Questions: Solution Thermodynamics: Partial Molar Quantities and Activity

Question 1

A chemist measures the vapor pressure of component A above a binary liquid mixture and finds it is 35% higher than Raoult's law predicts (P_A > x_A · P_A*). Which statement correctly interprets this result?

• A. The mole fraction x_A was calculated incorrectly; higher vapor pressure simply means more A is present
• B. The activity coefficient γ_A > 1, indicating A-B molecular interactions are less favorable than A-A interactions, so A molecules escape the liquid more readily
• C. The chemical potential of A in the mixture is lower than in pure liquid A, consistent with strong A-B attractions
• D. This measurement is impossible; Raoult's law holds exactly for all binary mixtures

Answer: (B)

A vapor pressure above the Raoult's law prediction is a positive deviation, quantified by activity coefficient γ_A > 1 (since P_A = γ_A x_A P_A*). It means A-B interactions are weaker than the average of A-A and B-B interactions — A molecules 'want to escape' the solution more than they would in an ideal mixture. Option C describes the opposite situation (negative deviation, γ < 1), which would result from unusually strong A-B attractions. This misconception — thinking stronger interactions always raise vapor pressure — is common.

Question 2

For an ideal solution formed by mixing two liquids at constant T and P, which thermodynamic statement is always true?

• A. The enthalpy of mixing is negative because forming unlike-molecule pairs releases energy
• B. The entropy of mixing is zero because the molecules are indistinguishable in an ideal solution
• C. The Gibbs energy of mixing is negative, driven entirely by the entropy of mixing since ΔmixH = 0
• D. The volume of the mixture is greater than the sum of pure component volumes

Answer: (C)

In an ideal solution, by definition ΔmixH = 0 (unlike-molecule interactions are identical to like-molecule interactions) and ΔmixV = 0. Therefore ΔmixG = −TΔmixS, which is always negative because mixing increases entropy (ΔmixS = −R Σ xi ln xi > 0 since ln xi < 0 for all xi < 1). Option A describes non-ideal behavior with negative deviations from Raoult's law. Option B reverses the truth — entropy of mixing is always positive (non-zero) even in ideal solutions.

Question 3

If the Gibbs energy of mixing (ΔmixG) is negative for two liquids, the solution should be behaving ideally.

• T. True
• F. False

Answer: False

All spontaneous mixing produces ΔmixG < 0 — this is simply the criterion for the mixing process to occur spontaneously. It says nothing about whether behavior is ideal or not. Ideal mixing additionally requires ΔmixH = 0 and ΔmixV = 0. A strongly non-ideal solution (e.g., acetone-chloroform with large negative excess Gibbs energy) still has ΔmixG < 0. The excess Gibbs energy G^E = ΔmixG − ΔmixG(ideal) is the quantity that captures deviation from ideality; G^E = 0 characterizes ideal solutions, not simply ΔmixG < 0.

Question 4

An activity coefficient γ_i = 0.6 means that component i behaves thermodynamically as if its concentration in the mixture is less than its actual mole fraction.

• T. True
• F. False

Answer: True

Activity is defined as a_i = γ_i · x_i, so γ_i = 0.6 means a_i = 0.6 x_i — the component's 'effective thermodynamic concentration' is 40% lower than its actual mole fraction. This represents a negative deviation from Raoult's law: A-B interactions are stronger than average, so molecules are less prone to escape into the vapor phase. Chemical potential, vapor pressure, and all thermodynamic equilibria depend on activity, not mole fraction, so in real solutions γ ≠ 1 changes equilibrium predictions significantly.

Question 5

Explain why activity coefficients can deviate substantially from 1 in real solutions, and describe the molecular-level interpretation of γ > 1 versus γ < 1.

Model answer: Activity coefficients deviate from 1 when unlike-molecule (A-B) interactions differ from the average of like-molecule (A-A and B-B) interactions. In an ideal solution, all molecular interactions are equivalent, so the thermodynamic tendency to escape the solution depends only on mole fraction. In real solutions, if A-B interactions are weaker than the A-A/B-B average (e.g., mixing polar and nonpolar molecules), A molecules escape more readily than predicted — giving γ_A > 1 (positive deviation from Raoult's law). If A-B interactions are stronger (e.g., hydrogen bonding between unlike species), molecules are held in solution more strongly than in the pure liquid, giving γ_A < 1 (negative deviation). The magnitude of γ reflects how different the real mixture is from the ideal case.

This molecular-level picture connects directly to excess enthalpy: positive deviations correspond to ΔmixH > 0 (energy cost to mixing), while negative deviations correspond to ΔmixH < 0 (energy released on mixing). The excess Gibbs energy G^E parameterizes this departure and is the foundation of engineering models like Margules and Wilson equations used in distillation design.