How to get the relative volatility

Relative volatility is defined as follows:

.......................................................... (1)

where p_a is the partial pressure

x_a is the mole fraction in the liquid phase

Now bring in Raoult’s law

.................................................................. (2)

where p_a is the partial pressure

P_a^o is the vapour pressure

Combine Raoult’s law with the definition for relative volatility to get:

.......................................................... (3)

This means that the relative volatility is the same as the relative vapour pressure or ratio of vapour pressures. If we know the vapour pressures we can determine a. This is very useful. We can get vapour pressures at fixed temperatures from books such as Perry’s Chemical Engineers Handbook or the CRC Handbook of Chemistry and Physics. The temperature range we’re interested in is from the boiling point of one component to that of the other. Relative volatility changes with temperature but one value only is required for the next section. That value is normally decided by averaging the relative volatilities at the two limits of the temperature range.

How to determine vapour and liquid mole fractions

Next, the mole fractions in the vapour and liquid phases can be determined from the relative volatility.

According to Dalton’s law

.................................................................. (4)

where P_T is the total pressure

y_a is the mole fraction in the vapour phase

Combine Dalton’s law with the relative volatility to give:

..................................................... (5)

This is an expression that contains a, which we know from above, and the vapour and liquid mole fractions. Pick one, e.g. the liquid mole fraction and we can determine the vapour mole fraction. So we have the equation we want. It could be rearranged, however, into a more useful form. See below:

..................................................... (6)

Now, for different values of x_a we can determine y_a. Use an average value for a (i.e. average of a at the higher and lower temperature limits).

Mole fractions using Raoult and Dalton’s laws

Dalton’s law and Raoult’s law can be easily combined to give an expression for y_a.

.................................................................. (7)

With this equation we can determine the vapour phase mole fraction if we know the liquid phase mole fraction. To get this we start with the fact that for a binary mixture the total pressure is equal to the sum of the partial pressures of the two components, i.e.

................................................................ (8)

Substitute the expression for partial pressure from Raoult’s law into (8) to get:

..................................................... (9)

Rearrange to give an expression in terms of x_a as follows:

............................................................... (9)

Now we can determine x_a using the vapour pressures and total pressure and from that we can use (7) to give y_a.