J . Phys. Chem. 1990, 94, 7321-7325 0, the result of Fourier and Laplace transformations of the equations of motion are
7321
To specify the solution completely, one must evaluate the terms A(s) and B(s). Evaluation of these constants requires substituting (A14) into (A12) and (A13) and the solution of the resultant algebraic equations. To conserve probability when the derivatives in (A12) and (A13) are being evaluated, it is necessary to use 0- and y 1’. The result is a 4 X the limiting procedure y 4 matrix which can be easily reduced to a 2 X 2 and solved for defect terms. Hence A ( s ) and B(s) are known. The solution of (A10) can be. written in a form that is similar to the solution of (A9). To proceed, we write q2in terms of a Green function that obeys the boundary conditions. Defining the analogous terms
-
where k is the variable conjugate toy and s is the Laplace variable conjugate to t . We have introduced new variables scaled with respect to the diffusion constant:
a
C(s) = -*,(O,s) ax
a
D(s) = - \ k 2 ( 1 , ~ ) ax
We have also defined Q = g + s and A = s + y. Equation A9 is completely uncoupled from (A10) so that its solution is easily found by Fourier inversion. To make the algebra a bit more transparent, we define the following quantities:
a
A ( s ) = -‘€‘~(O,S) ax
a
B(s) = -*,(l,s) ax
+ 2pC\ki(O,~)
(Ala
+ 2p*,(l,s)
(A131
The solution for the probability density in the first state is simply
-
+ (2p - B)*z(O,s)
(‘416)
+ (2p - F)\k2(l,s)
(A17)
F2Cv-y’~)= G 2 b - y ’ ~ )+ G~(,v,s)C(S) - G2(-V-l,S)D(S)
(A18)
we write the inversion of \k2 as *2Cv,s) = ~ 0I W Y - Y ’ J )*,Cv’,s)
dy’
(A191
To completely specify F2, we must find the constants C(s) and D(s). The result is another 2 X 2 matrix whose solution is easily found. The integral in (A19) is a bit tedious to perform for arbitrary y . However, the observable of interest is the quantum yield which according to (6) is obtained from (A19) by setting y = 1. This considerably simplifies the integration in (A19), and the result is given in the text as (8)-( 11). Registry No. Ht,12408-02-5; cytochrome oxidase, 9001-16-5.
Thermodynamics of the Unsymmetrical Mixed Electrolyte HCi-NiCI,. Pitzer’s Equations
Application of
Rabindra N. Roy,* Lakshmi N. Roy, Greg D. Farwell, Kristy A. Smith, Department of Chemistry, Drury College, Springfield, Missouri 65802
and Frank J. Millero Rosenstiel School of Marine and A tmospheric Science, University of Miami, Miami, Florida 331 49 (Received: December 18, 1989; I n Final Form: April 19, 1990)
Emf measurements were performed on solutions at temperatures from 278.15 to 3 18.15 K and at eight different ionic strengths from 0.1 to 3.0 mol kg-’, by using a Harned’s emf cell without liquid junction of the type Pt;H2(g, 1 atm)lHCl(m!),NiC12(m2))AgCI{AJ.From these measurements, the Pitzer mixing coefficients%H,Ni and J.H,Ni,,-i as well as linear representation of temperature derivatives bSeH,Ni/6T and 6 + H , N i , a / 6 T have been determined. The results are interpreted in terms of simple Harned’s rule and Pitzer’s formalism for mixed electrolyte solutions. The activity coefficients at ionic strength fraction y of NiClz at 298.15 are given.
Introduction There has been a great deal of interest, both theoretical and experimental, in recent years in the thermodynamic properties of aqueous binary and ternary electrolyte solutions especially at high concentration^.'-^ In view of the great importance of ~~
_____~________
( 1) Pitzer, K. S. In Aciiuiiy Coefficients in Elecirolyie Soluiions; Pytkowicz, R. M., Ed.; CRC Press: Boca Raton, FL, Vol. 111, Chapter 3, submitted for publication; also 1979; Vol. I, Chapter 3.
0022-3654/90/2094-7321$02.S0/0
Ni2+(aq) in hydrometallurgy, seawater, geochemical calculations in brines, pulp and paper chemistry, and other solutions of practical (2) Harvic, C. E.; Moller, N.; Weare, J. H. Geochim. Cosmochim. Acra
1984, 48, 723.
(3) Millero, F. J. Thalassia Jugosl. 1982, 18, 253. (4) Roy, R. N.; Gibbons, J. J.; Peiper, J. C.: Pitzer, K. S . J . Phys. Chem. 1983,87, 2365. ( 5 ) Plummer, L. N.; Parkhurst, D. L.; Fleming, G. W.; Dunkle, S. A.; Water Resources Investigations Report, 88-441 53; U S . Geological Survey, Reston, VA, 1988.
0 1990 American Chemical Society
7322 The Journal of Physical Chemistry, Vol. 94, No. 18, 1990 TABLE I: Ion-Interaction Parameters at 298.15 K HCI p(0)
P I 04(q3(o)/an 104(ap(1)/6n I 04(ac+/an
0.1775" 0.2945" 0.0008" -3.081' 1.419c 0.6213'
Roy et al.
NiCI, 0.3479" 1.5810" -0.003 7 2" -24.861 1 1 1.94Ib 13.3S6b
(0.3930)d (0.99773)d (-0.016S8)d
" From ref 34, used in this work for the calculation of 0 and 4. *From this work. work; results are given only for comparison interest, investigations of the thermodynamic properties for the system HCI + NiC12 + H 2 0 at finite concentrations are a matter of considerable interest. As a continuation of previous studiess-12to determine the activity coefficient of a strong electrolyte (e.g., NiCI2) in admixture with another electrolyte with a common ion (e.g., HCI), we have successfully applied emf methods to the unsymmetrical system HCI NiC12 H 2 0 . The emf technique using hydrogen and silver-silver chloride cells is the most precise and is particularly useful for work of this type, since the activity coefficients of the solute in which the electrodes are reversible can be evaluated directly. There are many common-ion mixtures of industrial and geochemical brines for which precise emf and activity coefficient data for a wide temperature and the ionic strength range are not available in the literature. Of particular importance have been those investigations in which m I and m2 were varied, but subject to the condition of constant total ionic strength (I = ml 3m2). Kim and Frederick'j have recently reported the values of Pitzer's interaction parameters for NiCI, (and many other electrolytes) at 298.15 K, based on the isopiestic results of Goldberg, Nuttal, and Staples.I4 Also Khoo, Lim, and ChanI5 have published emf data for the system HCI + NiC12 + H 2 0 , but only at 298.15 K. The properties of the system, however, are not well-defined at intermediate total ionic strengths of I = 0.25, 1.5, and 2.5, and at temperatures between 273.15 and 3 18.15 K. The Pitzer parameters for Ni-H interactions are also useful in interpreting the effect of Ni2+on the ionization of weak acids such as HS03-.16 Thus, we have supplemented and extended the emf measurements at constant total ionic strengths I = 0.1, 0.25, 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 mol kg-I in the temperature range of 278.15-318.15 K. In order to treat these experimental data, there are different available in the literature. The neutral-electrolyte equations of Scatchard et al.22-23 have not been used in the present study because of complexities of computations, formulas, little promise of simple physical interpretations of five interaction parameters, etc. Instead, we have successfully applied Pitzer's f ~ r m a l i s m ' ~which - ~ ~ contains only two mixing coefficients characterizing specific types of ionic interactions: 0, for binary
+
(0.3 190)' (1.2909)' (0.0407)'
From ref 35. dFrom refs 13 and 14. eFrom ref 36. /From this
interactions of H+-NiZ+, and $ for the ternary interaction of H+-Ni2+-CI-, for the mixtures at finite concentrations. These equations (Pitzer) are simple, convenient, and a d e q ~ a t e . ~ - ~ Experimental Section
All emf readings were made with a cell of the type
+
+
(0.3499)< (1.5300)c (-0.00471 I ) e
P~;H~(~)IHCI(~I),N~CIZ(~~)IA~CI,A~ (A) using a digital voltmeter (Keithley Model 191) and were precise to within fO.l mV or better from I = 0.1 to 3.0 mol kg-' at 278.15, 288.15, 298.15, 308.15, and 318.15 K. The emf was recorded at every 5 min until equilibrium was attained (the emf remained constant to within f0.08 mV after 30 min at ionic strengths below 1 .O mol kg-I and about 15 min above 1.0 mol kg-I). All the cells registered stable equilibrium emf values very quickly. The emf data, corrected to a hydrogen partial pressure of 1 atm (101.325 kPa), are reported in Table I . The silver-silver chloride electrodes were of the thermal electrolyte type.24 The cells were of the all-glass type described earlier25,26which provides a triple saturator for the hydrogen gas. The purification of the hydrogen gas, preparation of cell solutions, temperature control (fO.O1 K), the poisoning effect or irreversible behavior of the hydrogen electrode at high ionic strength, and other experimental details have been described p r e v i o ~ s l y . ' ~The -~~~~~ standard emf of the cell A was derived by using the usual standardization procedure (measurements of the emf of HCI solutions of molality 0.01 mol kg-I and molality of NiC12 = 0 mol kg-1).28 The standard potentials thus obtained were in excellent agreement with previously published values.29 Nickel chloride (ACS certified reagent grade) was recrystallized twice from water. Hydrochloric acid of reagent grade was diluted to the azeotropic composition and distilled twice, the middle third being retained. The molalities of the stock HCI and NiCI, solutions were determined by gravimetric chloride analysis. Triplicate analyses agreed to within f0.01 and f0.02% for HCI and NiCI2, respectively. Bouyancy corrections were applied to all weighings of the stock solutions. Theory and Equations
( 6 ) Roy, R. N.; Zhang, J. Z.; Millero, F. J. J . Solution Chem. Manuscript under preparation. (7) Roy, R. N . ; Gibbons, J. J.; Williams, R.; Baker, G.; Simonson, J. M.; Pitzer, K. S.; J . Chem. Thermodyn. 1984, 16, 303. (8) Roy, R. N.; Gibbons, J. J.; Roy, L. N.; Greene, M. A. J . Phys. Chem. 1986. 90, 6242. (9) Ananthaswamy, J.; Atkinson, G. J . Chem. Eng. Dura 1985, 30, 120. (IO) Robinson, R. A.; Roy, R. N.; Bates, R. G.; J . Solution Chem. 1974, 3, 837. ( I I ) Roy, R. N.; Wood, M. D.; Johnson, D.: Roy, L. N.; Gibbons, J. J.; J . Chem. Thermodyn. 1987, 19, 307. (12) Millero, F. J.; Schreiber, D. Am. J . Sci. 1982, 282, 1508. (13) Kim, H. T.; Frederick, W. J. J . Chem. Eng. Data 1988, 33, 177,278. (14) Goldberg, R. N.; Nuttall, R. L.: Staples, 8 . R. J . Phys. Ref. Data 1979, 8, 923. (15) Khoo, K. H.; Lim, T. K.; Chan, C. Y. J . Solution Chem. 1978,7,291. (16) Millero, F. J.; Zhang, J . Z.; Sibbles, M. A.; Roy, R. N. J . Solufion Chem. Manuscript under preparation. (17) Krumgalz, B. S.; Millero, F. J. Marine Chem. 1982, / I , 209. ( 1 8 ) Friedman, H . L. Ionic Solution Theory; Interscience: New York, 1962. (19) Pitzer, K. S. J . Phys. Chem. 1973, 77, 268. (20) Pitzer, K . S.; Kim, J. J. J . Am. Chem. SOC.1974, 96, 5701. (21) Pitzer, K . S. J . Solution Chem. 1975, 4, 249. (22) Scatchard, G . J . Am. Chem. Soc., 1961, 83, 2636. (23) Rush, R. hl.;Johnson, J. S. J . Phys. Chem. 1968, 72, 767.
The emf of cell A is given by the Nernst equation E = EO(Ag,AgCI) In
F
[
m(H+)r(H+y-)r(CI-)
(
(I)
where the values of E are the corrected emf of cell A and E O (Ag,AgCl) is the standard potential for the silver-silver chloride electrode; F , R, and T retain their usual meanings. The ionic activities are given by the product m,y,,where m has dimensions (24) Bates, R. G. Deferminationof p H , 2nd ed.; Wiley: New York, 1973; p 33. (25) Bates, R. G. NBS Tech. Note (LIS.)1965, No. 271, 28. (26) Gary, R.; Bates, R. G.; Robinson, R. A. J. Phys. Chem., 1964, 68, 1186. (27) Roy, R. N.; Gibbons, J. J.; Trower, J . K.; Lee, G. A. J . Solution Chem. 1980, 9, 535. (28) Bates, R. G.; Guggenheim, E. A.; Harned, H. S . ; Ives, D. J. G.; Janz, G. J.; Monk, C. B.; Prue, J. E.; Robinson, R. A,; Stokes, R. H.: Wynne-Jones, W . F. K . J . Chem. Phys., 1956, 25, 361; 1957, 26, 222. (29) Roy, R. N.; Gibbons, J. J.; Bliss, D. P.; Baker, B.; Casebolt, R. G. J . Solution Chem. 1980, 9, 12.
Thermodynamics of the Mixed Electrolyte HC1-NiCI2
The Journal of Physical Chemistry. Vol. 94, No. 18. 1990 7323
TABLE 11: Mixing Parameters from Isothermal Fits to Data of Table I 278.15 K 288.15 K With E8 and ‘ o ~ , ~ i / ( kmol-’) g $H,Ni,Cl/(kg2 mol-2)
0.0671 f 0.0144 0.0156 f 0.0139 0.34
0.0685 f 0.0123 0.0105 f 0.0109 0.33
-0.0861 f 0.0145 0.0758 f 0.0140 0.35
-0.0795 f 0.0125 0.0631 f 0.01 11 0.34
afit/mv
Without O H . N ~ / ( ~ mol-’) $H,Ni,c1/(kg2 mol-2) afit/mv
0.0690 f 0.0149 0.0056 f 0.0123 0.35
(2)
where I = m l + 3m2. Harned’s Equations.
1% Y1 = log YIO - atzv2 (3) where ylo is the activity coefficient of hydrochloric acid in a solution of the acid along (yl= 1, y 2 = 0) at the same total ionic strength and cyl2 is the Harned coefficient, which is independent of the composition y 2 of the solution but is a function of I . The nonlinear term of the complete Harned equation is given by 1% YI = 1% YIO - a192 - P12v22
(4) Table I1 contains values of cyl2 and P12at 298.1 5 K for each of the eight ionic strengths included in this study. It is clear from the values of rmsd that hydrochloric acid follows Harned’s rule. Thus, the quadratic term in eq 4 is not warranted. (Tables of the a I 2and PI2values are available as supplementary material.) Pitzer’s Equations The ion-interaction theory of Pitzer19 and Pitzer and Kim20forms the primary basis for the calculation of the activity coefficients of HCI and NiCI2. The ion-interaction expression for the activity coefficient of HCI (component 1) appearing in eq 1 in mixtures with the salt (NiCI2, component 2) is given by In
(YHCI)
= p + ( m H + m C l ) ( B H C I + mCICHC1) + m N i ( B N i , C l +
+ OH.Ni) + mHmCl(B’HCl + cHCI) + mNimCl(B’Ni,Cl + I/2$H,Ni,CI) + mHmNi(O’H,Ni + I/2$H,Ni,CI) (5) where the expressions for p,Bij, B!,, and Cjj are given in ref 4. The effects associated with the mixing of ions of the same sign are largely incorporated into the mixing parameter OH,Nj. These effects originate from differences in short-range interactions between H+ and Ni2+ from the appropriate mean of like pairs of the same sign, that is, H+-H+, and Ni2+-Ni2+. The unsymmetrical mixing coefficient BH,Ni can be represented as “JcNi,Cl
+
8H.Ni
cNi,Cl
=
EoH,Nj
+ ‘8H.Ni;
0’H.Ni
= Eo’H,Ni
308.15
K
318.15
K
0.0700 f 0.01 10 0.0080 f 0.0094 0.34
0.0793 f 0.0116 0.0033 f 0.0101 0.34
-0.0847 f 0.0120 0.0623 f 0.0095 0.36
-0.0845 f 0.0129 0.0640 f 0.01 12 0.38
and E8’ Included
of mol kg-I, and is the ionic activity coefficient. (Tables for the corrected emf values for cell A are available as supplementary material.) The quantities mo and Po are included to ensure dimensionless correctness and have the values 1 .O mol kg-I and 1 atm (or 101.325 kPa), respectively. The physical constants were obtained from Cohen and Taylor30 and the values for Eo(Ag,AgCI) were taken from Roy et aL2’ after considering the results of Bates et a1.28 An iterative computer calculation is used for the required correction to a constant hydrogen fugacity of 1 atm since the vapor pressure of water contributes to the total pressure over the experimental cell solutions. These are easily obtained by calculating the activity of water for each solution. The corrections for the hydrogen fugacity are made based on these calculated water activities and the experimental barometric pressures. The ionic strength fraction y 2 of NiCI2 is given by Y Z = 3m2/1
298.15 K
Included
+ ’O’H,Ni (5a)
The Debye-Huckel function,p, for the activity coefficient with parameter A,, has the value31 of 0.392 kgtl2mol-1/2for water at 298.15 K. The parameter B has the standard value 1.2 kg’l2 mol-’/2. The molality of ion i is given by mi and the ionic strength I both having dimensions mol kg-I. The second and third virial (30)Cohen, E. R.;Taylor, B. N. J . f h y s . Chem. Ref. Data 1973,2,663. (31) Bradley, D. J.; Pitzer, K . S.J . Phys. Chem. 1979,83, 1599.
-0.0978 f 0.0150 0.0666 f 0.01 25 0.36
coefficients for pure single-electrolyte ij (with i and j being charged ions, where i = H+ and j = Ni2+ in the present case) are Bijand Cijand have dimensions of kg mol-I and kg2 mol-2, respectively. The second virial coefficient is dependent on ionic strength with two adjustable parameters, and @ ( I ) , . which have the units of kg mol-’; ct is assumed to be 2.0 kgl/q’mol-t/2for 1:l or 2:l electrolytes.21 The ionic charges, in protonic units, enter into the formula for the third virial coefficient as zi and Zj and relate Cjj to the parameter Cmij (originally defined for the osmotic coefficient and normally tabulated). The single-electrolyte third virial coefficients, Cmij, account for short-range interaction of ion triplets and are important only at high concentrations. The terms EOH,Ni and EO’H,Nj represent contributions from higher order electrostatic effects of unsymmetrical mixing with the omission of short-range forces and ’OH,Ni and dependence of and is expected to be very small and hence is neglected. It IS important to emphasize once again that the quantities 0, O’, and $ are properties characteristic of the mixture, whereas B, B’, and @ are properties of single-electrolyte solutions. Higher Order Electrostatic Effects. The importance of the higher order electrostatic contribution for unsymmetrical mixing was demonstrated impressively by Harvie, Moller, and WeareZ in their calculations of mineral solubility in geochemical systems. The terms involving EOH,Nj and EO’H,Ni arise from incorporation of higher-order electrostatic effects of asymmetrical mixing (that is, higher charge-type electrolyte mixtures excluding the 1-1 type). Friedmant8predicted the existence of these significant effects for unsymmetrical mixtures. Pitzer2’ derived the equations for calculations of these effects based on cluster-integral theory.j2 These equations are given elsewhere in a more convenient form.8*z1 ’O’H,Ni account largely for the effects of short-range forces, as well as the effects due to the use of molarities instead of molalities in the Debye-Huckel term. The quantity ’O’H,Ni refers to the variation of ’OH,Ni with I and was set equal to zero according to the recommendatjons of Pitzer and Kimz0 The third virial coefficient $H,Ni,cl is the mixed-electrolyte parameter representing cation-cation-anion triplet interactions in mixed-electrolyte solutions. This parameter is assumed to be independent of ionic strength. ’6‘H,Ni defines a possible ionic strength. The terms EOH,Nj and EO’H,Nj are functions of total ionic strength, the electrolyte pair type (ion charge), and temperature. For anion pairs, similar equations can be derived. The univariant functions J and its derivatives J’ given by Pitzer4*8J.32are obtained from clusterintegral theory with omission of short-range forces. Since there are not simple integrals for J and J’, a variety of numerical techniques may be used to compute these integrals with modern computers. Pitzer2’ has given convenient forms for evaluating approximate values of the expressions, J ( X ) and J’(X). However, the authors prefer a 200-point Gauss-Legendre numerical integration method as used previo~sly.~,~’ For X > 0.10, numerical methods in integral form are most precise, whereas, for X I0.10, the numerical integration is difficult. Hence, the functions J and J’ in this low region are evaluated in series form for the present calculations, the detailed descriptions and equations for which have been published in earlier paper^.^,^' I t is important to mention an alternative method for computation of these functions by H a r ~ i who e ~ ~uses two Che(32) Peiper, J. C.; Pitzer, K. S.J . Chem. Thermodyn. 1982, 14, 613. (33) Harvie, C. E. Ph.D. Dissertation, University of California, San Diego; University Microfilm Int., Ann Arbor, MI, No. AAD82-03026, 1981.
Roy et al.
1324 The Journal of Physical C h e m i s t r y , Voi. 94, No, 18, I990 0.2
TABLE 111: Activity Coefficients for the [HCl(ml) + NiClz(mz) + HzO]System et Ionic-Strength Fraction y of NiQ' I/ mol kg-'
T/K 278.15
288.15
298. I 5
308.15
0.10 0.25 0.50 I .oo 1.50 2.00 2.50 3.00 0.10 0.25 0.50 I .oo ISO 2.00 2.50 3.00 0.10 0.25 0.50 I .oo I .50 2.00 2.50 3 .OO 0.10 0.25 0.50
1 .oo 1 .50
318.15
2.00 2.50 3 .OO 0.10 0.25 0.50 1 .00 I .50 2.00 2.50 3.00
y = 0.0 0.804 0.774 0.774 0.841 0.944 1.070 1.230 1.445 0.800 0.771 0.766 0.826 0.922 1.040 1.184 1.385 0.801 0.759 0.757 0.812 0.899 1.011 1.148 1.326 0.794 0.755 0.747 0.797 0.879 1.020 1.106 1.267 0.789 0.749 0.738 0.784 0.864 0.968 1.066 1.21 I
y = 0.50 0.801 0.760 0.752 0.790 0.858 0.948 1.060 1.200 0.797 0.757 0.745 0.779 0.842 0.927 1.028 1.159 0.797 0.748 0.737 0.766 0.825 0.905 0.999 1.1 I8 0.790 0.743 0.729 0.754 0.808 0.969 1.046 1.085 0.785 0.737 0.721 0.743 0.793 0.918 0.954 1.029
y =
y r
1.0 0.798 0.746 0.730 0.742 0.781 0.840 0.914 0.997 0.794 0.742 0.725 0.734 0.769 0.826 0.892 0.970 0.793 0.736 0.718 0.723 0.756 0.811 0.870 0.942 0.787 0.732 0.712 0.714 0.743 0.824 0.849 0.908 0.781 0.725 0.704 0.703 0.729 0.773 0.838 0.874
0.0 0.642 0.583 0.567 0.609 0.697 0.823 0.997 1.162 0.639 0.580 0.566 0.609 0.694 0.823 0.963 1.033 0.632 0.570 0.552 0.587 0.661 0.783 0.904 1.082 0.626 0.563 0.543 0.575 0.645 0.736 0.871 0.938 0.619 0.556 0.535 0.564 0.630 0.714 0.834 0.893
y= 0.50 0.633 0.563 0.531 0.535 0.580 0.642 0.734 0.808 0.630 0.562 0.531 0.535 0.575 0.642 0.708 0.741 0.625 0.555 0.521 0.523 0.557 0.619 0.680 0.768 0.618 0.545 0.510 0.507 0.535 0.589 0.639 0.663 0.612 0.538 0.502 0.498 0.521 0.566 0.609 0.630
1.0 0.625 0.546 0.499 0.475 0.490 0.507 0.559 0.581 0.623 0.544 0.497 0.475 0.488 0.507 0.551 0.573 0.616 0.536 0.488 0.462 0.463 0.485 0.507 0.540 0.611 0.530 0.482 0.454 0.454 0.441 0.498 0.526 0.605 0.523 0.465 0.446 0.443 0.417 0.470 0.497
byshev polynomial approximations, one for X 5 1 and the other for X 1 1. The appropriate equations for these regions are given e l ~ e w h e r e . ~Test , ~ ~calculations by Plummer et aLs showed that no significant differences were observed from more simplified approximations to J and J'as given by Pitzer.21 The temperature dependence of E 6 ~ ,and ~ i Eo'H,Niis determined mainly from the temperature dependence of A,. Thus, the computations of and $H,Ni,cl as functions of temperatures are accomplished from the information of A+ and the emf measurements interpreted and explained by employing both eqs 1 and 5. All the values of the virial coefficients appearing in eq 5 (except the unknown mixing parameter OH,Ni and $H,Ni,Ci without higher order electrostatic effect) must be known (or determined) over the entire range of interest. The parameters for the single electrolyte are assumed to have the representations =
p(O'Ni,ci(298.15
K)
+ (T-
l
y=
"Trace activity coefficients of HCI and NiCI2are given in columns 4 and 5 , respectively.
@(o)Ni,CI( T )
_______?
y,(NiCM
ya(HCI)
298.15 K)(6P'O'Ni,Cl/6T)
(6)
Similar expressions for the remaining pure-electrolyte parameters can be derived. The values for HCI and NiC12 parameters at 2 9 8 . 1 5 K and their temperature derivatives are presented in Table 111. Results and Discussion Linear least-squares isothermal fittings were made using eqs I and 5 with the emf data listed in the supplementary material. All the points were given equal weight as the measurement of E.
.-
z E
*= -
\
0.0
0
C
&
9
0 -0.2
HCI- N i C l 2
0
ml
With 0
E,,
25'~
E,'
W i t h o u t E,,
E,,
> E
\
0
0 I 3
w
I v) n w
O
**s
-
1 3 . D 0
I
-1
2
( m H + m C I ) / 2 mol kg-l
Figure 1. Fits of eq 13 for HCI + NiC12 + H 2 0 at 298.1 5 K The upper plot shows the concentration dependence of (A In YHC!/MN,) vs (MH +
c. * \,- * n e --1 _. __. . > - A L.-. ._ mcl)/ L. uara ror (mH 'r mcl)/ L \ u.1 mol ng . are nor inciuum m u s e of the large uncertainties and negative deviations which even become very are neglected (the open squares). The straight lines large if 'oH,Ni and '@' indicate the least-squares fits to the emf data. The lower plot refers to the values of the residual (Eobd - Ecrlcd)for the emf data at 298.15 K for least-squares calculations with and without higher order electrostatic terms, solid circles and open squares, respectively. It is clear that the definite dish-shaped curvature for the fit without E@H,Niand 'o'H,Ni is better represented at low ionic strength with the inclusion of these terms. \I*
...-I
I
--I
I
-_I
The pure-electrolyte parameters B, B', and 0 are readily calculable from Pitzer's tabulation^^^*^^ and are entered in Table I. For two isothermal fits, the only unknown parameters were OH,N~ and $H,Ni,cI (without the inclusion of E6H,Ni and %'H,Ni) and the corresponding terms %H,Ni and $H,yi,c1 (wlrh EOH,Ni and 'tJ'H,.,i). All these values of the mixing ion interaction parameters along with the standard deviation of the fit (unt) evaluated at each experimental temperature via the least-squares procedures are summarized in Table 11. PitzerZ0suggested simplified equations and a simple graphical procedure for evaluation of eH,Ni and $H,Ni,cl as a visual check on the least-squares calculations. This procedure defines the quantity A In yHC,as the difference between the experimental value of In yHClcalculated from eq 1 and that calculated for eq 5 with all the appropriate values for the single-salt parameters, but with %H,N~ and $H,Ni,cl set equal to zero, and, alternatively, with or without EOH,Ni and EI)'H,Ni. The final equation takes the form ( A In
YHcI)/"i
= 'OH,Ni 4-
f/3(mH 4- mCI)$H,Ni,CI
(7)
The quantity on the left-hand side of eq 7 is plotted against ( m ~ + m c l ) . Figure 1 shows a linear plot with intercept 'I)H,Ni and slope $H,Ni,cl for the data at 298.15 K. This type of presentation of the results, to which estimated errors can be attached, permits one to assess whether the values of and $ are significantly different from zero, and whether the data are consistent with Pitzer's equations within reasonable uncertainties. Kh00,~' (34) Pitzer. K. S.; Mayorga, G . ;J . Phys. Chem. 1973, 77, 2300. (35) Silvester, L. F.; Pitzer, K. S.1.Solution Chem. 1978, 7, 327. (36) Rard, J . A. J . Chem. Eng. Data 1987, 32, 334.
Thermodynamics of the Mixed Electrolyte HCI-NiCI,
The Journal of Physical Chemistry, Vol. 94, No. 18, 1990 7325
*H,Ni
0 0.04 O. 0 5
F
.
5
1 O
'
25
15
35
45
t /oc
Figure 2 Temperature dependence of the mixing parameters S9H,NIand The fitted lines are represented by eqs 14 and 15.
$H,N,,cl
however, pointed out the disadvantage of this procedure because of a nonlinear plot, particularly at small molalities of NX (NiCI2, in the present study). Pitzer and his c o - w o r k e r ~are ~ . ~aware of this problem and clearly indicated that care must be exercised when this procedure is used, since the uncertainty in ( A In YHCl)/", can be greatly magnified at lower molalities of Ni2+ which, in turn, reflects a constant uncertainty in the measured emf values. The validity of the above statements is clearly represented in Figure 1. Thus, it is always recommended to construct such a plot as a visual check on the least-squares calculations. The lower part of Figure 1 indicates the quality of the fit and of the measurements by plotting the residuals, that is, the deviation from the measured and the calculated values of E. The solid circles represent the inclusion of E O H , N I and EO'H,NI. The deviations are random (both positive and negative), rarely exceeding 0.6 mV. The empty squares indicate the results when higher order electrostatic terms are omitted. The systematic departures are apparent, primarily for dilute solutions. Figure 2 shows the graphical representation of the results of the temperature dependence of %H,NI and J,H,N~,cI.The error bars correspond to the standard deviations of the parameters as evaluated by the least-squares treatment. The mixing coefficients are well represented by the equations sOH,NI
+ 0.00026( T - 298.15)
(8)
0.0086 - 0.00027(T - 298.15)
(9)
= 0.0708
$ H , N ~ , C I=
The standard deviations of the fit for eqs 8 and 9 are 0.0031 and 0.0023, respectively. The temperature coefficients of these parameters are 6 6T
-('oH,Nl)
= 0.00026 f 0.00019
66T ( J , ~ , ~ I=, ~-0.00027 ~)
f
0.00015
(10) (11)
in which errors were determined via the least-squares fitting procedure. It is known from eqs 1 and 5 that the emf of the cell A is a function of m rather than O.lo Clearly, 0 varies with I . Khoo et al.38have convincingly shown that, by assuming 19' (the variation of 0 with I ) equal to zero, the Pitzer treatment did not alter the results significantly Thus, the common practice is to list a single ~
(37) Khm, K. H. J . Chem. Soc., Faraday Trans. I 1986, 82, 1. (38) Khm, K. H.;Lin, T. K.; Chan, C. Y.; J . Solution Chem., 1977, 7, 855
value of 8 with the range of ionic strength investigated. The earlier emf measurements" for the present system were limited only to 298.15 K. The values of %H,Ni and J,H,Ni,a at 298. 15 K are 0.082 and 0.005, respectively, and are compared with our results of 0.069 and 0.006shown in Table 11. The slight differences ~ are possibly due to (i) small differences in the reported in the values E values for pure HCI (hence, slight inconsistencies in the values of EoAg,ABCI), (ii) new experimental emf data in the intermediate range of the ionic strength, for example, I = 0.25, 1.5, and 2.5 mol kg-I in the present study, and (iii) the type of evaluation procedure for 0 and J,. For example Khoo et al.15determined the values of 0 and J, at each I by setting 0' = 0 and then weighing the values of the mixing parameters according to the molalities, or alternatively, the results were obtained from a single fitting using all the data points with 0' = 0. The results and general conclusions of Khoo et al.15 for 298.15 K, however, are the same as those of this study. As pointed out by Roy et al.,39 the values of '0 (0.06 to 0.09) and J, (-0.0014 to 0.006) for alkalineearth-metal chlorides fall in the narrow range which may also be valid for transition-metal chlorides as evident in this investigation. For most practical purposes, an average value of '0H.N could be defined to characterize H+ and M2+ interactions which are not specific to the M2+ ion. This single value may ultimately be used to make reliable estimates of activity coefficients for asymmetrical mixtures, provided precise values of activity coefficients are not needed. However, the values of the activity coefficients of HCI and NiC12 presented in Table I11 have been determined based on the precise mixing coefficients (with electrostatic effects) from Table 11. Caution must be exercised in using literature values of 0MN which should be compatible with the same single-salt parameters (p"O), @(I) etc.), as well as the type of treatments concerning the exclusion or inclusion of the higher order electrostatic terms. It is interesting to compare the results of the mixing parameters = 0.069, = 0.0056 from Table 11) obtained by using the older values of the pure electrolyte parameters34with = 0.076, $ = 0.0076; the magnitude of the difference in the resulting values of and $ are well within the standard deviation. With reference to the peculiarity of H+, it is of interest to compare our data (Table 11) for 0H,Ni = -0.0978, and $ ~ , ~ i ,=c l0.0666 with those Of &a,Ni = 0.059 17, and J,Na,Ni,CI = 0.01 1 52 obtained by Filippov and associatesa with the exclusion of higher order effects (the term). Forthcoming papers in this series deal with HCI BaC12 H 2 0 , HCI ThC14 H 2 0 , HCI + CoCI2 + H 2 0 , and HBr + MgBr2 H 2 0 in the temperature range 278.1 5-31 8.15 K (in some cases, 328.15 K), and the ionic strength range I = 0.01-4.0 mol kg-' (in some cases, 5.0 mol kg-'), using the Pitzer virial coefficient approach.
+
+
+
+
+
+
Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this work, and to the National Science Foundation for Grant No. CBT-8805882. F.J.M. acknowledges the support from the oceanographic section of NSF. Supplementary Material Available: Tables for the corrected emf for the cell (A) as functions of the ionic strength fraction of NiCI2, and temperatures as well as the parameters for Harned equations (3 pages). Ordering information is given on any current masthead page. (39) Roy, R. N.;Gibbons, J. J.: Ovens, L. K.; Bliss, G. A.; Hartley, J. J.; J . Chem. Soc., Faraday Trans. I 1981, 78, 1405. (40) Filippov, V. K.; Charykov, N . A.; Fedorov, Yu. A. Russ. J . Inorg. Chem. 1986, 31, 1071.
