A model of nonelectrolytic behaviour: activity coefficients

A model of nonelectrolytic behaviour: activity coefficients PREMP. SINGH'AND SANJEEV MAKEN Department of Chemistry, Maharshi Dayanand University, Rohtak-124001, Haryana, India Received January 3, 19912 PREMP. SINGHand SANJEEV MAKEN.Can. J. Chem. 70, 1631 (1 992). A model is presented for binary mixtures of nonelectrolytes that reproduces successfully the activity coefficient (y,, yj; xi) data for even those (i + j ) mixtures that are known to be characterized by the presence of 1 : 1 or 1 : 1 and 1 : 2 molecular complexes or have components that differ appreciably in their molar volumes when only limited (yr, y,'; xi = 0.5) or (y,"; ye, xi = 0.5) or (y:) data are available). MAKEN.Can. J. Chem. 70, 1631 (1 992). PREMP. SINGHet SANIEEV On prksente un modkle pour les mClanges binaires de non-Clectrolytes qui reproduit bien coefficients d'activitC (y, y,; x,). Le modkle s'applique mCme aux melanges (i + j ) qui sont complexes molCculaires 1 : 1 ou 1 : 1 et 1 : 2 ou ceux qui contiennent des composants dont ferent beaucoup et pour lesquels il n'y a que des donnCes fragmentaires (y:, y,'; x, = 0,5) ou de disponibles.

les donnees relatives aux connus pour contenir des les volumes molaires dif(yp"; y:, x, = 0,5) ou (yp") [Traduit par la rkdaction]

Introduction A binary (i + j ) mixture is characterized by the replacement of like contacts in the pure components by unlike i-j contacts in the mixture. The activity coefficient of component i in such a mixture would then depend primarily on the i-j interactions and can, in principle; be computed (1) from the conventional molar excess Gibbs free energy G E , provided a reliable equation of state is available for the mixture. The development (2) of the equation of state for a mixture via its canonical partition function involves the use of assumptions of doubtful validity (3-6) with the result that we are still far away (lb) from a general theory of liquid mixtures. Activity coefficients of the components of a binary mixture have, however, been evaluated from the compositional dependence of its G E data, but most of the equations that relate G~ to the composition of the mixture are empirical (1) in nature. This calls for an alternative approach. Theoretical aspects of the approach and results If we consider the component i in the (i + j ) mixture then the interaction of i with j would depend directly not only on the surface fraction of j that comes into effective i-j contacts but also on the magnitude of the molar i-j interaction energy, xii The thermal energy would also influence the magnitude of the xij energy. Based on these considerations we expressed (7) the activity coefficient of i, y,, in an (i + j ) mixture by the expression [ I ] [' I ]

In yi = xijxjV,/RT CxiVi

where the standard state of i is that of pure i. The activity coefficient of the jth component in the (i j ) mixture was then evaluated (7) via the Gibbs-Duhem equation, in the form

+

[2]

the j t h components of the (i + j ) mixture have nearly the same molar volumes so that the introduction of the j t h component into the ith component does not cause drastic changes in their surroundings. This assumption is bound to fail if the components of the binary mixture differ in their molar volumes. In such a situation the activity coefficient of component i would be determined not only by the i-j interactions but also by the work done in accommodating the jth component into the matrix of the ith component. The introduction of the jth component into the matrix of the ith component would require that an appropriate cavity be first created in the matrix of i. This would then bring the jth component into i-j contact with the surrounding i molecules. The work necessary to accommodate the j t h component into the matrix of the ith component would then be proportional not only to the difference (Vi - V,) in the molar volumes of i and j but also to the interaction of the jth component with the i components in the cavity. The interaction of the jth component with the surrounding i components in the cavity would evidently be dictated by the mole fraction xi of i so that if S, is the surface fraction of j that is brought into i-j contact with the surrounding i molecules then the work due to i-j interactions between j and i components in the cavity will be proportional to xisj. The net work done, G;, to accommodate the jth component into the matrix of the ith component over and above the i-i or j-j interactions could then be exxiSj(Vi - V,). But Sj = x,Vj/CxiVi (8), so pressed by G; that G; would be given by

[

In yj = (xijV,/RTVi) In - - xi V,/CxiVi

I

In deriving eqs. [ l ] and [2] it was asumed that the ith and Author to whom correspondence may be addressed. 2 ~ e v i s i o received n January 23, 1992.

where p is a const:ant. The activity coefficients of i and j in :, due to this effect, would then be given an (i j) by 141 RT(ln y i ) ~= l a ( n T ~ ~ > / a n i l n , . T . p +

and [51

RT(ln yj)w

=

[a(n~G;)/anjl,,,,~,~

If the effects due to i-j interactions above (neglecting the difference in molar volumes of i and j ) (eq. [I]) and that due to the difference in the molar volumes of i and j are assumed to make independent contributions to the activity

1632 coefficients of the components of a binary (i then In y, and In yj would be given by

CAN. J. CHEM.

+ j ) mixture,

and

Substitution of eq. [3] into eqs. [4] and [5] then yields (in view of eqs. [6] and [7])

and

It thus appears that if the two variables x,, and P of a binary (i + j ) mixture are evaluated from eqs. [8] and [9] using the activity coefficient data of its components at a single composition, then it should be possible to evaluate the activity coefficient data of these components of the binary mixture at any other composition. Such (y,, y,; x,) data3 at various x, values for some-selected (i + i ) mixtures are recorded in Table I and are also compared with their corresponding experimental (yr, yj') values (9- 13). Examination of Table I clearly shows that such (y,, yi; xi) data for the components of the various (i + j ) mixtures compare well with their corresponding experimental (y;, y:; xi) values, even for those (i + j ) mixture that contain I : 1 or 1 : 1 and 1 :2 molecular complexes though no direct cognizance has been taken of them in deriving eqs. [8] and [9]. The success of these expressions in reproducing well the (yyF, y;; xi) data of the components of these mixtures may be ascribed to the fact that the use of the (yyF, y;; xi = 0.5) data to evaluate them using eqs. [8] and [9] takes sufficient account of them. Alternatively, as xi + 0, xj + 1 so that the activity coefficient of i at infinite dilution in j would be given (in view of eq. [81) by The activity coefficients of the ith and the jth components of an (i + j ) mixture can then be expressed in terms of In y: by

and

Consequently if the y, datum of i at any x, is avaialble (or can be determined by appropriate experimentation) and as y: can 3~ablescomparing (yi, yj; xi) data (evaluated from the experimental (y;, y,'; xi = 0.5), (yy; y:, X, = 0.5), and (yy; yi = 1, xi = 0.9999) data) with their corresponding experimental (y:, y,'; x,) data for some 30 mixturees may be purchased from: The Depository of Unpublished Data, Document Delivery CISTI, National Research Council Canada, Ottawa, Canada K I A 0S2.

be determined easily (14, 15) by gas-liquid chromatography, eqs. [ l 1 ] and [12] should provide a convenient method to predict yi and yj data of both the components of an (i + j ) mixture at any xi value. (The functional form of eq. [9] is such that it is not possible to derive an expression for In yJm analogous to In y". Since it is the usual practice to report yi and yj data as functions of xi, ym has to first be evaluated for the mixture by either plotting the experimental y, data (9-13) for the various + j ) mixtures as functions of xi and then interpolating them to infinite dilution or from the compositional dependence of G~ data utilizing the reported compositional parameters. Such yy data for an (i + j ) mixture can next be combined with (y,; xi = 0.5, arbitrary composition) data to calculate (via eqs. [lo] and [ l 11) not only its x,; and p+ parameters but also the y, and yj data for both compp nents of the mixture at various other xi values. Such (y,, yj; xi) data for some selected (i + j ) mixtures are reported (as y', y;; xi) in Table 1 and are also compared with their corresponding experimental values (9- 13). Examination of Table 1 shows that such (y:, y; xi) data for an (i + j ) mixture compare as closely with the corresponding experimental (yr, yj'; xi) data as do those evaluated for it from the (yr, y;; xi = 0.5) data via eqs. [8] and [9], even for those binary mixtures that are known to be characterized by the presence of either 1 : I or I : 1 and 1 :2 molecular complexes. The success of eqs. [ l I] and [I 21 to reproduce the experimental activity coefficient data may again be traced to the same reasons as were advanced to deduce them from (yr, yj'; xi = 0.5) data using eqs. [8] and [9]. On the other hand, since the standard state of i in eq. [ I I has been taken (7) to be that of the pure i, i.e., y, = 1 at xi = 1, it may still be possible to predict (y,, yj; xi) data via eqs. [lo]-[I21 for an (i + j ) mixture when its yy datum alone is available or can be determined experimentally by gasliquid chromatography. To test this hypothesis we utilized (y;) with yi = I at xi = 0.9999 (the functional form of eq. [I 1] is such that we have to make this assumption about the y, value for an (i + j ) mixture) to evaluate its xjj and p parameters (via eqs. [lo] and [I I]), which were subsequently used to predict its (y,, yj; xi) data using eqs. [l 11 and [12]. Such (yi, y,; xi) data for some selected mixtures are recorded (as y,+, y,+) in Table I and are also compared with their corresponding experimental values, and it is apparent from Table I that such (y*, y?; xi) data compare well with the corresponding experimental data for those (i j ) mixturees that are not characterized by specific interactions between their components. Examination of Table 1 further reveals that, barring the two extreme ends of the composition scale, the (yT, y,+; xi) data calculated from eqs. [ 101-[12] compare reasonably well with the corresponding experimental (yr, yj'; x,) data even for those (i + j ) mixture that are known to contain 1 : 1 or I : I and 1:2 molecular complexes. Again, if the boiling temperatures of i and j in an (i + j ) mixture do not differ significantly, the xii and Pii parameters may be taken, for practical applications, to be independent of temperature; the effect of pressure on liquid phase properties is usually small except at higher pressures and at conditions near the critical temperature. In that event, the Vi and V, data at any temperature T coupled with the (y;, yj'; xi = 0,5) or y" yyF; xi = 0.5) or (y;; yr = 1, xi = 0.9999) data at moderate pressures should be expected to yield a reasonably good estimate of the activity coefficient data of the

+

1633

SINGH AND MAKEN

TABLE 1. Comparison of (y,, y,, x,) data (evaluated from eqs. [8]-[12] (see text) utilizing (y:, y,'; x, y,

=

1, x,

=

0.9999) data) with their corresponding experimental data for the components of (i responding x,,, etc. and P, etc. parameters.

=

0.5), (y:; y:, x,

=

0.5), and (y:;

+ j ) mixtures; also included are the cor-

Mole fraction I, xv

Property

0.1

0.2

0.3

0.4

Cyclooctane (i)

0.6

Pyridine (i) 0.6092 0.8883 0.6092 0.8890 0.6107 0.8900 0.5661 0.9008

1.0463 1.1141 1.0508 1.1168 1.0516 1.1125 1.0473 1.1 133

0.9

cal mol-'

P

7

cal cm-'

1.0265 1.1552 1.2087 1.1618 1.0298 1.1566 1.0262 1.1563

1.0102 1.2071 1.0131 1.2164 1.0142 1.2110 1.0115 1.2076

1.0032 1.2654 1.0036 1.2831 1.0043 1.2804 1.0028 1.2675

+ chloroform ( j ) , T = 303.15 K 0.8996 0.6517 0.9062 0.6458 0.9056 0.6480 0.8304 0.6584

0.7188 0.8127 0.7172 0.8143 0.7179 0.8159 0.6583 0.8306

0.9560 0.5827 0.9727 0.5666 0.9718 0.5689 0.9007 0.5661

0.9874 0.5297 1.0124 0.5035 1.01 14 0.5056 0.9546 0.4756

0.9989 0.4978 1.0218 0.4826 1.0211 0.4835 0.9884 0.3903

+ chloroform ( j ) , T = 323.15 K

Acetone (i) 0.6938 0.9235 0.6890 0.9285 0.6810 0.9241 0.6529 0.9304

0.8

+ 2,3-dimethylbutane ( j ) , T = 288.15 K

1.1145 1.0483 1.1 169 1.0507 1.1 157 1.0486 1.1111 1.0495

1.1573 1.0272 1.1625 1.0283 1.1594 1.0271 1.1552 1.0277

0.7

0.8993 0.7494 0.9057 0.7428 0.9089 0.7305 0.8639 0.7386

0.7721 0.8719 0.7678 0.8760 0.7639 0.8687 0.7273 0.8778

Chloroform (i) 0.5306 0.9632 0.5473 0.9401 0.5769 0.9613 0.6818 0.9357

0.5918 0.9076 0.6061 0.8896 0.6202 0.9244 0.7523 0.8874

0.9331 1.3212 1.8142 1.3788 1.7222 1.2848 1.3996 1.2925

1.4992 1.5130 1.4549 1.5504 1.4310 1.4179 1.2007 1.4018

+

0.9760 0.6203 0.9918 0.6031 0.9972 0.5912 0.9629 0.5729

0.9942 0.5587 1.8068 0.5566 1.0105 0.5516 0.9904 0.4881

1,4-dioxane ( j ) , T = 303.15 K 0.7542 0.7074 0.7323 0.7335 0.7210 0.7913 0.8786. 0.7596

n-Pronanol (i) 1.1402 1.9725 1.1621 1.9277 1 .I700 1.7259 1.0509 1.5914

components of the mixture at any composition. Since the boiling temperatures for the components of the n-propanol (i) + water ( j ) mixture vary from 87.8 to 100°C at a constant pressure of 1.013 bar (1 bar = 100 kPa), it would be an ideal mixture to test the validity of this hypothesis. For this purpose we utilized density data (16, 17) at 298.15 K to evaluate Vi and V, data for the n-propanol (i) water ( j ) mixture. These values of Vi and V, were next coupled with

+

0.9444 0.6844 0.9571 0.6705 0.9622 0.6573 0.9197 0.6575 0.8426 0.5753 0.7985 0.6238 0.7799 0.6829 0.9290 0.6848

0.9218 0.4388 0.8658 0.4876 0.8454 0.5336 0.9674 0.6063

0.9786 0.3319 0.9333 0.3131 0.9184 0.3262 0.9917 0.5269

1.0253 2.5077 1.0510 2.4346 1.0582 2.1842 1.0084 1.7448

1.0055 2.7963 1.0210 2.8802 1.0249 2.6313 1.0018 1.8103

+ water ( j ) 1.0662 .2.2322 1.0951 2.1518 1.1041 1.9217 1.0230 1.6722

the (yp, y,"; xi = 0.5) or (y" yyF, xi = 0.5) or (y" yi = 1, xi = 0.9999) data (in the manner explained above) to yield activity coefficient (y,, y,; xi) or (y', y;: xi) or (y*, y;*; xi) data. Such data are recorded in Table 1 , and are also compared with the corresponding experimental data (13). It is apparent that the activity coefficient data evaluated by either of the present approaches reproduce reasonably well the corresponding experimental activity coefficient data.

1634

CAN. J. CHEM. VOL. 70. 1992

The present study thus provides, for the first time, a very simple method that successfully reproduces the experimental (yf, y;; xi) data for the components of an ( i + j) mixture when only limited (y;, yj'; xi = 0.5) or (y" y:, xi = 0.5) or (y;; yi = 1, xi = 0.9999) data are available.

Acknowledgements The authors express their thanks to the Head of the Chemistry Department for providing the necessary research facilities. S.M. thanks the Council for Scientific and Industrial Research (CSIR), New Delhi, for the award of a Research Associateship. 1. J. M. Prausnitz, R. N. Lichtenthaler, and E. Gomez de Azevedo. In Molecular thermodynamics of fluid phase equilibria. 2nd ed. Prentice-Hall, Englewood Cliffs, N.J. 1986. (a)p. 291; (b) p. 361. 2. T. L. Hill. In An introduction to statistical thermodynamics. Addison-Wesley, Reading, Mass. 1976. 3. G. C. Maitland, M. Rigby, and W. A. Wakeham. In Intermolecular forces. Clarendon, Oxford. 198 1. p. 5 19. 4. J. W. Leland, J. S. Rowlinson, and G. A. Sather. Trans. Faraday Soc. 64, 1447 (1968).

5. G. H. Hudson and J. C. McCoubrey. Trans. Faraday Soc. 56, 761 (1960). 6. B. E. F. Fender and G. D. Halsey. J. Chem. Phys. 3, 1881 (1962). 7. P. P. Singh, H. P. Dahiya, and S. Dager. Z. Phys. Chem. (Leipzig), 269, 8 17 (1988). 8. M. L. Huggins. J . Phys. Chem. 74, 371 (1970). 9. M. B. Ewing and K. N. Marsh. J. Chem. Thermodyn. 2 , 6 8 9 (1970). 10. S. M. Byer, R. E. Gibbs, and H. C. van Ness. A.I.Ch. E. J. 19, 238 (1973). 11. M. M. Abbott and H. C. van Ness. A.I.Ch. E.J. 21, 62 (1975). 12. W. H. Severns, A. Sesonske, R. H. Perry, and R. L. Pigford. A.I.Ch. E.J. 1 , 401 (1955). 13. T. A. Gadwa. Dissertation, Massachusetts Institute of Technology. 1936. Quoted by H. C. Carlson and A. P. Colburn. Ind. Eng. Chem. 34, 581 (1942). 14. H. Hackenburg and A. P. Schmidt. In Gas chromatographic head space analysis. Heydon and Sons, Ltd., London. 1976. 15. T. G. Kiechbusch and C. J. King. Chromatogr. Sci. 17, 273 (1979). 16. R. F. Brunel. J. Am. Chem. Soc. 45, 1334 (1923). 17. P. Chappuis. Trav. Mem. Bur. Int. Poids Mes. 13, D-40 (1907). '