354
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
Infinite Dilution Activity Coefficient Values from Total Pressure VLE Data. Effect of Equation of State Used Patrick J. Maher and Buford D. Smlth' Thermodynamics Research Laboratory, Washington University, St. Louis, Missouri 63 130
The accurate determination of the infinite dilution activity coefficient values from binary total-pressure VLE data is demonstrated in conjunction with the Mixon-Gumowski-Carpenter data reduction method. For highly accurate data, the procedure is superior to the method which requires the assumption of a correlation form for GE. The sensitivity of the infinite dilution values to the equation of state used to predict the vapor-phase properties is
demonstrated.
Introduction Infinite dilution activity coefficients play an important role in separations technology. They determine the performance of distillation columns designed to produce high-purity products and have been used as the experimental data base for correlations used to predict activity coefficients over the entire composition range. Unfortunately, they are difficult to measure directly (particularly the isothermal values which are needed), and the simple extrapolation of the binary yi curves to the endpoints seldom gives accurate values. This paper discusses the determination of the ylmand yzmvalues when total pressure VLE data are being reduced by the Mixon-Gumowski-Carpenter method (Mixon et al., 1965). That data reduction method involves a finitedifference solution which does not provide a value for y1 a t x1 = 0 and y2 at x1 = 1.0. A simple quadratic extrapolation of the Q = GE/RT values usually provides yim values which appear reasonable, but obviously such an extrapolation often will not predict accurately the rapidly changing yi values as the endpoints are approached. An alternative to the approach of Mixon et al. is the method which assumes an analytical function for the GE curve and then determines that set of constants which best reproduces the experimental P vs. x1 curve. (That approach is often termed the Barker method despite the fact that the procedure was in widespread use before Barker's (1953) paper appeared.) An independent determination of the yimvalues is not necessary because they are fixed by the correlation equation constants obtained from a fit of the entire P vs. x1 curve. Such an approach is much simpler than the method of Mixon et al. and it can be used for data which scatter too much to permit use of the approach of Mixon et al. However, the yimvalues determined from a fit of the entire P-xl data set may not reflect the true behavior at the endpoints. Also, the analytical form may not reproduce the experimental P vs. x1 values within experimental error and hence the "Barker" approach will often degrade highly accurate data. For these reasons, the approach of Mixon et al. is preferred for highly accurate, smooth total pressure VLE data. A reliable method for the independent determination of the ylmand y2mvalues then becomes essential. Equations The equations used to calculate the yimvalues from isothermal data have been presented (Gautreaux and Coates, 1955). 0019-7874/79/1018-0354$01.00/0
The standard states are the pure liquids at the system T and P. It is important to note the sensitivity of the calculated yimvalues to the various quantities in eq l a and lb. The liquid molar volumes VIL and V2L are usually so much smaller than the vapor molar volumes VlV and V2v that extreme accuracy is not needed; if experimental values are not available some predictive method such as that of G u n and Yamada (1971) usually provides sufficiently accurate values. An equation of state must be used to provide the VlV and V', and the various fugacity coefficient values (q+pt,, &,pi, r#~lp~:, and &p(). The fugacity coefficients appear as ratios which decreases somewhat the impact of errors in the individual values because both coefficients in a given ratio will probably be off in the same direction. However, it should be noted that the coefficients in each ratio represent endpoint values on the vs. x1 curve; hence the calculated ratios are usually more sensitive to shortcomings in the equation of state than are the values for intermediate compositions. Unfortunately, the Vlv values do not appear in ratios and yimbecomes very sensitive to the Viv value when the (dP/dx1)" value is such as to cause the subtraction of two numbers of approximately equal size in eq l a and lb. It can be seen that the calculated T~~and 72m values can be strongly dependent upon the equation of state used. This is demonstrated by an example later. The experimental quantities needed in eq l a and l b are the pure compound vapor pressures P1'and Pi and the slopes of the total pressure P vs. 3c1 curve at x1 = 0 and x1 = 1.0. The P1' and P i values must be an integral part of the P-xl data set; i.e., they must have been measured along with all the other P vs. x1 values. Reliable values of the
ai
0 1979 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
endpoint slopes can then be determined from the P-xl data set as described below, if the P vs. x1 values do not exhibit excessive scatter. Determination of the Endpoint Slopes In most cases, a least-squares cubic spline fit of the P vs. rl data with well-chosen knot points will provide good values for the (dP/dxl)" values. However, polynomials tend to weave when fitted to points which are not completely smooth. If weaving occurs, it can cause significant errors in the values of the terminal slopes. Elimination of the weaving can require a tedious trial-and-error choice of the data point weights and the knot points specified for the splined fits. A more reliable graphical method described by Ellis and Jonah (1962) and similar to a procedure described by Mrazek and Van Ness (Mrazek and Van Ness, 1959; Van Ness, 1964) can be used to obtain the terminal slope values. The P vs. x1 data are converted to P D vs. x1 values using P D = P - [Pi (P
