between y and x are known, both the vapor and the liquid compositions can be calculated. In order to determine the equilibrium relationships either the temperature or the total pressure must be known. Given either of these, the other can be determined from the equilibrium relationship for all the components involved.
Differential Distillation. This type of distillation is usually carried out as a batch operation although continuous units may also-operate in this manner. Considering first a batch distillation, if a mixture of liquid is distilled, the distillate contains a greater portion of the more volatile material than the residue, and as distillation proceeds both the distillate and the residue become poorer in the more volatile components. This change in composition may be estimated quantitatively if the relation of the composition of vapor to that of the liquid is known. Consider W parts of original mixture containing x0 fraction of component A. Allow a differential amount — dW to be vaporized of a composition, y, under such conditions that the vapor is continually removed from the system.
By material balance,
This equation was developed by Rayleigh (Ref. 2) and is often termed the Rayleigh equation. It can be used with W as weight and x as weight fraction, or with W as mois and x as mol fraction. It is usually applied on the basis that, at any given instant, y is in equilibrium with x, but the derivation does not require this condition. A similar equation applies to each component in a mixture.
The use of Eq. (6-3) requires the relationship between y and x and, even if they are assumed to be in equilibrium with each other as the vapor is formed, it is usually difficult to express the equilibria mathematically for a general integration. The integration can be performed graphically if the relationship between y and x is available.
Batch Distillation Example. As an illustration of the use of this equation, consider the experiment performed by Rayleigh. 1,010 g. of a 7.57 mol per cent solution of acetic acid in water was distilled until the still contained 254 g. whose com position was 11 mol per cent acetic acid. Assuming that the vapor leaves in equilibrium with the liquid, calculate the final composition to be expected on the basis of Eq. (6-3). Equilibrium data for the system acetic acid-water are given in Table 6-1.
There are several approximate methods that may be followed in integrating Eq. (6-3). First, for small temperature and composition ranges, the relation between the vapor and liquid may be approximately represented by a straight line, or y — ex, where c is a constant.
In it „ fx dx ^ i* dx = 1 Jn £ Wo Jxo cx — x Jxo x(c — 1) c — 1 x0
Mol fraction of acetic acid in liquid, x
Mol fraction of acetic acid in vapor, y
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