Integral method for infinite dilution activity coefficients calculation

Transcription

1 Integral method for infinite dilution activity coefficients calculation A. C. CHIALVO Programa de Electroqrcimica Aplicada e Ingenieria Electroquimica (PRELINE), Facultad de Ingenieria Quimica (UNL), Santiago del Estero 2829, (3000) Santa Fe, Argentina Received September 27, 1990 A. C. CHIALVO. Can. J. Chem. 70, 1645 (1992). An integral method to determine infinite dilution activity coefficients (7:) from data sets of finite concentration activity coefficients is proposed for binary systems. This method is based upon the integration of the experimental dependence of In y,/x,' over different composition ranges. The limiting behavior of this function is analysed, in order to improve the precision of its integration. The accuracy of this method is demonstrated by the calculation of y: for several systems. A. C. CHIALVO. Can. J. Chem. 70, 1645 (1992). On propose une mkthode dlintcgration permettant, pour les systkmes binaires, de determiner les coefficients d'activit6 B dilution infinie (y:) a partir des donnees relatives aux coefficients d'activitc B concentrations finies. La methode est basee sur I'intCgration de la relation exp6rimentale de In yi/xj2 sur diffkrents ensembles de composition. Afin d'arnkliorer la prccision de I'intCgration, on a analysc le comportement limitant de cette fonction. On a dcrnontrc I'exactitude de la. mcthode par le calcul du y"e plusieurs systkmes. [Traduit par la redaction] Introduction One of the sources more commonly used to obtain the infinite dilution activity coefficients (y:) is undoubtedly the vapor-liquid equilibrium (VLE). The reduction of VLE data to obtain the y," values varies from the graphical or analytical extrapolation of a In yi -xi data set (1) to the determination of the limiting slope of temperature or pressure with respect to the liquid phase composition (2-6). All of these processes require experimental determinations in the composition range near to that of the pure components. However, a method that allows us to verify the thermodynamic consistency of these data is not available. The present work intends to cover these important aspects by the development of a process that permits us to evaluate the y," value starting from an entire or partial In yi - xi data set. Fundamentals The Gibbs-Diihem equation for activity coefficients at constant temperature and pressure in a binary system can be written in the form: The integration of eq. [I] can be carried out using different limits. Taking into account the limiting behaviour of the activity coefficients (7), it is possible to arrive at the following useful expressions: venient to analyze the limiting behavior of the integrand function. Limiting behaviour of the function In yi/xj2 Taking into account that, when xi = 1, In yi = 0 and d In yi/dxi = 0: The right-hand side of eq. [4] can be evaluated by derivation of the classical form of the Gibbs-Diihem equation and application of limiting conditions: Substituting into eq. [4]: The behaviour of the integrand function on the other integration limit is Besides, by derivation of the integrand function with respect to xi, where ge = G~/RT and GE is the excess Gibbs energy. the limiting slopes can be found. For x, + 0: These equations allow us to determine the infinite dilution activity coefficients. Nevertheless, in order to improve [9] lim the evaluation of the integrals in eqs. [2] and [3] it is con-

2 1646 CAN. J. CHEM. VOL. 70, 1992 TABLE 1. Limiting behavior of the integrand functions lirn I dxl System - lim x.-i dx2-6 Ldr:] TABLE 2. Calculated and experimental infinite dilution activity coefficients In r;" (Y;) In r7 (r7) Present work Experimental f Present work Experimental lim g E /~2 Dichloromethane(1) (ref. 8) methanol(2) 25 C (ref. 8) (2.920) (2.938) (2.945) (10.115) (ref. 8) (10.322) re0 For xi + 1, the right-hand side of eq. [8] becomes indeterminant. Through the application of L'Hopital's rule twice, the following limit is obtained: Estimate of the limiting values As stated before, the evaluation of eqs. [2] and [3] can be improved if the limiting values of the integrand function are known. For such an estimate, the supplementary function g E / ~is used. It provides the following relationships:

3 CHIALVO FIG. 1. Dependence of the function ge/xi upon the composition for the system 1-propanol(1)-n-heptane(2) at 45OC. FIG. 2. Dependence of the integrand function upon the composition for the system 1-propanol(1)-n-heptane(2) at 45OC. (a) i = 1; (0) i = 2. I ' [ [12] lim- = d g 1 dlny, A summary of the limiting equations derived here is shown Table 1. These expressions can be applied to evaluate the x,+o dx, x, y: from experimental isobaric (P) or isothermic (T) activity coefficient data, obtained from VLE studies. These data re- Hence, eq. [9] can be rewritten as follows: quire a point-to-point temperature or pressure correction and therefore information about the excess enthalpy (he) or excess volume (ve) function under the same conditions as the VLE measurements is necessary. However, the weak de- 5 [_la= -z [?I in

4 CAN. 1. CHEM. VOL. 70, 1992 I I I I slope: -4.9 slope: I I x1 I 1 FIG. 3. Dependence of the function ge/x, upon the composition for the system methanol(1)-cyclohexene(2) at 45OC. x2 FIG. 4. Dependence of the integrand function upon the composition for the system methanol(1)-cyclohexene(2) at 45 C. (e) i = 1; (0) i = 2. pendence of the activity coefficient on pressure makes the correction for isothermic data quite negligible. For this reason, isothermic data were used in this paper to illustrate the expressions obtained. Applications The following general procedure is proposed for the calculation of the infinite dilution activity coefficients on the basis of the expressions deduced above: (i) Estimation of the limiting behavior of the integrand function starting from the limiting behavior of ge/~i, through eqs. [Ill-[13]. (ii) Evaluation of the integrand function from the corrected In yi -xi data set. (iii) Calculation of y>hrough eqs. [2] or [3] according to the data set used. The proposed procedure was used to calculate yl for several systems. The experimental data were obtained from the literature and the results are summarized in Table 2. The calculations for two of these systems are described here in

5 CHIALVO 1649 order to illustrate the procedure. From data of Van Ness et al. (9) for the system 1-propanol(1)-n-Heptane(2) at 45"C, the auxiliary plots were drawn for the estimation of the limiting integrand functions (Fig. 1). With these values, the In y,/xf vs. x, plot was completed (Fig. 2). The following values were obtained from the integration: In y;" = (y;" = ) and In yy = (yy = 7.228). The second example is the methanol(1)-cyclohexane(2) system at 45 C reported by Madhovan and Murti (1 1). This mixture exhibits phase separation with the compositions (16): xi = and x; = The,g E/x, vs. x, and In y,/x? vs. x, plots are shown in Figs. 3 and 4, respectively. The following integration limits were used: (i) for In yy, x2 = (,ge = ) andx, = 1, and (ii) for In y;, x, = (,ge = ) and x, = 1 (Fig. 4). The application of eq. [3] gave the following results: In y;" = (y; = ) and In y; = (yy = ). Discussion The proposed method is based on a suitable solution of the Gibbs-Diihem equation at constant temperature (T) and pressure (P). It provides thermodynamically consistent y:(t,p) values from corrected y, (T,PcxpP,x,) or y, (TexpP,P,x,) data sets. In this work the calculations that illustrate the proposed procedure were made from isothermic data without pressure correction. The error of this approximation can be estimated with the following expression: For the different systems analysed, eq. [14] shows that this correction term is less than and therefore negligible compared to the In y, values. Nevertheless, in the case where the pressure variation is appreciable, the correction should be made. It is important to note that the limiting values were estimated from the,ge/x, VS. X, plot, which exhibits a limiting slope value that is one-half of that of the In y, vs. x, plot (eq. [12]) and consequently represents a better extrapolation condition. The accuracy obtained in the analysed systems was good, as can be observed in Table 2, for the final evaluation of the v10. A different approach could be made in order to take into account, for example, the pressure effect. In this case, eq. [l] becomes The use of this expression is similar to that derived above and, in this case, information about the ve(xi) at similar conditions of the activity coefficient data and the total vapor pressure dependence upon the composition of VLE is necessary. The corresponding form of eq. [6] is now It can be concluded, on the basis of the results obtained, that the proposed method allows us to evaluate the activity coefficient at infinite dilution with accuracy and reliability. 1. L. B. Schreiber and C. A. Eckert. Ind. Eng. Chem. Process Des. Develop. 10, 572 (1971). 2. M. F. Gautreaux and J. Coates. AIChE J. 1, 496 (1955). 3. S. R. M. Ellis and P. A. Jonah. Chem. Eng. Sci. 17, 971 (1962). 4. G. M. Loblen and J. M. Prausnitz. Ind. Eng. Chem. Fundam. 21, 109 (1981). 5. E. R. Thomas, B. A. Newman, G. L. Nicolaides, and C. E. Eckert. J. Chem. Eng. Data, 27, 233 (1982). 6. K. A. Pividal and S. Sandler. J. Chem. Eng. Data, 33, 438 (1988). 7. A. C. Chialvo. An. Asoc. Quim. Argent. 73, 531 (1985). 8. J. R. Khurma, 0. Muthu, S. Munjal, and B. 0. Smith. J. Chem. Eng. Data, 28, 412 (1983). 9. H. C. Van Ness, C. A. Soczak, G. L. Peloquin, and R. L. Machado. J. Chem. Eng. Data, 3, 217 (1967). 10. C. A. Eckert, B. A. Newman, G. L. Nicolaides, and T. C. Long. AIChE J. 27, 33 (1981). 11. S. Madhavan and P. S. Murti. Chem. Eng. Sci. 21, 465 (1966). 12. J. Gmehling and U. Onken. Vapor-liquid equilibrium data collection. DECHEMA, Vol. 1. Part 2.a p R. Srivastava and B. D. Smith. J. Chem. Eng. Data, 31, 94 (1986). 14. J. R. Khurma, 0. Muthu, S. Munjal, and B. 0. Smith. J. Chem. Eng. Data, 28, 100 (1983). 15. R. R. Davison. J. Chem. Eng. Data, 13, 348 (1968). 16. T. Nitta, S. Takeuchi, and T. Katayama. Chem. Eng. Sci. 29, 2213 (1974).