DETERMINATION OF ACTIVITY COEFFICIENTS AT INFINITE DILUTION FROM DEW POINT AND/OR BUBBLE POINT CURVES"

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1 DETERMINATION OF ACTIVITY COEFFICIENTS AT INFINITE DILUTION FROM DEW POINT AND/OR BUBBLE POINT CURVES" MASAHIRO KATO AND MITSUHO HIRATA Dept. of Ind. Chem., Tokyo Metropolitan University, Tokyo A new method for determining activity coefficients at infinite dilution from dew point and/or bubble point data in binary systems is proposed. The extrapolated values of dew point and bubble point parameters, respectively, defined as D=aia2/(l-ai) (l-a2) and B= (l-bi) (l-b2)/bib2, agree thermodynamically with the activity coefficients at infinite dilution. The values of 'a' and *b', respectively, can be calculated by a^pc = {n/p)y from dew point data and by b= (y/f) = (P/w)x from bubble pointdata, where x, y, P, n and Yt respectively, express mole fraction in liquid phase, that in vapor phase, vapor pressure, total pressure and activity coefficient. Introduction The activity coefficients at infinite dilution are important for elucidating the properties of liquid mixtures, and the constants of van Laar, Margules and Wilson equations can be calculated by application of the activity coefficients at infinite dilution. Thus knowing the activity coefficients at infinite dilution is equivalent to knowing vapor-liquid equilibrium data over the rest of the composition range. It is therefore of importance to be able to calculate activity coefficients at infinite dilution from dew point and/or bubble point data which can in many instances6~9) be more readily and reliably obtained than a direct measurement of vapor-and liquid-phase compositions in the terminal regions of the composition range. In this paper, a simple method is presented for determining the values of activity coefficients at infinite dilution from the data of dew point and/or bubble point curves of binary systems. Determination of Activity Coefficients at Infinite Dilution from the Dew Point Curve A method for determining the activity coefficients at infinite dilution from the dew point curve was proposed by Gautreaux et al.5) previously. Their equations involve the limiting slopes of dew point curves at infinite dilution, but it is not always easy to determine the slopes accurately. Bellemansl;> discussed the correlation between the dew point data and the excess free energy in binary systems by the following equation, derived from the Gibbs-Duhem relation. where activities a\ and a* are expressed as follows. ax = T\xi = {-jtjyi fn\ (2) The activity coefficients at infinite dilution, ft0 and fc0, may be calculated from the dew point data by Eq. (l). But it is rather difficult to draw accurately the tangent lines as shown by the dotted lines in Fig. 1. In the present paper, a newparameter D, which can be calculated only from the dew point data, is proposed as follows. D- (1-aOd-fl,) Wo Values of activity coefficients at infinite dilution, T\ and fc0, agree theoretically with the extrapolated values of the dew point parameter D when the latter is plotted against the vapor phase composition, j>i, as shown in Fig.2. n = limd yi=0 (4) r20 = limd yi=l (5) Received on June 6, 1969 Presented at the 34th Annual Meeting of Soc. Chem. Eng., Japan, Tokyo, April 2, 1969 Fig. I Correlations proposed by Bellemans1 VOL.3 NO.l 1970

2 The method proposed by Gautreaux et al.5) is characterized by the determination of slopes of bubble point curves as in the case of dewpoint curves. Ellis et al.3) somewhat improved the method suggested by Gauteraux et al.5) In the present paper, a new parameter B, which can be calculated from bubble point data, is proposed as follows. B =(i-&,)(i-feo_ ( ) a - bibt w where, b\ and bi are defined as follows. Fig. 2 Determination of activity coefficients by use of dew point data \ =*-(p* 72 V 7T Like the dew point curve, the values of activity coefficients at infinite dilution, n and r20 agree theoretically with the extrapolated values of the bubble point parameter B when it is plotted against the composition in liquid phase, X\, as shown in Fig. 3. ri = lim (10) r20 = lim5 (ll) ;ci=0 The derivations of Eqs. (10) and (ll) are described in Appendix. In ideal solutions b\ and b*, respectively, agree with the compositions in vapor phase, y-i and j>2, since the values of B are equal to unity all over the range of concentration. (9) Examples of Calculation Fig. 3 Determination oh activity coeh-icients by use of bubble point data The derivations of Eqs. (4) and (5) are described in Appendix. In ideal solutions the activities, d\ and a^ respectively, agree with the compositions in liquid phase, xi and xi, and the values of D are equal to unity over the whole range of composition. Determination of Activity Coefficients at Infinite Dilution from the Bubble Point Curve Methods for determining activity coefficients at infinite dilution from the bubble point curve were proposed by Carlson et al.2) as well as Gautreaux et al.5) and Ellis etal.3) The method proposed by Carlson et al.2) is based on the following Eqs. (6) and" (7). Tl= PlXl ' T2= Pix2 (6) rio = Hmn/, r20 = iimr2; (7) xi=0 j:i =0 The values of 7\ and?v can be calculated by Eq. (6) from the bubble point data. By plotting these values against xi, extrapolation of the curves gives the values of activity coefficients at infinite dilution, as shown in Eq.(7). ' The vapor-liquid equilibrium data for four binary systems4>11>12'14) have been compared with values calculated by the method proposed above. They are the isothermal data for Acetone-Chloroform14) and Benzene- Toluene1^ systems and the isobaric data for Ethyl acetate- Ethanol4) and Ethanol-Watern) systems. Fig. 4 shows the relations between a\ and a<i for these systems, where the values of a\ and ai have been calculated from the dew point data by Eq. (2). Fig.5 shows the relations between b\ and bi for these systems, where the values of b\ and b2 have been calculated from the bubble point data by Eq. (9). For the calculation of Eqs. (2) and (9), the values of vapor pressures, Pi and P2, have been calculated by Antoine equations8>9>10). At high pressures, it seems necessary to correct the deviation13) from ideal gas law in Eqs. (2) and (9). Figs. 6 to 9 show the relations among 7i, 7*2, D and B for the four typical systems, where 7\ and 7i have been calculated from the vapor-liquid equilibrium data by the usual method, and the values of D and B, respectively, have been calculated from dew point data by Eq. (3) and from bubble point data by Eq. (8). Fig. 6 shows the relations for Acetone-Chloroform system14), which has negative values of excess free energy. The values of D and B are smaller than unity. Fig. 7 shows the relation for Benzene-Toluene system12) which has nearly zero values of excess free energy. All values of Tu 7i, D and B appear to be almost unity all JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

3 Fig. 4 Typical examples of Qi vs. 02 potl Fig. 6 Acetone (I)-Chloroform (2) system1 at 35.I7 C Fig. 7 Benzene (l)-toluene (2) system12) at 79.7 C Fig. 5 Typical examples of h\ vs. 62 potl over the range of concentration. Fig. 8 shows the relation for Ethyl acetate-ethanol system4} which has positive values of the excess free energy. Values of both D and B are larger than unity. Fig. 9 shows the relation for Ethanol-Water system11\ which are similar to Ethyl acetate-ethanol system. Since the excess free energy is large, a semilogarithmic plot is made. In Fig. 9, values of both D and B axe much larger than those for other examples. In Figs. 6 to 9, smoothed curves or 7*1> T2, D and B were drawn, by considering that all the extrapolated values would cometo agree with each other. Conclusions A new method is proposed for determining activity coefficients at infinite dilution by use of dew point and/ or bubble point data in binary systems. Advantages of this method are as follows : ( 1 ) The extrapolated values of dew point and bubble point parameters defined by Eqs. (3) and (8) agree thermodynamically with the activity coefficients at infinite dilution. ( 2 ) The procedures are the same both for isothermal and for isobaric data. This method can be used by itself, but also is useful together with other methods for the determination of the activity coefficients at infinite dilution. Acknowledgement The authors wish to thank Teruo Kato, who was helpful to these calculations. VOL.3 NO

4 Fig. 8 Ethyl acetate (l)-ethanol (2) system0 at Iatm Fig. 9 Ethanol (l)-water (2) systemn) at Iatm Appendix Derivation of Eqs. (4) and (5) The following equation is derived from Eq. (3), because the values of a\ and a2, respectively, are equal to zero and unity when yx is equal to zero. limd = lim-/, x/dai\, / ^\/da\ Ui-1)(-j-1+(a2-1)1-j- \ dyi J \ dyi = -Up.) ' (a_d Differentiating Eq. (2), dai=t\{dxx+xidint\) \ (A 9n da2=t2(dx2+x2dinh) I l Z) therefore, lim(^-) s.-ovdot) = tr^olvft/i_ lim[7a)j, (d\nh\jj 1 1] (A_3) 1 + \~d^r) Based on the generalized Gibbs-Duhem equation, X i=fdhhr1\+jduu \ dxx J \ dxx AH(dT\. AV(diz\,A A. ~ " -RT^{-dx7) + -RT{-dx7) (A"4) the following equation is obtained, because both AHand AV are equal to zero at the end point of the composition. UnYi^U y1=0\ dxx J o (A-5) Substituting Eq. (A-5) into Eq. (A-3), and combining with Eq. (A-l), gives: lim > = lim(aw]o (A-6) Thus, Eq. (4) is obtained. In the same way, Eq. (5) can be derived easily. Derivation of Eqs. (10) and (ll) The following equation is derived from Eq. (8), because the values of b\ and bi, respectively, are equal to zero and unity when x\ is equal to zero. lim B = lim 7-77 v 7-j-7-v = - Si(-t-) (A-7) Differentiating Eq. (9), 7 7 _ dy\-y\dinft db\- therefore, rf^2= /^2_\ _ /d\nt2\ /^2\ v fti\ \dxij \dxx) /AQx v lim -^r-1=lim M- / 7 N (A-9) «!=o\dbi) ^=0 \72) Idyi\ \dxj Substituting Eq. (A-5) and the relation dy2- -dyi into Eq. (A-9), and combining with Eq. (A-7), gives: lim.b - limf^ ^ ri0 (A-10) Thus, Eq. (10) is derived. In the same way, Eq. (ll) is obtained easily. Nomenclature a -activity given by Eq. (2) [ ] B = bubble point papameter defined by Eq. (8) [-] b =function given by Eq. (9) [ 1 D =dew point parameter dinned by Eq. (3) [ ] Z/H = heat of mixing [atm-//mol] P = vapor pressure [atm] R - gas constant [atm-//mol K] T = absolute temperature [ K] AV= volume change on mixing [//mol] x - mole fraction in liquid phase [-] y =mole fraction in vapor phase [ ] 7 = activity coefficient [ ] u (A-8) JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

5 activity coefficient at infinite dilution parameter total pressuregiven by Eq. (6) Subscripts 1 light component 2 = heavy component Literature Cited 1) Bellemans, A. : Bull. Soc. Chim. Berg., 68, 355 (1959) 2) Carlson, H. C. and A. P. Colburn: Ind. Eng. Chem.y 34, 581 (1942) 3) Ellis, S.R.M. and D.A. Jonah: Chem. Eng. Set., 17, 971 (1962) 4) Furnas, C. C. and W. B. Leighton: Ind. Eng. Chem., 29, 709 (1937) 5) Gautreaux, 496 (1955) M.F. and J. Coates: A. I. Ch. E. Journal, 1, 6) Hiranuma, M., M. Hashiba and M. Kugo: Kagaku Kogaku, 30, 613 (1966) 7) Kojima, 149 (1968) K., K. Tochigi, H. Seki and K.Watase: ibid., 32, 8) Kojima, K., 32, 337 (1968) M. Kato, H. Sunaga and S. Hashimoto: ibid., 9) Kojima, K. and M. Kato: ibid., 33, 769 (1966) 10) Lange, N.A.: "Handbook of Chemistry", 10th ed., Mc- Graw-Hill, New York (1967) ll) Rieder, R. M. and A. R. Thompson: Ind. Eng.Chem., (1949) 12) Rosanorf, M. A., C. W. Bacon Chem. Soc, 36, 1993 (1914) and J. F. W. Schulze: J. Am 13) Scheibel, E.G.: Ind. Eng. Chem., 41, 1076 (1949) 14) Zawidzki, J. : Z. physik. Chem., 35, 129 (1900) ESTIMATION OF CRITICAL CONSTANTS An estimation method for critical properties that is easily available compounds with considerable accuracy is proposed. In this method pressure and critical volume can be estimated solely from data of the liquid density of pure substance by means of succesive approximations TOSHIKATSU HAKUTA AND MITSUHO HIRATA Department of Industrial Chemistry, Faculty of Engineering, Tokyo Metropolitan University for both organic and inorganic critical temperature, critical normal boiling point and the that are repeated until calculated critical pressure (Pc) converges. The average errors of Tc, Pc and Vc for about 24O substances studied were 1.39, 2.61 and 2.O4 percent, respectively, but the errors for nitrogen compounds, bromide compounds, metallic elements, and compounds associated in the vapor phase were sometimes larger. Introduction The critical constants of pure substances are very important because they are often utilized in estimating other physical properties for chemical process design. It is sometimes difficult to determine critical constants directly with accuracy from experimental measurements, so reliable and easy methods for estimation of critical constants are of interest. Recently manyexcellent estimation methods have been proposed in the literature. For instance, Lydersen12), Riedel16>17>18) and Eduljee5>6) proposed group contribution methods. Thodos22) has developed a remarkably accurate methodfor estimation of Tc and Pc for saturated and unsaturated aliphatic hydrocarbons. Stiel and Thodos20) have proposed an estimation methodof critical constants and normal boiling point of saturated aliphatic hydrocarbons by using Wiener number (path number) and polarity number. Nakanishi13) found a simple correlation between critical constants and carbon number for normal alkanes. However there are no satisfactory methods for estimation of critical constants of both organic and inorganic compounds. Fishtine4} has proposed a method involving * Received on July 1, 1969 successive approximations for organics and inorganics. For application of his method the molecular weight, the normal boiling point, a liquid density at or below the normal boiling point and the heat of vaporization at T& are required. But critical pressures estimated by his method often have relatively large error, mainly because Lydersen's correlation12) between Zc and AHvbis applied. Further, measured values of the latent heat of vaporization that have to be known are rather scarce. Considering these defects, a fairly accurate and easy method was developed for estimation of critical constants solely from molecular weight, normal boiling point, and one value of liquid density. Development of the Estimation Method Normal boiling points and liquid densities for very large numbers of compoundsare given in the literature8>11>14'21), and with rare exceptions these data are very accurate. The proposed method uses only these accurate data. The critical temperatures generally can be estimated more accurately than other critical properties, that is, critical pressure and volume. So it is advisable to estimate critical temperature first. In the proposed method, critical temperature is calculated from normal boiling point by using the following equations proposed by VOL.3 NO