19.2 Thermodynamics of solutions and mixtures

Thermodynamic Properties and Laws

Solutions can behave as ideal or non-ideal, and the distinction comes down to how their molecules interact with each other. Ideal solutions follow Raoult's law for vapor pressure, while non-ideal solutions deviate from it and require activity coefficients to describe their behavior accurately. These differences matter because they determine how you predict properties like vapor pressure, solubility, and mixing behavior in real chemical systems.

Properties of ideal vs. non-ideal solutions

An ideal solution forms when the components have similar molecular sizes and similar intermolecular forces. Because the interactions between unlike molecules are roughly the same as those between like molecules, there's no net change in volume or enthalpy upon mixing. A classic example is a benzene-toluene mixture, where the two molecules are structurally similar enough that mixing is essentially "seamless."

A non-ideal solution forms when the components differ significantly in molecular size or intermolecular forces, leading to measurable changes in volume or enthalpy upon mixing. These solutions deviate from Raoult's law, and you need activity coefficients to describe their vapor pressures accurately.

Non-ideal solutions show two types of deviation:

• Positive deviation: Intermolecular forces between unlike molecules are weaker than those between like molecules. The molecules "escape" into the vapor phase more easily, so the observed vapor pressure is higher than Raoult's law predicts. Example: ethanol and cyclohexane.
• Negative deviation: Intermolecular forces between unlike molecules are stronger than those between like molecules. The molecules hold each other in the liquid phase more tightly, so the observed vapor pressure is lower than Raoult's law predicts. Example: chloroform and acetone, which form a hydrogen-bond interaction between the C–H of chloroform and the C=O of acetone.

Vapor pressure and solubility laws

Raoult's law applies to ideal solutions (or as an approximation for the solvent in dilute solutions). It states that the total vapor pressure of a solution is the sum of each component's contribution:

$P_s = \sum x_i P_i^*$

where $x_i$ is the mole fraction of component $i$ in the liquid phase and $P_i^*$ is the vapor pressure of pure component $i$. This law is useful for determining the composition of the vapor phase above a liquid mixture using Dalton's law alongside it.

Henry's law applies to the solute in dilute solutions, particularly for dissolved gases. It states that the solubility of a gas is directly proportional to the partial pressure of that gas above the solution:

$C = k_H P$

where $C$ is the concentration of dissolved gas, $P$ is the partial pressure of the gas, and $k_H$ is Henry's law constant (which is specific to each gas-solvent pair and depends on temperature). This is why opening a carbonated beverage causes fizzing: reducing the pressure above the liquid lowers the solubility of $CO_2$, so dissolved gas escapes.

Partial Molar Quantities and Chemical Potentials

Partial molar quantities in mixtures

In a pure substance, adding one mole increases the total volume by exactly the molar volume. In a mixture, though, adding one mole of a component can change the total volume by a different amount because of how the molecules pack together. Partial molar quantities capture this effect.

The partial molar quantity of component $i$ with respect to an extensive property $M$ (such as volume, enthalpy, or Gibbs energy) is defined as:

$\bar{M}_i = \left(\frac{\partial M}{\partial n_i}\right)_{T, P, n_{j \neq i}}$

This is the rate of change of the total property $M$ when you add an infinitesimal amount of component $i$, holding temperature, pressure, and the amounts of all other components constant.

• Partial molar volume quantifies each component's contribution to the total volume. For example, adding one mole of ethanol to a large volume of water increases the total volume by about 54 mL, not by ethanol's pure molar volume of 58.4 mL. The difference arises because ethanol and water molecules pack differently together than they do in their pure states.
• Partial molar enthalpy describes the heat effect when one mole of a component is added. Dissolving sulfuric acid in water, for instance, is highly exothermic, and the partial molar enthalpy of $H_2SO_4$ varies with concentration.

Activity in non-ideal solutions

The chemical potential $\mu_i$ is the partial molar Gibbs free energy of component $i$. It measures the tendency of that component to undergo change (transfer between phases, react, etc.). For an ideal solution:

$\mu_i = \mu_i^* + RT \ln x_i$

where $\mu_i^*$ is the chemical potential of pure component $i$ at the same temperature and pressure.

For non-ideal solutions, the mole fraction $x_i$ alone doesn't capture the true thermodynamic behavior. Instead, you replace it with the activity $a_i$:

$\mu_i = \mu_i^* + RT \ln a_i$

The activity coefficient $\gamma_i$ bridges the gap between the actual activity and the mole fraction:

$a_i = \gamma_i \, x_i$

The value of $\gamma_i$ tells you the type and degree of deviation from ideality:

• $\gamma_i = 1$: Ideal behavior. The component interacts with its neighbors the same way it does in the pure state. Example: benzene and toluene.
• $\gamma_i < 1$: Negative deviation. Unlike molecules attract each other more strongly than like molecules do, stabilizing the liquid phase. Example: chloroform and acetone.
• $\gamma_i > 1$: Positive deviation. Unlike molecules interact more weakly than like molecules, making it easier for them to escape into the vapor. Example: ethanol and cyclohexane.

As a solution becomes more dilute ($x_i \to 1$ for the solvent), $\gamma_i \to 1$ and Raoult's law is recovered. This is why Raoult's law works well for the solvent in dilute solutions even when the overall mixture is non-ideal.