Pure and mixed phases

Vapor pressure over a solution

If one considers a two-phase system, consisting of a liquid and a gaseous mixed phase, this system is only in thermodynamic equilibrium if the chemical potential in the solution and in the gas phase is the same for each component i of the overall system.

One designates the liquid phase with $\alpha$ and the gas phase with $\beta$, the following applies in equilibrium for component i:

$${{\mu }_{\mathit{i}}}^{\alpha }={{\mu }_{\mathit{i}}}^{\beta }$$

With the condition of a permanent equilibrium, the relationship between the vapor pressure of a component i and the composition in the liquid phase can be established from this.

Raoult Law

Raoult's law results for the simplest case of ideal behavior in both the liquid and the gas phase.

$${\mathit{p}}_{\mathit{i}}={\mathit{x}}_{\mathit{i}}\cdot {\mathit{p}}_{\mathit{i}}*$$

It is ${\mathit{p}}_{\mathit{i}}$ as partial pressure of component i in the gaseous mixed phase, ${\mathit{x}}_{\mathit{i}}$ as the mole fraction of the component in the liquid phase and ${\mathit{p}}_{\mathit{i}}*$ as the vapor pressure of the pure component.

If the liquid mixture does not behave ideally, deviations from Raoult's law are obtained, which can be taken into account by introducing activity coefficients:

$${\mathit{p}}_{\mathit{i}}={\mathit{p}}_{\mathit{i}}*\cdot {{\mathit{x}}_{\mathit{i}}}^{\alpha }\cdot {{\mathit{f}}_{\mathit{i}}}^{\alpha }=\mathit{p}*\cdot a_{\text{i}}^{\alpha }\text{with}\underset{{{\mathit{b}}_{\mathit{i}}}^{\alpha }\to 1}{lim}{\mathit{f}}_{\mathit{i}}=1$$

Henry Law

For ${{\mathit{x}}_{\mathit{i}}}^{\alpha }\to 0$ the activity coefficient approaches a constant value ($\underset{{{\mathit{x}}_{\mathit{i}}}^{\alpha }\to 0}{lim}{{\mathit{f}}_{\mathit{i}}}^{\alpha }=\text{const.}$). Henry's law follows for the component in infinite dilution.

$${\mathit{p}}_{\mathit{i}}={\mathit{p}}_{\mathit{i}}*\cdot {{\mathit{x}}_{\mathit{i}}}^{\alpha }\cdot \text{const.}=k_{\text{i}}\cdot {{\mathit{x}}_{\mathit{i}}}^{\alpha }$$

Real mixes

When describing the vapor pressure of real mixtures, the Henry or the Raoult Straight Line of Limits.