The liquid phase activity coefficients y1 and y2 depend upon temperature, pressure and concentration. Typical values taken from Perry's Chemical Engineers' Handbook(14) are shown in Figure 11.8 for the systems n-propanol-water and acetone-chloroform. In the former, the activity coefficients are considered positive, that is greater than unity, whilst in the latter, they are fractional so that the logarithms of the values are negative. In both cases, y approaches unity as the liquid concentration approaches unity and the highest values of y occur as the concentration approaches zero.
The fundamental thermodynamic equation relating activity coefficients and composition is the Gibbs-Duhem relation which may be expressed as:
This equation relates the slopes of the curves in Figure 11.8 and provides a means of testing experimental data. It is more convenient, however, to utilise integrated forms of these relations. A large number of different solutions to the basic Gibbs-Duhem equation are available, each of which gives a different functional relationship between log y and x . Most binary systems may be characterised, however, by either the three- or four-suffix equations of Margules, or by the two-suffix van Laar equations, given as follows in the manner of Wohl(15,16) . The three-suffix Margules binary equations are:
Constants A12 and A21 are the limiting values of log y as the composition of the component considered approaches zero. For example, in Equation 11.22, A12 = log y1 when x1 = 0.
The four-suffix Margules binary equations are:
log Y1 = x|[A12 + 2X1 (A21 - A12 - Ad) + 3Adx2] (11.24)
log Y2 = x2[A21 + 2X2(A12 - A21 - Ad) + 3Adx|] (11.25)
A12 and A21 have the same significance as before and AD is a third constant. Equations 11.24 and 11.25 are more complex than equations 11.22 and 11.23 though, because they contain an additional constant AD, they are more flexible. When AD becomes zero in equations 11.24 and 11.25, they become identical to the three-suffix equations.
The two-suffix van Laar binary equations are:
These equations become identical to the three-suffix Margules equations when A12 = A21, and the functional form of these two types of equations is not greatly different unless the constants A12 and A21 differ by more than about 50 per cent.
The Margules and van Laar equations apply only at constant temperature and pressure, as they were derived from equation 11.21, which also has this restriction. The effect of pressure upon y values and the constants A12 and A21 is usually negligible, especially at pressures far removed from the critical. Correlation procedures for activity coefficients have been developed by Balzhiser et al.(17), Frendenslund et al.(18), Praunsitz et al.(19), Reid et al.(20), van Ness and Abbott(21) and Walas(22) and actual experimental data may be obtained from the PPDS system of the National Engineering Laboratory, UK(23). When the liquid and vapour compositions are the same, that is xa = yA, point xg in
Figures 11.3 and 11.4, the system is said to form an azeotrope, a condition which is discussed in Section 11.8.
From curve a of Figure 11.4 it is seen that, for a binary mixture with a normal y — x curve, the vapour is always richer in the more volatile component than the liquid from which it is formed. There are three main methods used in distillation practice which all rely on this basic fact. These are:
(a) Differential distillation.
(b) Flash or equilibrium distillation, and
(c) Rectification.
Of these, rectification is much the most important, and it differs from the other two methods in that part of the vapour is condensed and returned as liquid to the still, whereas, in the other methods, all the vapour is either removed as such, or is condensed as product.
The simplest example of batch distillation is a single stage, differential distillation, starting with a still pot, initially full, heated at a constant rate. In this process the vapour formed on boiling the liquid is removed at once from the system. Since this vapour is richer in the more volatile component than the liquid, it follows that the liquid remaining becomes steadily weaker in this component, with the result that the composition of the product progressively alters. Thus, whilst the vapour formed over a short period is in equilibrium with the liquid, the total vapour formed is not in equilibrium with the residual liquid. At the end of the process the liquid which has not been vaporised is removed as the bottom product. The analysis of this process was first proposed by Rayleigh(24) .
If S is the number of moles of material in the still, x is the mole fraction of component A and an amount dS, containing a mole fraction y of A, is vaporised, then a material balance on component A gives:
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