K Values, Relative Volatility, and x-y Diagrams

A convenient way to rearrange vapor-liquid equilibrium equations is through the use of K values.

DEFINITION

K values, also known as equilibrium ratios or distribution coefficients are ratios of the mole fraction in one phase to that in a different phase, and are functions of temperature and pressure (and composition as well in non-ideal systems). For vapor-liquid systems it is the ratio in the vapor phase to that in the liquid phase.

We can use K values in several ways:

(1) to re-write the Raoult's Law expression as:

K = yi/xi = pi*(T)/P

and Henry's Law expression as:

K = yi/xi = Hi(T)/P

(2) We can use K-values directly, by looking values up (DePriester Charts) or using correlations.

(3) For a binary mixture, we will often take a ratio of K-values in order to try to eliminate (most of) the temperature dependence:

DEFINITION

The relative volatility is a ratio of the K value for one component to that of another. This is useful because it will often be only a weak function of temperature, and thus depends almost exclusively on pressure.

$\alpha_{AB} = \frac{K_A}{K_B} = \frac{y_A/x_A}{y_B/x_B} = \frac{y_A(1-x_A)}{(1-y_A)x_A}$

NOTE

The relative volatility is harder to use in non-binary systems, as the last version of this equation is no longer true (can you prove this to yourself?)

This last equation can be rearranged to yield:

$y_A = \frac{\alpha_{AB}x_A}{1+(\alpha_{AB}-1)x_A}$

which is useful for plotting x-y diagrams like the following:

Outcome:

Define and use K values, relative volatility, and x-y diagrams

Test Yourself

An azeotropic mixture is one where the vapor and liquid phases have the same composition. What would an x-y diagram for such a mixture look like?