## Vapour Pressure Diagrams

These diagrams consider two component systems at a constant temperature. The two remaining variables that affect the stabilities of the phases are the pressure and the composition of the mixture, so a vapour phase diagram represents diagrammatically the compositions and pressures at which the condensed and vapour phases of the mixture are stable.

The partial vapour pressures of the components of an ideal solution of two volatile liquids are related to the composition of the liquid mixture by Raoult’s Law:

The total vapour pressure, p, of the mixture is therefore given by:

(The last step follows from the fact that in a binary mixture, x_{A} + x_{B} = 1 .) This shows that the total vapour pressure (at fixed T) shows a linear dependence upon the composition:

Note that in the above diagram, p_{B}* < p_{A}* (i.e. A is more volatile than B). If B is more volatile than A, the slope of the line is negative. (The gradient is equal to p_{A}* – p_{B}* when it is the mole fraction of A being altered.)

The liquid and vapour that are in equilibrium do not necessarily have the same composition.Note that in the above diagram, p_{B}* < p_{A}* (i.e. A is more volatile than B). If B is more volatile than A, the slope of the line is negative. (The gradient is equal to p_{A}* – p_{B}* when it is the mole fraction of A being altered.)

It is common sense to think that the vapour phase would be richer in the more volatile component of the system. (i.e. the substance with a higher p* value would be a greater mole fraction of the vapour phase than it is of the liquid phase.)

This hypothesis can be confirmed by using Dalton’s Law of Partial Pressures (which states that the pressure exerted by a mixture of perfect gases is equal to the sum of their partial pressures) to manipulate the expressions for the partial pressures of the components. The resulting expressions (with y_{A} and y_{B}being the gas phase mole fractions of A and B respectively) are:

If we make the assumption that the mixture is ideal, we can substitute the expressions for p and p_{X} given at the top of the page into these equations to obtain:

If p_{A}* > p_{B}* (i.e. A is more volatile than B) then the above expression always gives y_{A} > x_{A} , making the vapour phase richer in the more volatile component than the liquid phase, in line with our expectations.

Note that if either component is non-volatile (if the pure substance has a zero partial pressure at the temperature under consideration) then it makes no contribution to the vapour.

We have derived a relationship between the gas and liquid phase mole fractions of each substance in a two component mixture, and a relationship giving the dependence of the total vapour pressure upon the composition of the mixture. By combining these, we can obtain the following equation giving the relationship between the total vapour pressure and the composition of the vapour:

(The analogous expression for y_{B} may obtained by the substitution y_{B} = 1 – y_{A}.) The above equation gives graphs of the following form (the lines in the plot are labeled with the value of the ratio p_{A}*/p_{B}*):

It proves convenient to combine information about the compositions of the liquid and vapour phases into one diagram – such diagrams are termed pressure composition diagrams.