9.2 Ideal and non-ideal solutions

Ideal vs Non-ideal Solutions

An ideal solution is one where the components mix without any change in volume or enthalpy. This happens when the intermolecular forces between solute and solvent are essentially identical to the forces within each pure component. Think of mixing hexane and heptane: both are nonpolar hydrocarbons of similar size, so molecules don't "care" whether they're surrounded by hexane or heptane.

Non-ideal solutions deviate from this because the intermolecular interactions between unlike molecules differ from those between like molecules. When solute-solute, solvent-solvent, and solute-solvent interactions aren't equivalent, you get measurable changes in volume, enthalpy, or both upon mixing. Ethanol and water are a classic example: hydrogen bonding patterns change significantly when these two liquids combine.

The thermodynamic mixing properties for ideal solutions can be calculated from composition alone:

• Enthalpy of mixing: $\Delta_{mix}H = 0$
• Volume of mixing: $\Delta_{mix}V = 0$
• Gibbs free energy of mixing: $\Delta_{mix}G = nRT \sum x_i \ln x_i$ (always negative, so mixing is spontaneous)
• Entropy of mixing: $\Delta_{mix}S = -nR \sum x_i \ln x_i$ (always positive)

For non-ideal solutions, these quantities pick up additional "excess" terms that reflect the real molecular interactions at play.

Raoult's Law for Ideal Solutions

Partial Vapor Pressure and Mole Fraction

Raoult's law states that the partial vapor pressure of each component in an ideal solution is proportional to its mole fraction in the liquid phase:

$P_i = x_i \cdot P_i^*$

where:

• $P_i$ is the partial vapor pressure of component $i$ above the solution
• $x_i$ is the mole fraction of component $i$ in the liquid phase
• $P_i^*$ is the vapor pressure of pure component $i$ at the same temperature

The total vapor pressure above the solution is the sum of all partial pressures:

$P_{total} = \sum x_i \cdot P_i^*$

For a binary solution of components A and B, this simplifies to:

$P_{total} = x_A P_A^* + (1 - x_A) P_B^*$

This is a linear equation in $x_A$, which is why the liquid line on a pressure-composition diagram for an ideal solution is always a straight line connecting $P_B^*$ to $P_A^*$.

Vapor Phase Composition and Phase Diagrams

The composition of the vapor above the solution differs from the liquid composition. You find it using Dalton's law:

$y_i = \frac{P_i}{P_{total}}$

The component with the higher pure vapor pressure (more volatile) is always enriched in the vapor phase relative to the liquid. This difference between liquid and vapor composition is what makes distillation work.

For ideal solutions, no azeotropes form. The vapor composition is always different from the liquid composition at every point, so you can in principle achieve complete separation by distillation.

Deviations from Raoult's Law

Positive Deviations

Positive deviations occur when the total vapor pressure is higher than Raoult's law predicts. This happens when solute-solvent interactions are weaker than the average of solute-solute and solvent-solvent interactions. Molecules "escape" into the vapor phase more readily because they're less stabilized in the mixture.

A standard example is carbon disulfide ($CS_2$) and acetone. The nonpolar $CS_2$ disrupts the dipole-dipole interactions between acetone molecules, and acetone can't interact as strongly with $CS_2$ as $CS_2$ molecules do with each other.

Negative Deviations

Negative deviations occur when the total vapor pressure is lower than predicted. Here, solute-solvent interactions are stronger than the like-like interactions in the pure components. Molecules are more stabilized in the mixture, so they have less tendency to escape into the vapor.

Acetone and chloroform are the textbook example: chloroform's C-H can hydrogen bond with acetone's carbonyl oxygen, creating an attractive interaction that doesn't exist in either pure liquid.

Azeotrope Formation

Large enough deviations from Raoult's law produce azeotropes, compositions where the liquid and vapor phases have identical composition. At an azeotrope, distillation cannot achieve further separation because the vapor you boil off has the same composition as the liquid.

• Positive deviations can produce a minimum-boiling azeotrope (pressure maximum on a P-x diagram). Ethanol-water at 95.6% ethanol (by mass) and 78.2 °C is the classic case.
• Negative deviations can produce a maximum-boiling azeotrope (pressure minimum on a P-x diagram). Hydrochloric acid and water at about 20.2% HCl and 108.6 °C is an example.

Activity Coefficients for Non-ideal Behavior

Concept and Definition

Activity coefficients ($\gamma_i$) quantify how much a component's behavior deviates from ideal. The modified form of Raoult's law for non-ideal solutions is:

$P_i = \gamma_i \, x_i \, P_i^*$

The activity of a component is defined as:

$a_i = \gamma_i \, x_i$

Activity replaces mole fraction in all thermodynamic expressions when dealing with non-ideal solutions. You can think of it as the "effective" mole fraction that accounts for real molecular interactions.

• In an ideal solution, $\gamma_i = 1$ for all components, and activity equals mole fraction.
• $\gamma_i > 1$ means the component behaves as if it's at a higher effective concentration than its actual mole fraction (positive deviation).
• $\gamma_i < 1$ means the component behaves as if it's at a lower effective concentration (negative deviation).

Determining Activity Coefficients

Activity coefficients can be obtained from:

• Experimental vapor pressure data: Measure the actual vapor pressure above a solution and compare to Raoult's law predictions.
• Empirical models: The Margules equation, van Laar equation, and Wilson equation each provide different functional forms for how $\gamma$ depends on composition. These are fitted to experimental data.
• Excess Gibbs free energy: Activity coefficients are formally related to the excess Gibbs energy by $\ln \gamma_i = \left(\frac{\partial (n_T G^E / RT)}{\partial n_i}\right)_{T, P, n_{j \neq i}}$

Activity coefficients are essential for accurate calculations of vapor-liquid equilibrium, chemical reaction equilibrium, and solubility in real systems.

Solution Ideality and External Factors

Temperature and Pressure Effects

Temperature influences ideality by changing the relative importance of intermolecular interactions compared to thermal energy. As temperature increases, $kT$ grows relative to the interaction energies, and specific attractions or repulsions matter less. Solutions generally become more ideal at higher temperatures. Ethanol-water mixtures, for instance, show smaller deviations from Raoult's law as temperature rises.

Pressure has a negligible effect on ideality for liquid solutions under normal conditions because liquids are nearly incompressible. For gaseous mixtures, though, increasing pressure forces molecules closer together, amplifying intermolecular interactions and driving deviations from ideal gas mixing behavior.

Solute Concentration and Nature of Components

At low solute concentrations, solutions tend toward ideal behavior because solute molecules are mostly surrounded by solvent, and the few solute-solute interactions that would cause deviation are rare. As concentration increases, deviations become more pronounced.

Electrolyte solutions are a major source of non-ideality. When an ionic compound like NaCl dissolves in water, the strong ion-dipole and ion-ion interactions create large deviations from ideal behavior even at low concentrations. These systems require specialized models (like Debye-Hückel theory) rather than simple activity coefficient equations.

The general rule for predicting ideality: components with similar polarities, molecular sizes, and shapes form more ideal solutions. Hexane and heptane mix nearly ideally. Ethanol and water, despite being fully miscible, deviate significantly because their molecular interactions (hydrogen bonding networks vs. hydrocarbon portions) are quite different.