Questions: Solution Thermodynamics and Activity Coefficient Models

Question 1

A mixture of benzene and ethanol shows positive deviation from Raoult's law. A student claims this means the vapor pressure of each component is lower than predicted by Raoult's law. What is wrong with this claim, and what does positive deviation actually imply about molecular interactions?

• A. The student is correct — positive deviation means lower vapor pressure than ideal
• B. Positive deviation means higher vapor pressure than Raoult's law predicts, because unlike interactions are weaker than like interactions, so molecules escape solution more easily
• C. Positive deviation means the Gini coefficients of the activity are greater than 1, which lowers vapor pressure
• D. Positive deviation occurs only when the mixture forms azeotropes, not due to interaction differences

Answer: (B)

Positive deviation means vapor pressures are *higher* than Raoult's law predicts, corresponding to γᵢ > 1. This occurs when the unlike molecule interactions (benzene-ethanol) are weaker than the like interactions (benzene-benzene, ethanol-ethanol). Molecules in this mixture 'prefer their own kind' and escape solution more readily than in an ideal mixture, raising the vapor pressure above the Raoult prediction. The student reversed the direction: a higher activity coefficient means the component acts as if it is more concentrated — it wants to escape.

Question 2

Regular Solution theory successfully predicts activity coefficients for benzene-cyclohexane mixtures but fails for ethanol-water. What fundamental assumption of Regular Solution theory explains this limitation?

• A. It assumes that all molecules have the same size and shape, which fails for mixtures with very different molecular geometries
• B. It assumes that excess entropy of mixing is zero, meaning all non-ideality comes from enthalpy differences — this fails for systems where specific interactions like hydrogen bonding create asymmetric entropic effects
• C. It assumes γᵢ = 1 for all components, which is valid only for ideal mixtures
• D. It uses quantum mechanical calculations that are only accurate for aromatic systems like benzene

Answer: (B)

Regular Solution theory assumes that the excess entropy of mixing (beyond the ideal entropy of mixing) is zero — all non-ideality enters through the enthalpy of mixing. This works for nonpolar molecules of similar size (like benzene-cyclohexane) where the mixing is nearly random and enthalpy differences are small. For systems with hydrogen bonding (like ethanol-water), molecules do not mix randomly — they preferentially cluster with energetically favorable neighbors, creating significant excess entropy. NRTL and UNIQUAC models address this by modeling non-random local compositions.

Question 3

An activity coefficient γᵢ < 1 for a component in solution means that component exerts a higher vapor pressure than Raoult's law predicts.

• T. True
• F. False

Answer: False

γᵢ < 1 corresponds to *negative* deviation from Raoult's law, meaning the component's vapor pressure is *lower* than the Raoult prediction. When γᵢ < 1, the component is stabilized in solution — unlike interactions are stronger than like interactions, so molecules are less inclined to escape. The chloroform-acetone system is the classic example: a weak hydrogen bond forms between components, stabilizing each in the mixture and reducing vapor pressure below the ideal value.

Question 4

Activity coefficients approach 1 as a solution becomes more dilute in a given component, reflecting ideal (Raoult's law) behavior at infinite dilution for the solvent.

• T. True
• F. False

Answer: True

By definition, Raoult's law holds for the solvent in the limit of high concentration (the solvent approaches its pure state). As a component's mole fraction approaches 1, its local environment consists mainly of like molecules, the unlike interactions become negligible, and γᵢ → 1. This is the Raoult's law limit. At the other extreme (infinite dilution), a component surrounded entirely by unlike molecules may have γᵢ very different from 1, described by Henry's law. Activity coefficient models must correctly capture both limits.

Question 5

Why does a real solution with γᵢ > 1 have a higher vapor pressure than Raoult's law predicts, and what does this reveal about the molecular interactions in that mixture?

Model answer: When γᵢ > 1, the actual chemical potential of component i in solution is higher than the ideal chemical potential at the same mole fraction. This means the component has a greater 'escaping tendency' — it is energetically less stable in the mixture than it would be in its pure form. Physically, this occurs when unlike molecule interactions (between the two components) are weaker than the like interactions (within each pure component). Molecules in the mixture 'miss' their preferred interactions and escape into the vapor phase more readily, driving vapor pressure above the Raoult prediction.

The activity aᵢ = γᵢxᵢ enters the chemical potential as μᵢ = μᵢ° + RT ln(γᵢxᵢ). A higher activity means a higher chemical potential, which drives the component toward the phase with lower chemical potential — the vapor. The physical intuition is that if unlike molecules interact weakly, mixing them creates a less stable liquid phase than either pure component alone, and the excess free energy manifests as increased volatility (positive deviation from Raoult's law).