About Entropy of Mixing
When two ideal gases are separated by a partition and the partition is removed, they mix irreversibly. The thermodynamic driving force is purely entropic: the number of accessible microstates increases enormously when the molecules can explore the entire volume. For n moles of an ideal binary mixture, the entropy of mixing is ΔS_mix = −nR(x_A·ln x_A + x_B·ln x_B), where x_A and x_B = 1 − x_A are the mole fractions.
Because ln(x) < 0 for all 0 < x < 1, ΔS_mix is always positive. The Gibbs free energy of mixing ΔG_mix = ΔH_mix − TΔS_mix = RT(x_A·ln x_A + x_B·ln x_B) is always negative for ideal gases (ΔH_mix = 0), confirming that mixing is spontaneous at any temperature. For non-ideal mixtures, the Margules model adds an enthalpic term ΔH_mix = ε·x_A·x_B, which can cause phase separation when the repulsive interaction ε is large enough.
This simulation shows 200 particles (red = gas A, blue = gas B) undergoing random walks in a 2D box. Initially gas A is confined to the left half and gas B to the right. When the partition is removed, particles diffuse freely and mix. The right panel shows thermodynamic plots: ΔS_mix vs composition (top) and ΔG_mix vs composition at the selected temperature (bottom).
Frequently Asked Questions
What is the entropy of mixing?
The entropy of mixing ΔS_mix quantifies the increase in disorder when two substances mix. For ideal gases, ΔS_mix = -nR(x_A·ln x_A + x_B·ln x_B) where x_A and x_B are mole fractions. Since ln(x) < 0 for 0 < x < 1, ΔS_mix is always positive — mixing always increases entropy.
Why does mixing always happen spontaneously for ideal gases?
The Gibbs free energy of mixing for ideal gases is ΔG_mix = RT(x_A·ln x_A + x_B·ln x_B) < 0 at all compositions (since ln x < 0 for 0 < x < 1). Because ΔG < 0, mixing is thermodynamically spontaneous. There is no enthalpy change (ΔH_mix = 0) for ideal gases — the driving force is purely entropic.
What is the Gibbs paradox?
The Gibbs paradox asks: if we mix two samples of the same gas, does entropy increase? By the mixing formula, ΔS_mix = -nR(0.5·ln0.5 + 0.5·ln0.5) > 0 would suggest it does. But this contradicts our expectation that mixing identical gases changes nothing observable. The resolution is quantum statistics: identical particles are indistinguishable, so the entropy formula doesn't apply — ΔS = 0 for mixing identical gases.
What makes a gas mixture ideal?
An ideal gas mixture has no interactions between molecules of different species — the only thermodynamic effect of mixing is configurational entropy. In practice, noble gas mixtures (He+Ar, Ne+Kr) are nearly ideal. Real solutions with strong interactions between unlike molecules (alcohol+water, HCl+water) show large non-ideal ΔH_mix and deviate significantly from the ideal mixing formula.
What is the Margules parameter ε?
The Margules model describes non-ideal mixing with a single interaction parameter: ΔH_mix = ε·x_A·x_B. If ε > 0 (endothermic mixing, unfavorable), the two species repel each other and may phase-separate. If ε < 0 (exothermic, favorable), they attract and mix more readily. For |ε| > 2RT, the ΔG_mix curve develops double minima and the mixture phase-separates.
What is the relationship between ΔG and phase equilibrium?
A mixture is thermodynamically stable when the ΔG_mix vs composition curve is concave up (positive second derivative d²G/dx²). When the curve has a concave-down region, the system lowers its free energy by separating into two phases with compositions at the common tangent — the binodal. The inflection points (d²G/dx²=0) define the spinodal, where the mixture is mechanically unstable.
How is configurational entropy derived statistically?
The Boltzmann entropy S = k_B·ln(Ω) counts the number of microstates Ω. Distributing N_A molecules of A and N_B of B among N = N_A + N_B sites gives Ω = N!/(N_A!·N_B!). Taking the logarithm and using Stirling's approximation ln(N!) ≈ N·ln(N) - N gives exactly ΔS_mix = -nR(x_A·ln x_A + x_B·ln x_B) — entropy is the log of the number of ways to arrange the molecules.
Why does ΔS_mix peak at x_A = 0.5?
The mixing entropy -R(x·ln x + (1-x)·ln(1-x)) is maximum at x = 0.5 by symmetry and can be verified by differentiation: d/dx[-x·ln x - (1-x)·ln(1-x)] = -ln x + ln(1-x) = 0 when x = 0.5. At equimolar composition, there are the most possible arrangements of the two species, giving maximum configurational disorder.
What is the difference between entropy and enthalpy of mixing?
Entropy of mixing ΔS_mix is always positive for ideal gases and solutions — it counts the increase in disorder. Enthalpy of mixing ΔH_mix measures the heat released or absorbed — zero for ideal mixtures, negative (exothermic) when unlike molecules attract, positive (endothermic) when they repel. The Gibbs free energy ΔG = ΔH - TΔS: at high T the entropic term dominates; at low T the enthalpic term can cause phase separation.
How is entropy of mixing used in materials science?
In alloy design, entropy of mixing drives the formation of high-entropy alloys (HEA) — multi-component metal alloys with 5+ elements in near-equimolar proportions. The large configurational entropy ΔS_mix stabilizes a single-phase solid solution against decomposition into competing phases. The ΔG_mix = ΔH_mix - TΔS_mix criterion predicts which HEA compositions are stable single phases vs phase-separated mixtures.