The Thermodynamics of Mixing

Ideal gases mix spontaneously because the Gibbs energy of mixing is negative at constant temperature and pressure. Since mole fractions are always less than one, the logarithmic terms in the Gibbs energy equation become negative, driving spontaneous mixing in all proportions. This occurs regardless of the initial amounts of each gas.

The entropy of mixing is positive for ideal gases at constant temperature and pressure. Since mole fractions are less than one, the logarithmic terms are negative, making the entropy change positive. This increase in entropy is the driving force for spontaneous mixing, as the enthalpy of mixing remains zero.

The Gibbs energy of mixing depends on the partial pressures of each gas component. Using the relationship between partial pressure and mole fraction, the Gibbs energy change can be expressed as ΔGmix = RT(nA ln xA + nB ln xB). Since mole fractions are less than one, this yields negative Gibbs energy, confirming spontaneous mixing occurs.

The enthalpy of mixing is zero because unlike and like molecular interactions are similar in ideal gases. From ΔG = ΔH − TΔS, with negative ΔG and positive ΔS, the enthalpy term must equal zero. This means no energy is absorbed or released during mixing; entropy increase alone drives the spontaneous process.

Before mixing, the total Gibbs energy equals the sum of chemical potentials of separated gases at their pure pressures. After mixing, each gas has a lower chemical potential due to reduced partial pressure. The difference between final and initial Gibbs energy gives the Gibbs energy of mixing, which is negative for ideal gases.

Temperature affects the entropy contribution to Gibbs energy through ΔG = ΔH − TΔS. Since enthalpy of mixing is zero and entropy of mixing is positive, the TΔS term is always positive. This makes ΔG negative at any temperature, ensuring ideal gases mix spontaneously regardless of temperature changes at constant pressure.

Mole fractions replace partial pressures in the Gibbs energy equation using the relationship pJ/p = xJ for each component. This simplification yields ΔGmix = RT(nA ln xA + nB ln xB), where logarithms of mole fractions are negative. This expression reveals why thermodynamic properties of ideal solutions show spontaneous mixing in all proportions.