## Info

As 0.9992 is near enough to 1.000, the dew point may be taken as 372.0 K.

### 11.2.3. Relative volatility

The relationship between the composition of the vapour yA and of the liquid xA in equilibrium may also be expressed in a way, which is particularly useful in distillation calculations. If the ratio of the partial pressure to the mole fraction in the liquid is defined as the volatility, then:

Xa Xb

The ratio of these two volatilities is known as the relative volatility a given by:

XaPb

Substituting PyA for PA, and PyB for PB:

yAxB

yBXA

yB Xb

This gives a relation between the ratio of A and B in the vapour to that in the liquid. Since with a binary mixture yB = 1 ā yA, and xB = 1 ā xA then:

This relation enables the composition of the vapour to be calculated for any desired value of x, if a is known. For separation to be achieved, a must not equal 1 and, considering the more volatile component, as a increases above unity, y increases and the separation becomes much easier. Equation 11.14 is useful in the calculation of plate enrichment and finds wide application in multicomponent distillation.

From the definition of the volatility of a component, it is seen that for an ideal system the volatility is numerically equal to the vapour pressure of the pure component. Thus the relative volatility a may be expressed as:

This also follows by applying equation 11.1 from which PA/PB = yA/yB, so that:

Whilst a does vary somewhat with temperature, it remains remarkably steady for many systems, and a few values to illustrate this point are given in Table 11.1.

Table 11.1. Relative volatility of mixtures of benzene and toluene

Temperature (K) 353 363 373 383

It may be seen that a increases as the temperature falls, so that it is sometimes worthwhile reducing the boiling point by operating at reduced pressure. When Equation 11.16 is used to construct the equilibrium curve, an average value of a must be taken over the whole column. As Frank(13) points out, this is valid if the relative volatilities at the top and bottom of the column differ by less than 15 per cent. If they differ by more than this amount, the equilibrium curve must be constructed incrementally by calculating the relative volatility at several points along the column.

Another frequently used relationship for vapour-liquid equilibrium is the simple equation:

For many systems K is constant over an appreciable temperature range and Equation 11.11 may be used to determine the vapour composition at any stage. The method is particularly suited to multicomponent systems, discussed further in Section 11.7.1.

11.2.4. Non-ideal systems

Equation 11.4 relates xA, yA, Pa and Pā¢ For a non-ideal system the term y, the activity coefficient, is introduced to give:

or in Equation 11.18:

0 0