J. Chem. SOC., Faraday Trans. 1, 1986, 82, 69-75 An Application of the Gibbs-Duhem Relation to the Binary Mixed Electrolytes Tiong-Koon Lim Chemistry Department, University of Malaya, Kuala Lumpur 22- 11, Malaysia Numerical calculations of the activity coefficients of CoCl, in the binary mixture HCl-CoC1,-H,O at 25 "C using different methods are reported. From the results it is found that the recently derived general Gibbs-Duhem relation is useful in computing the activity coefficients of one electrolyte from another electrolyte whose activity coefficients can be experimentally fitted. For binary mixed electrolytes with a common ion it is well known that the activity coefficients can only be determined experimentally for one of its e1ectrolytes.l The cross-differential condition as required by the Gibbs-Duhem equation prevents the Harned coefficients being adjusted independently to give the best fit with experimental data. Theoretical methods are therefore required for the calculation of the activity coefficients of the other electrolyte which were not experimentally fitted. McKay2 was able to calculate Harned's coefficients for an electrolyte, using the Gibbs-Duhem condition, when it is known that Harned's rule3 applies to the other electrolyte.In another more convenient approach, Scatchard4 and Pitzer5y expressed activity coeffi- cients of both electrolytes in terms of the same set of parameters, so that the activity coefficients of both electrolytes can be calculated simultaneously from fitting the experimental activity coefficients of one of the electrolytes only.The same approach has also been used by Lim et al.'? to express activity coefficients of both electrolytes in terms of the mixing coefficients of A,Gex with an additional requirement that the higher-order limiting law9 must be satisfied. Recently'O it has been shown that the cross-differential condition leads to a general Gibbs-Duhem relation which interrelates the Harned coefficients of the two electrolytes; this enables the Harned coefficients of one electrolyte to be computed from the Harned coefficients of the other electrolyte. This paper produces some numerical results for this new method and compares the results with those obtained from the other methods. The Methods The activity coefficients of a binary electrolyte mixture with a common negative ion can be given by the Harned equation:ll or the following equivalent equations :' + M$' g ; if( YI)" +;( YI)?z+l n-o 6970 Gibbs-Duhern Equation for Binary Mixed Electrolytes + Mi1 g; (; (YI).-g YI)n+l . (4) n -o )I Here we assume that electrolyte A contains vA+ ion 1 and v, ion 3 with valency charges z , and z3, respectively, while similarly electrolyte B contains v$ ion 2 and vg ion 3 with charges z2 and z,, and vA = vA+ + v ~ etc. I is the total ionic strength, y is the fraction of the ionic strength due to electrolyte B, and Y = 1 -2y. M is the order of the equation and is taken as 2 throughout this paper. The coefficients gn are the mixing coefficients in the expression of the changes of excess Gibbs free energies, their statistical-mechanical expressions have been given elsewhere ; 8 1 their expressions in some semi-empirical methods are given as follows.In Pitzer's method, we have whereas in the Scatchard method they are defined by In the higher-order limiting-law (HOLL) method these parameters were forced to satisfy the higher-order limiting law and the consistency condition so that go = A In ~ + p ~ + v f i g , = constant. (7) Since g, is necessarily an I-independent constant in order to satisfy the Gibbs-Duhem cross-differential condition,l0 either the Pitzer parameters CMx, C,, and vMNX must be I-independent, as is usually assumed in the practical calculations, or their sum [which equals g, as defined in eqn (5)] must be a constant. In the above methods the Harned coefficients for both electrolytes A and B were expressed by the same set of parameters, so that both aAn and aBn can be obtained simultaneously from fitting experimental logy, data only. Recently a new method was proposed from which aBn can be obtained from the experimentally fitted aAn coefficients according to a general relation as required by the Gibbs-Duhem cross-differential condition.lO In this method aRn are related to a,, as follows: whereT- K .Lirn 71 Therefore, after all the aAn coefficients were obtained from fitting the experimental log yA values, daA,/dI can be computed, and a,, can be calculated from eqn (8) successively beginning with n = 1. In the calculation of the derivativesf(1) of a function f(1), a three-point Lagrangian formula may be used for all the intermediate ionic strengths I, : where whereas for the first and the last ionic strengths the usual two-point Newtonian formula can be applied.The above procedure of successive evaluation of a,, results in the following explicit relation : n=o r=j-n k=1 where ETo = r F$=O if r < O and D(lk) 1 The principal aim of this paper is to check the reliability of our a,, coefficients as computed from the Gibbs-Duhem general relation, eqn (8). The results thus obtained can be used to compute logy, and then compared with those calculated from other methods. The mixture being tested is HCl-CoCI, at 25 "C for I = 0.01, 0.03, 0.05, 0.08, 0.1, 0.5, 1, 2 and 3 mol kg-l. Experimental activity coefficients of HCl for this mixture have been reported elsewhere, and the calculated results using the Scatchard method, the Pitzer method and the HOLL method have also been r e p ~ r t e d .~ ? ~ ~ The calculated4 td Table 1. Calculated values of -log yB from various methodsa for the mixture HC1-CoC1,-H,O at 25 "C - ~- - Y R Scatchard Pitzer HOLL present average 0.0 0.064 371 0.130 609 0.248 701 0.368 853 0.498 856 0.625 563 0.753 900 0.0 0.080 617 0.126 678 0.269 531 0.366 31 1 0.495 172 0.631 279 0.758 658 0.878 071 0.935 600 0.0 0.063 189 0.154 162 0.226 463 0.376 733 0.500 640 0.610 933 0.758 761 0.897 366 0.087 03 (-0.4) 0.087 03 (-0.5) 0.087 02 (-0.4) 0.087 02 (-0.4) 0.087 02 (-0.5) 0.087 01 (-0.5) 0.087 01 (-0.5) 0.087 00 (-0.6) 0.135 7 (- 1.7) 0.135 7 (- 1.9) 0.135 7 (-2.1) 0.135 7 (-2.4) 0.135 8 (-2.4) 0.135 8 (-2.5) 0.135 9 (-2.3) 0.135 9 (-2.1) 0.135 9 (- 1.8) 0.136 0 (- 1.5) 0.163 4 (-8.4) 0.163 5 (-7.5) 0.163 6 (-6.4) 0.163 6 (-5.6) 0.163 8 (-4.2) 0.163 9 (-3.3) 0.164 0 (-2.6) 0.164 l (-2.0) 0.164 2 (- 1.7) I = 0.01 mol kg-l 0.090 26 (2.8) 0.090 08 (2.7) 0.089 89 (2.5) 0.089 59 (2.1) 0.087 74 (0.3) 0.087 68 (0.3) 0.087 62 (0.2) 0.087 51 (0.1) 0.089 31 (1.8) 0.089 04 (1.5) 0.088 82 (1.3) 0.088 62 (1.0) 0.087 42 (- 0.1) 0.087 32 (-0.2) 0.087 23 (-0.3) 0.087 14 (-0.4) I = 0.03 mol kg-l 0.141 2 (3.9) 0.137 l (-0.3) 0.140 8 (3.2) 0.137 0 (-0.6) 0.140 6 (2.8) 0.136 9 (-0.9) 0.140 1 (2.0) 0.136 7 (- 1.4) 0.139 7 (1.5) 0.136 5 (- 1.7) 0.139 3 (1.1) 0.136 4 (- 1.9) 0.139 0 (0.8) 0.136 2 (-2) 0.138 8 (0.8) 0.136 1(-1.9) 0.138 6 (1.0) 0.1360(-1.7) 0.138 6 (1.1) 0.1360(-1.5) I = 0.05 mol kg-' 0.170 2 (- 1.6) 0.165 3 (-6.5) 0.169 8 (- 1.2) 0.165 2 (-5.8) 0.169 4 (-0.6) 0.165 0 (-5.0) 0.169 0 (-0.2) 0.164 9 (-4.3) 0.168 5 (0.5) 0.164 7 (-3.3) 0.168 l ( l .0 ) 0.164 6 (-2.6) 0.167 9 (1.3) 0.164 5 (-2.1) 0.167 6 (1.5) 0.164 4 (- 1.7) 0.167 5 (1.6) 0.164 4 (- 1.5) 0.084 63 (-2.8) 0.084 91 (-2.5) 0.085 20 (- 2.2) 0.085 69 (- 1.8) 0.086 17 (- 1.3) 0.086 67 ( - 0.8) 0.087 13 (- 0.4) 0.087 57 (-0.01) 0.135 4 (-2) 0.137 0 (-0.6) 0.137 9 (0.1) 0.139 9 (0.8) 0.140 8 (2.6) 0.141 5 (3.3) 0.141 6 (3.4) 0.141 l(3.1) 0.140 1 (2.5) 0.139 4 (1.9) 0.188 l(16) 0.185 4 (14) 0.181 9 (12) 0.179 4 (10) 0.174 9 (6.9) 0.172 0 (4.9) 0.170 0 (3.4) 0.168 2 (2.1) 0.167 5 (1.6) 0.087 4 0.087 4 0.087 4 0.087 4 0.087 5 0.087 5 0.087 6 0.087 6 0.137 4 0.137 6 0.137 8 0.138 1 0.138 2 0.138 3 0.138 2 0.138 0 0.137 7 0.137 5 0.171 8 0.171 0 0.170 0 0.169 2 0.168 0 0.167 2 0.166 6 0.166 1 0.165 9T-K.Lim 73 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 z oo 8 II 4 0 E 0 II d v! II n 0 3 II +-4 P G F b YB Scatchard Pitzer HOLL present a1:erage x $ 3 h 0.177 1 (6.5) 0.171 8(1.2) 0.180 8 (10) 0.152 9 (- 18) 0.170 7 9 !i. 0.218 5 (- 10) 0.228 5 0 0.681 635 0.272 3 (1.7) 0.2718 (1.2) 0.273 2 (2.6) 0.265 3 (- 5.4) 0.270 7 3 0.286 9 (- 3.4) 0.2928 ? 0.303 5 bcr 0.0 0.037 2 (22) 0.036 1 (20) 0.046 8 (3 1 ) -0.057 6 (- 73) 0.015 7 G 0.121 202 0.063 6 (1 6) 0.063 8 (1 7) 0.068 9 (22) -0.007 3 (-55) 0.047 3 % 0.390 199 0.124 4 (7.9) 0.123 8 (7.3) 0.123 4 (6.9) 0.094 3 (-22) 0.1 16 5 g- 0.637 729 0.183 4 (2.2) 0.185 4 (4.2) 0.180 3 (-0.9) 0.175 7 (-5.5) 0.181 2 a h % 2 2 2 Table 1.(cont.) I ~~ ____ ___ ~- _ _ ~~ ~~~ ~- ______________~__ - I = 2 mol kg-' x 0.0 0.135 548 0.194 6 (5.2) 0.191 O(1.8) 0.197 7 (8.3) 0.174 4 (- 15) 0.189 4 0.405 646 0.231 7 (3.2) 0.230 2 (1.7) 0.233 6 (5.1) 0.805 475 0.291 4 (1.1) 0.290 9 (0.6) 0.291 9 (1.6) 0.887 185 0.304 3 (0.8) 0.303 7 (0.2) 0.304 6 (1.1) 0.301 3 (-2.2) 2. I = 3 mol kg-' Q 0.785 655 0.220 1 (0.7) 0.221 3 (1.9) 0.217 3 (-2.1) 0.218 9 (-0.5) 0.219 4 0.247 4 (0.4) 0.247 6 (0.6) 0.245 6 (- 1.5) 0.247 6 (0.6) 0.247 1 0.892 980 ~~ ~~ ~ ~ ______-- ~~ ~ ~ ~~ a CD were computed from the Scatchard parameters for single electrolytes in the HOLL method; M = 2 for all methods. 2T-K. Lim 75 log yB results as obtained from the present method using general Gibbs-Duhem relation are listed in table 1 together with those calculated from other methods.The average values of the four methods are shown in the last column of table 1. We multiply the difference between a calculated value and the average value by a factor of lo3 and placed it inside parentheses following the calculated value. From these values we know that the values of log yB calculated from the present method are just as good as those obtained from other methods, except at high concentrations. The disagreements at high concentrations are probably due to the following reasons. (a) At high concentration the interaction between ions are larger and thus the higher-order parameters with M 3 3 may be significant and cannot be neglected; this requires g , to be I-dependent at higher concentrations.(b) The separations between two ionic strengths are larger; this makes the calculation of the derivatives less accurate. Furthermore, we also observe that the agreements between various methods in the calculation of log yH are not as good as in the calculation of log yA. However, if we study the variation of the values of -log yR from large y , to small yB at constant I, we know that Pitzer’s method behaves similarly to the HOLL method, while the present work and the average values are also similar, with Scatchard’s method different from all the rest. Nevertheless, if we consider the changes from high I to low I then these variations in all the methods seem to follow a same pattern: generally all the methods show decrease at high I ; then whcn I falls, the present method shows a change from a decrease to an increase at I = 0.5, whereas the Pitzer method and the HOLL method show this change at I = 0.1, and the Scatchard method at I = 0.01.When I is further reduced, a second change is observed at I = 0.08 for all the methods except that of Scatchard; this suggests that this second change may exist in the Scatchard method only at a lower I value. There is also a change from an increase to a decrease at I = 0.03 for the present work and the average values; this change can probably be observed in the other methods only at very small I. Other mixtures have also been tested, but since the results are similar to those given here they will not be reported. We may therefore conclude that the Gibbs-Duhem relation, eqn (S), is useful in calculating aBn from the experimentally fitted aAn.However, at higher concentrations higher-order parameters and/or a closer separation between two ionic strengths may be needed. References 1 R. A. Robinson and R. H. Stokes, Electrolyte Solutions (Butterworths, London, 2nd edn, 1959). 2 H. A. C. McKay, Trans. Faraday Soc., 1955, 51, 903; Discuss. Faraday SOC., 1957, 24, 76. 3 H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolyte Solutions (Reinhold, New York, 4 G. Scatchard, J . Am. Chem. Soc., 1961, 83, 2636; 1968, 90, 3124. 5 K. S. Pitzer, J . Phys. Chem., 1973, 77, 268. 6 K. S. Pitzer, in Activity Coeficients in Electrolyte Solutions, ed. M. R. Pytkowicz (C.R.C. Press, Boca 7 T-K. Lim, C. Y. Chan and K. H. Khoo, J . Solution Chem., 1980,9, 507. 8 T-K. Lim, Int. J. Quantum Chem., 16, 247. 9 H. L. Friedman, Ionic Solution Theory (Butterworths, London, 2nd edn. 1959); H. L. Friedman and 3rd edn, 1958). Raton, Florida, 1979), chap. 7. C. V. Krishnan, J . Phys. Chem., 1974, 78, 1929. 10 T-K. Lim, J . Chem. SOC., Faraday Trans I , 1985, 81, 1195. 11 R. H. Harned and R. A. Robinson, Multicomponent Electrolyte Solutions (Pergamon, Oxford, 1968). 12 K. H. Khoo, T. K. Lim and C. Y. Chan, J . Chem. SOC., Faraday Trans. I , 1978, 74, 2037. Paper 5/279; Receiued 18th February, 1985