, so y_N2 = 0.7 and y_O2 = 0.3. M_mix = 0.7(0.028)+0.3(0.032) = 0.0292 kg/mol.

The nitrogen mass fraction is 0.7(0.028)/0.0292 = 0.67123; oxygen mass fraction is 0.32877. Partial pressures are 140 kPa and 60 kPa. The specific gas constant is 8.314/0.0292 = 284.73 J/(kg K).

The mixture is 70% nitrogen by mole but only about 67.1% by mass. Mixing up those bases changes calculated density and heat capacity.

Entropy of ideal-gas mixing

When initially separated different ideal gases at the same temperature and pressure mix to a final equilibrium mixture at that temperature and pressure:

ΔS_mix = -R_u sum(n_i ln y_i).

For the one-mole example, ΔS_mix = -8.314[0.7 ln(0.7)+0.3 ln(0.3)] = 5.079 J/K. Ideal-gas mixing at the same temperature has zero enthalpy of mixing, but positive entropy of mixing. This expression is for distinguishable species; removing a partition between identical gas samples at the same state does not create a composition-mixing entropy.

The entropy of a species in the ideal mixture uses its partial pressure. Summing pure-species entropies all evaluated at total pressure would miss the mixing contribution.

Beyond the ideal-gas model

Real mixtures may show nonzero enthalpy and volume changes on mixing. Liquid solutions may need activity coefficients, and real gases may need fugacities or a mixture equation of state. Mole fractions still satisfy their definitions, but Dalton's simple ideal-gas property model may no longer describe the actual thermodynamics accurately.

Quick check

For equal mole amounts of species with molar masses 20 and 40 g/mol, the mass fractions are 1/3 and 2/3. Their mole fractions are both 1/2. At fixed composition, use one consistent mass or mole basis throughout a property calculation.