Questions: Partial Molar Properties and Solution Thermodynamics

Question 1

One liter of ethanol is mixed with one liter of water at constant temperature and pressure. The resulting volume is approximately 1.93 liters, not 2 liters. Which concept directly explains this observation?

• A. The density of water increases when ethanol is added, compressing the water into a smaller volume
• B. The partial molar volumes of ethanol and water in the mixture differ from their respective pure-component molar volumes
• C. Conservation of mass is violated when polar and nonpolar molecules mix
• D. The activity coefficients are greater than 1, indicating positive deviations from Raoult's law

Answer: (B)

The partial molar volume V̄ᵢ is the actual effective volume contribution of component i in the context of the mixture — not the molar volume of the pure substance. When ethanol fits into the hydrogen-bonding network of water, the ethanol molecules occupy less effective space than they would in pure ethanol, and vice versa. This is precisely why partial molar properties exist: pure-component molar volumes are not additive in real solutions. Activity coefficients (option D) describe chemical potential deviations, not volume deviations directly.

Question 2

At equilibrium between a vapor phase and a liquid phase in a two-component system, the chemical potential of component i must satisfy which condition?

• A. μᵢ(liquid) > μᵢ(vapor), so that molecules are driven from liquid to vapor
• B. μᵢ(liquid) = μᵢ(vapor) for each component, since no net transfer occurs at equilibrium
• C. The chemical potential of each component equals zero in both phases
• D. μᵢ depends only on temperature and is equal across phases only when T is uniform

Answer: (B)

The fundamental criterion for phase equilibrium is equal chemical potential across all coexisting phases: μᵢ(α) = μᵢ(β). Chemical potential is the partial molar Gibbs free energy, and systems at constant T and P minimize total G. If μᵢ is higher in one phase, molecules will spontaneously transfer to the lower-μ phase until equality is reached. This is the driving force behind distillation, extraction, and crystallization — not concentration gradients per se, but chemical potential gradients. Concentration equalization (option A's implication) and zero values (option C) are wrong; temperature alone (option D) doesn't determine μᵢ in mixtures.

Question 3

The partial molar volume of a component in a mixture equals the molar volume of that component in its pure state.

• T. True
• F. False

Answer: False

This is only true for ideal solutions. In general, V̄ᵢ = (∂V/∂nᵢ)_{T,P,nⱼ} — the rate of change of total volume when adding a differential amount of i to the mixture. In real solutions, the intermolecular interactions between unlike molecules change how each component's molecules pack together. Ethanol's partial molar volume in water (~55 mL/mol in water-rich mixtures) differs significantly from pure ethanol's molar volume (~58.4 mL/mol) precisely because it sits in a water hydrogen-bonding network rather than an ethanol environment.

Question 4

For an ideal solution, mixing two components produces no change in total volume and no change in enthalpy.

• T. True
• F. False

Answer: True

The defining properties of an ideal solution are ΔV_mix = 0 (no volume change) and ΔH_mix = 0 (no enthalpy change on mixing). This occurs when all molecular interactions — A–A, B–B, and A–B — are essentially identical, so molecules experience the same environment whether surrounded by like or unlike molecules (e.g., benzene-toluene mixtures). Note that the Gibbs energy of mixing is still negative (ΔG_mix < 0) due to the entropy of mixing — ideal solutions mix spontaneously even with no enthalpy driving force.

Question 5

Why does the chemical potential of a component, rather than its molar concentration, determine when two phases are in equilibrium with each other?

Model answer: Chemical potential μᵢ = (∂G/∂nᵢ)_{T,P} measures the tendency of component i to leave a phase — it incorporates both concentration and all intermolecular interaction effects (captured by the activity coefficient). Two phases can have equal concentrations of a component yet still not be in equilibrium if the molecular environments differ (e.g., a solute that strongly prefers one solvent over another). Equilibrium requires that the escaping tendency be equal in both phases; since chemical potential is the thermodynamic measure of escaping tendency, equality of μᵢ across phases is the correct criterion, not equality of concentrations.

Concentration-based criteria fail because the 'desirability' of a phase for a given molecule depends on molecular interactions, not just how crowded it is. A molecule surrounded by favorable neighbors (low activity coefficient, γ < 1) has a lower chemical potential and less tendency to escape than the same concentration in an unfavorable environment (γ > 1). The activity coefficient γᵢ captures this interaction effect, entering as μᵢ = μᵢ° + RT ln(γᵢxᵢ). Equal chemical potential ensures that no molecule has a net incentive to change phases — the true thermodynamic equilibrium condition.