where y is the mole fraction of B in the vapour phase, p*A and p;*B are the vapour pressures of pure A and of pure B, p is the total pressure, V1,*A and V1,*B are the molar volumes of pure liquid A and of pure liquid B, BAA and BBB are the second virial coefficients of A and of B, δAB = BAB - (BAA + BBB)/2 where BAB is the second virial coefficient for AB interactions, and V1A and V1B are the partial molar volumes of A and of B at mole fraction x of B. The last term of equations (2) and (3) is generally less than 0.1 J moJ-1 if the excess volume is less than 1.0 cm3 mol-1 and (p[??] - p) is less than 100 kPa.

Measurements on second virial coefficients of pure substances and mixtures made prior to 1969 have been compiled by Dymond and Smith and by Mason and Spurling.

Cox and Lawrenson have listed more recent measurements on pure gases in the first volume of this series and Knobler discusses gas mixtures in Chapter 7 in this volume. Second virial coefficients for both pure components and mixtures can also be estimated from various empirical correlations.

Until recently, the most common method for obtaining GEm consisted of the independent measurement of the three quantities p, x, and y by means of a recirculating still. The method has the major advantage that GE can be calculated directly from equations (1), (2), and (3) by straightforward arithmetic. The two major problems in using recirculating stills are the difficulty in establishing a steady state which differs insignificantly from true equilibrium and the difficulty in analysing the liquid and condensed vapour samples. A second advantage of measuring all three quantities is that one can test for thermodynamic consistency because the three variables are not independent but are related by the Gibbs-Duhem relationship, which at constants p and T reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

This equation is rarely used for testing the consistency of p, x, and y data, the claim being made that it is difficult to obtain the gradients with sufficient accuracy except where there are a considerable number of points spread over the entire composition range. The most common method for testing consistency is to use an integrated form of the Gibbs-Duhem equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Substituting from equations (3) and (4) and assuming the gas phase to be ideal gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The equal-area consistency test, proposed independently by Coulson and Herington and Redlich and Kister, involves plotting ln{(1-y)xp*B/[y(l-x)p*A)} against x and if the areas above and below the x axis are equal, then the data are considered to be consistent. Other equations, based on equations (3), (4), and (5) but including a term for non-ideality of the gas phase, have been proposed. 34 However, a more detailed examination of equation (6) shows that it is a consistency test only in the most restricted sense. The only data needed to construct the plot are x, y, p*A, and p*B Except for the pure-component values, the pressure measurements are not involved. The only test made by the area method is to determine whether the vapour and liquid compositions are self-consistent and if the ratio p*A/p*B is appropriate for a given set of x, y data. In other words, no matter how badly the pressure is measured, as long as x, y, and the pure component vapour pressures are measured correctly, the equal-area test would show the data to be consistent, even though the individual values of µEA and µEB may be completely wrong. In fact any plot which is based on the Gibbs-Duhem equation in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

will give rise to a ratio in which the total pressure will cancel. This statement is only correct when the total pressure p and y are measured. If the partial pressures were measured individually by some method which does not involve a knowledge of the total pressure, then the equal-area test and other related tests would be a valid test of consistency.

Recently Herington has extended the area test in a different form by considering the integrals

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

To analyse for consistency over parts of the composition range he considers three properties of the integrals, (a) IA + IB = constant; (b) the curves of IA and IB against x are mirror images about (IA + IB)/2; and (c) the properties (a) and (b) apply to any restricted range of composition.

The question arises as to the best method for reducing the simultaneous measurements of p, x, and y to obtain GE. The advantage of simple arithmetic in using p, x, and y directly is now minimal with the advent of high-speed computers. There are three alternative methods available, each making use of only two of the variables and the Gibbs–Duhem relation. With a perfect set of data one should be able to calculate the same chemical potentials using all the variables or any two of the variables. Alternatively one uses one pair of the data to predict the third variable, which is then compared with the experimental value. This method of data reduction and testing has been considered in detail by Van Ness et al.

The excess Gibbs function GE and the total pressure p can be calculated from (x,y) data by assuming GE to have some functional form, for example:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

whence:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

For the special case of m = 1

µEA/RT = x2{A0 + A1(3 - 4x)}, (13)

µEB/RT = (1 - x)2{A0 + A1(l - 4x)}. (14)

From equations (2), (3), (13), and (14) and assuming ideal-gas behaviour, it follows that

(µEA - µEA)/RT = ln[(1 - y)xp*B/{y(l - x)p*A = A0(2x - 1) - A1(1 - 6x + 6x2).(15)

If y, x, p*A, and p*B are known, the coefficients A0 and A1 can be determined by applying a least-squares analysis to equation (15). Once the coefficients are determined, the total pressure p can be calculated from equations (2) and (13) or (3) and (14). This approach can be readily extended to the general case as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

The gas-imperfection terms can be easily incorporated as a rough estimate of the total pressure would be sufficient. If pressure measurements have been made, then consistency requires that the differences between the measured and calculated pressures scatter randomly about zero. Where there is a distinct bias the data can be judged to be inconsistent.

There are two distinct ways of reducing (p,y) data. The first involves assuming a specific form for GEm(x), for example, equation (10). The resulting equations (11) and (12) can be combined with equations (2) and (3) and the 2N equations can be solved for the N + M) unknowns by some systematic improvement of trial values for the N values of x and M values of A. The usual method minimizes the pressure residuals. The second method assumes no explicit form for GEm(x). Brewster and McGlashan make a least-squares fit to the pressure in the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

where p1d and x1d are defined by the relations:

pid = p*A p*B/{p*B - y(p*B - p*A)}, (18)

xid = (pid - p*A)/(p*B - p*A). (19)

The derivative dp/dy is obtained by differentiation of equation (17) and the mole fraction x is obtained by substituting equations (2) and (3) into (4) to give the relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

where βc = (Bcc - V1,*c)/RT. If experimental values of x have been determined they can be compared with the calculated values to determine consistency.

In the analysis of (p, x) measurements, equations (11) and (12) can be combined with equations (2) and (3) and the 2M equations can again be solved for the (N + M) unknowns by trial for the N values of y and M values of A. Alternatively equation (20) can be rearranged to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

which can be integrated by a marching procedure starting at an appropriate limit. Usually p is expressed as a function of x over the whole or part of the composition range by a series expansion. Some care must be exercised in selecting the appropriate starting point for the integration. This method of analysing (p, x) data has been discussed in detail by Van Ness. As well as testing for consistency of (p, x, y) data, these alternative methods can be used to calculate G when only (x,y), (p, x), or (p,y) data are available.