J . Phys. Chem. 1990, 94, 5109-5114 is regulated by the nozzle, whose shape guides the expansion. An analytical representation of the nozzle diameter D ( x ) is stored in the program either as a simple polynomial in x or, more appropriately for some nozzles, as a polynomial in X I / * . It is assumed that the Mach number is unity at the nozzle's minimum diameter (throat), and the program computes the pressure PI and temperature T I at the throat corresponding to an isentropic expansion of an ideal gas from the initial pressure Po and temperature To. Because the heat capacity of the subject gas may contribute significantly and be a strong function of temperature, the usual expression in which the heat capacity ratio y = C p / C vis taken as constant is inappropriate. At the nozzle throat at x = xo, where PI and T I are now known, a small reference volume is chosen and followed as it propagates downstream in accord with the governing relations. The element is short, Axo in length and A ( x o ) in area, and contains P I A ( x o ) A x o / R Tmoles. , As the flow accelerates with increasing area A ( x ) , the length of the element becomes [ u ( x ) / u ( x o ] A x o . A small time interval I , is selected over which the flow advances by I,u and over which the differential relations (A2), (A3), (A6), (A7), (AlO), (A1 I ) , (A13), (A14), (A16)-(A20), (A24), and (A25) are combined and used to update the flow parameters.
5109
Unless the derivative d A ( x ) vanishes at the throat, however, the slope dT/dx is singular at xo. Therefore, for the first brief interval downstream of the throat, an analytical solution of the equations is applied, where nucleation, cluster growth, and frictional terms can be safely neglected. In subsequent intervals the difference equations for flow are solved for changes in temperature, speed, and degree of saturation. When supersaturation is achieved, the nucleation rate in the advancing reference element is computed until it reaches Jo,the onset of nucleation, usually considered to 3 ~ After ~ that, clusters born in each interval be 1015nuclei ~ 1 1 s-I. are kept track of and allowed to grow in accordance with the option selected. Individual and accumulated cluster mass and area are computed and recorded. Once the nucleation rate subsides to a value below Jo, additional clustering is trivial and is neglected. On a mainframe computer a time interval of 5 x IO4 s is usually adequately short for simple summations to suffice for integrations of the governing relations when a nozzle similar to our nozzle 6 (throat diameter 0.0115 cm) is adopted. Quite good runs can even be made on a microcomputer with a mathematics coprocessor in approximately a minute with time intervals set initially at 5 X IO-* s and slowly increased to 3 X IOT7 s after nucleation and cluster growth abate.
Intradlffuslon Coefficients for Zinc and Water and Shear Viscosities in Aqueous Zinc( II)Perchlorate Solutions at 25 "C William E. Price,* Lawrence A. Woolf, Diffusion Research Unit, Atomic and Molecular Physics Laboratories, Research School of Physical Sciences, Australian National University, Canberra, A . C.T . 2601, Australia
and Kenneth R. Harris Department of Chemistry, University College, University of New South Wales, Australian Defence Force Academy, Canberra, A.C.T. 2601, Australia (Received: June 29, 1989; In Final Form: January 2, 1990)
Measurements have been made of the intradiffusion of zinc ions and water in solutions of zinc perchlorate at 25 O C up to a concentration of 2.8 M. The results have been used to compare the diffusional behavior of zinc perchlorate with that of zinc chloride. Another method of establishing differences was afforded by calculating distinct diffusion coefficients for the two systems. In this way the effects of interactions between complex ions on the diffusion for zinc chloride have been highlighted. I n addition, a simple hydration model has been used to obtain a dynamic hydration number for the zinc ion in zinc perchlorate solutions.
Introduction Many different experimental techniques have been employed to shed light on the dynamics of water molecules in ionic solutions, ~ ~ ~transport such as neutron-scattering,' IR s p e c t r o ~ c o p y ,and In some recent work, Easteal and c o - w ~ r k e r studied s~~~ tracer diffusion of the cation and water in a number of trivalent metal chloride and perchlorate solutions, and of the anion in the chlorides, as well as obtaining viscosity and conductivity data. The present study investigates the intradiffusion (subsequently referred to as diffusion unless otherwise distinguished) of zinc ions and water in zinc perchlorate solutions. Its primary purpose is to increase the available pool of accurate transport data in nonsymmetric salt solutions. This is of considerable use to theoretical groups for the further understanding of ionic solutions, particularly concentrated ones. Quantities obtained from diffusion studies, for example, distinct diffusion coefficients, form an ideal link To whom correspondence should be addressed. Present address: Department of Chemistry, University of Wollongong, PO Box 1144, Wollongong, N.S.W. 2500, Australia.
0022-3654/90/2094-5 109$02.50/0
between experiment and t h e ~ r y . ~ The new data for the zinc perchloratewater system also enable a useful comparison with zinc chloride-water.8 This latter system has been shown to exhibit unusual transport behavior which is attributed to complex formation. We have made direct comparison of the experimental tracer diffusion coefficients of the cation and water in the two systems. The Onsager mobility coefficients (lu)g*10 (1) Salmon, P. S.; Howells, W. S.; Mills, R. J. Phys. C. 1987, 20, 5727. (2) Symons, M. C. R.; Waddington, D. J. Chem. SOC.,Faraday Trans. 2 1975, 71, 22. (3) Stranjret, J.; Kostrowicki, J . J. Solution Chem. 1988, 17, 165. (4) Easteal, A. J.; Mills, R.; Woolf, L. A. J. Phys. Chem. 1989, 93, 4968. (5) Easteal, A. J.; Price, W. E.; Woolf, L. A. J . Phys. Chem. 1989, 93, 7517.
(6) Eriksson, P.-0.; Lindblom, G.; Burnell, E. E.; Tiddy, G. J. T.J . Chem. SOC.,Faraday Trans. I 1988, 84, 3129.
(7) Friedman, H. L. In Water and Aqueous Solutions; Neilson, G . W.; Enderby, J. E., Eds.; Adam Hilger: Bristol, U.K., 1986; pp 117-132. (8) Weingartner, H.; Muller, K. J.; Hertz, H. G.; Edge, A. V. J.; Mills, R . J. Phys. Chem. 1984,88, 2173. (9) Agnew, A.; Paterson, R. J. Chem. SOC.,Faraday Trans. 1 1978, 74, 2885.
0 1990 American Chemical Society
Price et ai.
5110 The Journal qf Physical Chemistry, Vol. 94, No 12, 1990
for the two systems also have been used by other workers to highlight differences in ion-ion interactions."-13 With the tracer data now available, we have derived distinct diffusion coefficients from a combination of the intradiffusion coefficients with the I,, and used these to contrast the two zinc systems. I n doing so we are able to probe the ion-ion interactions in the zinc chloride system. The work also forms part of a collaborative effort with the Bristol neutron-scattering group.' Accurate diffusion data are essential for part of the analysis of their scattering profiles and the estimation of hydration parameters. The results have been used here to obtain hydration numbers Nh for ZnZ+in zinc perchlorate by using a simple hydration m ~ d e l ,previously ~.~ applied to Cr3+and Fe3+. This model is independent of neutron-scattering analyses. being based on proportioning the water among a number of different environments.
Experimental Section Preparation ofZn(ClO,), Solutions. Zinc perchlorate supplied by the G.F. Smith Chemical Co. (Columbus, OH) was used without further purification. A stock solution was prepared with deionized, distilled water yielding a zinc concentration o f about 3.5 mol kg-'. This stock solution was filtered through a 0.45-fim Millipore hydrophilic system to remove a small amount of insoluble white powder. On application of a few simple tests the residue was found to be iinc oxide. The filtered stock solution was analyzed by EDTA titration for zinc using standard technique^'^.^^ and was used to prepare, by accurate weight dilution. further solutions with approximate concentrations of 2.5, 2.0, 1 .O. 0.5, 0.25, 0.1. and 0.05 mol kg-' The accuracy of the solution concentrations is estimated to be & O . l % . Zinc D i f f s i o n Experiments. lntradiffusion experiments for zinc were performed on these solutions by the diaphragm cell method4.l6 at 25 (f0.02) "C using 6SZn (Amersham Radiochemicals, U.K.) as the radioactive tracer species and standard gamma-counting techniques for the analysis.I6 A diffusion cell with a platinum sinter was used to avoid adsorption problems. Approximatell mol of 65Zn per cm3 of solution was used for an experimental run. The cell was calibrated by using standard techniques16 both before and after the series of experiments. The precision of the zinc diffusion coefficients so obtained is estimated to be f0.5'3. Water DifjCusion Experiments. Measurement of the intradiffusion coefficient of water was made by two independent methods. The primary one was the N M R spin echo methodi7 using standard technique^.'*^^^ The measurement frequency was 19.8 MHz and the calibration constant for the quadrupole coil used was 0.4545 T / ( A m rad). Measurements were made on KCI solutions as a check on procedures: these reproduced earlier results'* within the experimental precision of f 1 % . Second, for comparison, a number of jHHO tracer diffusion runs were carried out using diaphragm cells with glass sinters at 25 "C; about I0-l: mol of jHHO (Amersham Radiwhemicals) per cm3 of solution was used in each experiment and liquid scintillation technique for the analyses.16 The cells were calibratedt6before and after the series of runs. The accuracy of the water diffusion coefficients from (10) Agnew, .A; Paterson, R. J. Chem. Soc., Faradqv Trans. 1 1978. 7 4 . 2896 ( 1 I ) Miller. D. G . J. Phys. Chem. 1966, 70, 2639. (12) Miller. D.G.: Pikal. M . J. J. Solution Chem. 1972, I . I1 I . (13) Pikal, M.J. J. Phys. Chem. 1971, 75. 3124. (14) Vogel. A . Quantitatiue Inorganic Analysis, 4th ed.; Longman: Harlow, U.K., 1978. ( 1 5) Flaschka. FI. A . EDTA Titrations, 2nd ed.;Pergamon: Oxford, U.K.. 1964 (16) Mills, R . : Woolf, L. A The Diaphragm Cell: A.N.U. Press: Canberra, 1965, (17) Tyrrell. H . J. V.; Harris, K . R. Diffuusion in Liquids: Butterworths London, 1934. (18) Harris, K R : Mills, R.; Back. P .I Webster. D. S. J . Mann. Reson. 1978. 29, 473. (19) The apparatus described in ref 18 has been upgraded by the replacement of the original Rruker P20 unit and ancillary electronic equipment.
TABLE I: Tracer Diffusion Coefficients for Zinc in Aqueous Zinc Perchlorate Solutions, Together with Related Density and Viscosity Data at 25 O C
mlmol k g ' clmol dm-' 0.0 0.0821 0.1067 0. I200 0.2498 0.5045 I ,0040 2.0120 2.5830 3.6730
0.0 0.0814 0.1067 0.1 186 0.2447 0.4854 0.9320 1.7404 2.1478 2.8270
p / g cm-'
TjmPa s
0.99705" 1.01296
0.8903b
1.02005 1.04452 1.09032 1 . I7462 1,32499 1.39917 1.51681
0.926
I09D(Zn2+) / m2 s-l
0.705' 0.696 0.692 0.688 0.630 0.549 0 432 0.356 0.255
1.026 1.203 1.664 2.001
"Density of pure water, ref 23. bViscosity of pure water, ref 24. 'Value obtained from limiting molar conductivity of ZnZC,ref 25.
TABLE 11: Water Diffusion Coefficients in Aqueous Zinc Perchlorate Solutions Obtained by Two Methods, NMR and Diaphragm Cell ____ diffusion coeff(H20)/10-9 m2 s-' m/mol kg-' c/mol dm-3 NMR NMRZ2 DC" 0.0 0.0821 0. I200 0.2498 0.5045 0.996 1.0040 2.0120 2.02 2.583 2.95 3.673 3.98
0.0 0.0814 0.1 186 0.2447 0.4854 0.925 0.9320 1.7404 1.746 2.1478 2.39 2.8270 2.99
2.30 2.22 2.19 2.14 2.01
2.03 1.85
1.82/1.79b 1.38/ I .40b
1.42 1.40
1.21
I .06 0.85, 0.76
Trace diffusion coefficients for HTO by diaphragm cell converted to corresponding H 2 0 values using the Mills factor20~21 for pure water. Duplicate runs.
the diaphragm cell runs is estimated as f0.5%. Comparison between the N M R and the diaphragm cell values was afforded by using the conversion factor (1.028) established by Mills20.2' for normal water. Shear Viscosity Measurements. Shear viscosities of the zinc perchlorate solutions were determined at 25 OC by using a flared capillary viscometer which had been calibrated by using a variety of solutions and liquids of known viscosity. Flow times were reproducible to *0.01'% and the overall accuracy of measurements is estimated to be better than &l.O%. Density Measurements. The densities of the perchlorate solutions, for use in calculating both molar concentrations and shear viscosities, were determined with an Anton-Paar DMA 60/602HT vibrating tube densimeter (Graz, Austria) and a series of temperature controllers/circulating baths to improve temperature control. The densities were obtained by measuring the period and monitoring the temperature during each measurement. The average temperature was calculated and then the procedure was repeated ten times. A linear regression was performed to obtain the density at the exact temperature of interest ( i l mK). This procedure was repeated a number of times to improve the accuracy. The overall accuracy in the temperature was better than f0.02 K. The densities obtained for the perchlorate solutions were g cm->. reproducible to better than f l X The densimeter method was checked by standard pycnometry using matched pycnometers: The accuracy is estimated to be f5 X g c d , which was the level of agreement between the two techniques. __-
(20) Mills, R. J . Phys. Chem. 1973, 77, 685. (21) Easteal, A. .J.; Price, W. E.; Woolf, L. A. J . Chem. Soc.. Faraday Trans. / 1989. 85. 1091
The Journal of Physical Chemistry, Vol. 94, No. 12, 1990 5111
Intradiffusion Coefficients for Zinc and Water
,
TABLE 111 Coefficients of Eq 1 sDecies
an
D(ZnZt)
0.705
a,
a,
lO2rmsdo
-0.240709
0.030926 -0.077295
0.9 1.4
ar
0.050424
D ( H 2 0 ) 2.299 -0.171863 -0.276913
__
7 1
Root mean square deviation of the fitted X values 1 ct
v
c 51, 0
!
2 c
0C
c
L_
3
mal
Figure 2. Water intradiffusion coefficients in zinc perchlorate solutions at 25 " C . Symbols: (V)NMR data; ( 0 )diaphragm cell data; (m) ref 22 data.
\
I
1
+dl
I - - L - _ L
di
ncl 3m
2 5t
' -
0.5 , 0
1
2
c
'I
Figure 1. lntradiffusion coefficients for a number of cations in divalent metal salt solutions at 25 OC. Symbols: ( 0 )ZnZt in ZnCI,; (V) ZnZt in Zn(C104)2; (m) Mg2+ in MgCI2.
Results and Discussion The shear viscosities, densities, and zinc ion tracer diffusion coefficients for the zinc perchlorate solutions are presented in Table 1. The values of the water diffusion coefficients obtained by the two methods, N M R and diaphragm cell, are given in Table I1 together with other water diffusion data in zinc perchlorate solutions from Weingartner's group in Karlsruhe.2z This latter set of data was also obtained by the spin-echo N M R technique, allowing a direct comparison with the present N M R results. The present transport data may be represented by polynomials in the form D = xajcJ12 (j= 0, 1, ...) J
(1)
The coefficients for these equations are given in Table I11 together with the root mean square deviation for the fit. Zinc Ion Diffusion. Figure 1 shows the concentration dependence (expressed as the square root of the ionic strength (Z)) of intradiffusion coefficients, D(Zn2+), for the cation in zinc perchlorate and literature data for the cations in zinc chlorides and magnesium c h l ~ r i d e . ~It~is* known ~ ~ from spectroscopic studiesz7 that CIO[ has greater structure-breaking properties than chloride. This can also be inferred from, for example, the difference in the viscosity E coefficients28(-0.056 dm3 mol-' for Clod- compared with -0.007 dm3 mol-I for Cl-). At high salt concentrations, D(Zn2+) in the chloride solutions is much higher than in the corresponding perchlorate ones, by as much as 100% at 2.8 M. This suggests that in this high concentration region the diffusing species are different: namely in the perchlorate solutions there is little, if any, complexation between the cation and the anion whereas in the chloride solutions species of the type [Zn(H20)6-nCln]2-nare formed which have higher mobilities than the uncomplexed zinc ion. This finding for the cation at high salt concentrations supports conclusions for zinc chloride solutions drawn from data for a number of different transport proper(22) Weingartner, H. Unpublished data, University of Karlsruhe. (23) Kell, G.S.J . Phys. Chem. Ref Data 1977, 6, 1109. (24) Stokes, R. H.; Mills, R. Viscosities of Electrolytes and Related Properties; Pergamon Press: Oxford, U.K., 1965. (25) Mills, R.; Lobo, V.M.M. SeljDiffusion in Electrolyte Solutions; Elsevier: Amsterdam, 1989. (26) Harris, K. R.; Hertz, H. G.;Mills, R. J . Chim. Phys. 1978, 75, 391. (27) Walrafen, G.E. J . Chem. Phys. 1970, 52, 4176. (28) Millero, F. J. In Water and Aqueous Solutions: Structure, Thermodynamics and Transport Processes; Horne, R. A,. Ed.; Wiley: New York, 1959; Chapter 13.
4 2
t
3
i ma: d n ' - 3
Figure 3. Water intradiffusion data for the numbers divalent metal salt solutions at 25 OC. Symbols: ( 0 )ZnCI,; (V)Zn(CI04)z; (m) MgCI2.
ties.8*10*29 The comparison between the present zinc perchlorate and the zinc chloride diffusion data is a better indicator of the effect of complexation on the cation intradiffusion than that used previously,8 as the cation is common to both. The diffusion coefficient of Mgz+ in MgCIz data is very similar to that for D(Zn2') in the zinc perchlorate solutions above 0.5 M. As MgC12 is usually regarded as exhibiting typical properties of unassociated 2: 1 electrolyte^^^ (strong electrolyte), it may be concluded that Zn(C104)2also behaves as a strong electrolyte in this high-concentration region. This is entirely consistent with findingslO-M based data calculated from transport properties for the two on ",I systems. Comparison of the data for zinc diffusion in Zn(C104)2and ZnCI, at low salt concentrations (below 0.2 M) provides some useful and surprising insights. The data for ZnC12 show a maximum value of D(Zn2+) occurring at about 0.1 M.8 At this point the diffusion coefficient is some 2% higher than the limiting value. Weingartner and co-workers8 examined the possible formation of inner-sphere complexes in this regime, by analogy with the case of solutions of cadmium halide^.^',^^ However, as they point out themselves, this is inconsistent with data from other transport and equilibrium properties of the ZnCI2-H20 system ~ ~ * ~about ~ which suggest that the salt is fully d i s s o ~ i a t e dbelow 0.2 M. The present data for zinc in perchlorate solutions throw new light on this enigma: below concentrations of approximately 0.5 M definite divergence from the behavior of D(Mg2+)in MgC1, is shown despite the fact that zinc and magnesium have almost (29) Rard, J. A,; Miller, D.G. 2. Phys. Chem. (Munich) 1984,142, 141. (30) Miller, D.G. In Proceedings of the 2nd Australian Thermodynamics Conference,Melbourne, Australia, 1981; Royal Australian Chemical Institute: Melbourne, Australia, 1981; p 490. (31) Mills, R.; Hertz, H. G. J . Chem. Soc., Faraday Trans. I 1982, 78, 3287. (32) Hertz, H. G.; Mills, R. J . Chem. SOC.,Faraday Trans. 1 1983, 79, 1317.
5112
The Journal of Physical Chemistry, Vol. 94. No. 12, 1990
identical limiting ionic conductances. Instead, there is more similarity between the two sets of zinc data within this concentration range. Perchlorate ions do not participate in any innersphere c ~ m p l e x i n gin' ~contrast ~ ~ ~ to chloride ions. This, therefore, suggests that any explanation for the observed diffusion behavior of zinc in ZnC12 does not require invocation of ideas of complexation at low salt concentration. It may be concluded therefore that the observed intradiffusion behavior of zinc ions in chloride and perchlorate solutions at low concentration is due to the zinc ion itself and not to any complexation effects. Water Diffusion Resulrs. The present diaphragm cell and N M R results for water diffusion in zinc perchlorate solutions compare most favorably as is shown in Figure 2. The limits of the difference between the two is less than *2%, which is within the combined experimental error. The direct comparison of D ( H 2 0 ) data from NMR measurements with those from Weingartner's group22shows excellent agreement. The curve of D ( H 2 0 ) for Zn(CIO,),, depicted in Figure 3 , follows closely that26 for MgCI,, which exhibits typical 2:1 electrolyte behavior.)O However, comparison with the data for D ( H 2 0 ) in ZnCI, again shows a large difference, with a consistently higher value for the latter at high concentrations, for example, over 60% at 3.5 M. The diffusion coefficient measured for water in the zinc chloride solutions is an average of contributions from those water molecules which diffuse with the zinc, chloride, and complex chloro-zinc ions ( [Zn(H20)6-nCln]2-")as well as that water which can be regarded as bulk (in the sense that it cannot be identified as moving with another ionic species on the time scale of a single diffusive step). The formation of the hydrated chloro-zinc complexes is thought to be accompanied by the displacement of one or more water molecules from the hydration layer of the cation. The singly charged complex ion would be expected to have a higher mobility than that of the doubly charged free zinc ion. The number of coordinated water molecules attached to the complexed species may be substantial. Those in the zinc ion primary hydration layer are likely to assume the diffusion coefficient of the complex; in addition, the water molecules coordinated to the chloride ligands must also be considered. Recent neutron-scattering experiments on zinc chloride solutions by Salmon34 have shown that the number of primary layer water molecules per chloride in the zinc complexes is independent of concentration, and equal to about 3.5. However, the residence time of primary layer water molecules on "free" chloride ions is known to be very short.35 In addition, as D ( H 2 0 ) is similar in value to D(CI-) in these solutions the effect of the displacement of some of the coordinated water of the chloride ions on the measured water diffusion coefficient is not likely to be significant. The major contributors to the enhanced D(H,O) values in ZnCI, are therefore the water molecules displaced from the zinc primary hydration shell by the formation of the complex and those coordinated in the primary layer of the singly charged chloro-zinc ions, which at sufficiently high concentrations will represent a significant proportion of the total number of water molecules present. The formation of the chloro-zinc complexes, therefore, should produce the overall result that the measured value for D(H,O) will be greater than in the corresponding situation where such complexes are not present; this is evident in Figure 3 for D ( H 2 0 ) in the zinc perchlorate and zinc chloride solutions. Distinct Diffusion Coefficients. An alternative approach for establishing the differences between zinc chloride and perchlorate diffusion properties, is to examine the distinct diffusion coefficients. These coefficients Ddjj (where i j = 1,2 with 1 = cation and 2 = anion) have been proposed as direct measures of particle interthe dominance of repulsive interactions tend actions in to make the Ddij negative while that of attractive forces tends to result in positive Dd,]. (33) Jones, M. M.; Jones, E. A.; Harmon, D. F.; Seames. R. T J . Chem SOC.,Faraday Trans. I 1961, 83, 2038. (34) Salmon, P. S. J . Phys. Chem. 1989, 93, 1182. 135) Friedman. H . L. Chem. Scr. 1985, 25. 42
Price et al. c
J".
v
c,
'v 5t,-+
-
_i_--_i-_--l__-_+-__~ -2.-
1 1
,:!
TO:
g
SK8-38
Figure 4. Calculated cation-cation distinct diffusion coefficient, for zinc chloride and perchlorate solutions a t 25 O C . Symbols: ( 0 )ZnCI,; ('I) Zn(C10,)*.
In the solvent-fixed reference frame,36 the distinct diffusion coefficients Ddjj for a general electrolyte M,,Y,2, with charges z l and z 2 on the cation and anion, are given by3'
+
I,/N = b, = rj/rlzlRf16ijfYj+ rjDdjj/(ri r j ) ]
(2)
where 8, is the Kronecker delta, R is the universal gas constant, and Ds, is the measured intradiffusion coefficient of ion i in a solution of concentration N equiv dm-3 at temperature T. (To illustrate the use of the charges and the subscripts from the stoichiometric formula in the case of ZnCI, r, = I , r2 = 2, z , = f 2 , and z2 = -1 .) The iij are the calculated Onsager transport coefficients, which may be calculated" by eq 3 . M* is the I,/N = t , f , A / ( 1 0 3 ~ z , z ,+ ) r,rjD,,,/( 1O3RTrrIzlM*) ( 3 )
+
thermodynamic factor in terms of molality m (M* = 1 m {d In r i d m i ) . The other new symbols in eq 3 have the following meanings: t, is the transference number of ion i, A is the equivalent conductivity of the solution at any particular concentration, F is the Faraday constant, D, is the measured mutual diffusion coefficient. for the volume-fixed reference frame, and r is simply r, rz. The three distinct diffusion coefficients may be determined for a system from the following expressions:
+
D d ,I = 2 ( b , l ~ l R T- PI)
(4)
Ddli = z,rb,,RT/r2
(The use of the absolute value of z2 in eq 6 should be carefully noted; otherwise the expressions 4 and 6 would not be equivalent.) For ZnCI, and Zn(C10,)2 the I , have already been calculated by Agnew and Pater~on.~*'O Consequently the Ddl, may be determined directly by combining the l,, with the intradiffusion data using eqs 4-6. The Dd, are directly related to the Onsager transport coefficients, particularly for the cation-anion case (Le., l12/N). Miller predicted" that I l 2 / Nis a measure of ion pairing. This was confirmed subsequently from both theoreticalI3 and experimentalt2studies which also showed that I l 2 / Nincreases with increased ion pairing. The same qualitiative conclusions about association in the zinc chloride and perchlorate systems that have been madeeI3 analyzing the I t 2 / Ndata should be expected from considering the Dd information. The use, however, of Ddv obtained from experimental data in interpreting the properties of electrolyte solutions is not obvious and largely untried. Problems such as the possibility of using a standard system to normalize the Dd values, and the influence of the chosen reference frame, need to be addressed. It should be noted that the use of the solvent-fixed reference frame for the DdlJ effectively excludes the information ( 3 6 ) Friedman, H. L.; Raineri, F. 0.;Wood, M D. Chem. Scr., in press. ( 3 7 ) Miller, D. G . Private communication. This was as a result of corre-
spondence between D.G.M. and Dr. F. 0. Raineri. SUNY, Stonybrook, NY.
The Journal of Physical Chemistry, Vol. 94, No. 12, 1990 5113
Intradiffusion Coefficients for Zinc and Water
l
e
i
e “E
: ‘
e
e
I
e
e e
e
e
e
i
a
c
5L
3 t l ;
m o ~ 3 m - 3 1
Figure 5. Calculated cation-anion distinct diffusion coefficient, Dd12,for zinc chloride and perchlorate solutions at 25 OC. Symbols: ( 0 )ZnCI,;
(W Zn(C10&; (m) MgCI,. on solute-solvent interactions that is reflected in the measured diffusion coefficient of the solvent. It is therefore necessary to be cautious in one’s interpretation of the present set of derived data. Errors accrue in their calculation, the largest of which stems from determining the activity coefficient derivative. The overall accuracy of the Dd, is estimated to be about *lo%. However, certain qualitative effects may be found in Figures 4-6. The values of Ddll, in Figure 4,are similar in form for both the chloride and the perchlorate. The negative values indicate repulsive interaction between the positively charged Zn species; the differences between the ZnC12 and Zn(C10,)2, although probably within the combined error of the data, may reflect the effect on the interactions of the lower charge density of the positively charged chloro-zinc complex. The picture for Ddl2 is dramatic: the values for Zn(C104), reach a small maximum at around 0.2 M and then diminish while, in contrast, those for ZnC12increase monotonically to large quantities. This is a strong indicator for the formation of complexes between the zinc and chloride ions. The decreasing slope with increasing concentration may be interpreted in terms of an increasing proportion of negative-negative interactions which repel (e.g., due to species such as Zn(H20)3C13-and CI-) and hence a negative contribution to PI,.Figure 5 also show Dd12for MgCI,. Interestingly, there is a great deal of similarity between this curve and that for Zn(C104)2. This supports the idea that zinc perchlorate acts like a standard unassociated 2:l electrolyte in contrast to zinc chloride. These findings match earlier conclusions based ~’~ on I l z / N data.M Onsager mobility data for ZnC12 i n d i ~ a t e ~that it acts as a strong electrolyte below about 0.2M and is complexed above that value. This conclusion has also been reached from examination of both interdiffusion and thermodynamic diffusion coefficients for the zinc chloride system.29 Figure 6 shows Dd22data for MgCI,, the archetypal 2:l electrolyte for comparison with zinc chloride. These were calculated from data critically reviewed by Miller and c o - w o r k e r ~ . As ~ ~ may be seen large values of DdZ2are not observed for MgCI,; in contrast to the large positive values for zinc chloride. This may be qualitatively rationalized in terms of a dominance of attractive forces between positive chloride-containing species and free chloride, as CI- displaces water molecules from hydrated zinc ions to form complexes of the type Zn(H20)6-nCl:-”. The linear portion of the curve may then indicate the region where the majority of CI-CI interactions are between ZnCl+(aq) and Cl-(aq). At higher concentrations dDd,,/d11/2 decreases. This suggests the effects of higher uncharged and negatively charged chloro-zinc species. These conclusions, using Dd2, data, again support earlier ones9,I0based on I2,/N data for the two zinc salt systems. Comparison of Density Data with Literature Data. The densities obtained may be compared with literature data.9-39 Very (38) Miller, D. G.;Rard, J. A.; Eppstein, L. B.; Albright, J. G. J . Phys. Chem. 1984, 88, 5139. ( 3 9 ) Pogue, R. F.; Atkinson, G. J . Solution Chem. 1989, 18. 249.
__ \li
2
’
3
mcl d v 3 r
Figure 6. Calculated anion-anion distinct diffusion coefficients, 0d2,, for zinc chloride and magnesium chloride solutions at 25 OC. Symbols: ( 0 ) ZnCI,; (m) MgC1,.
poor agreement was found between all three sets of data, at all concentrations. For example, at 1 .OO m the data of Pogue and those of Paterson9 and A t k i n ~ o nyield ~ ~ a value of 1.1779g and our own 1.1746 g Our co-workers 1.1799 g densimeter results were checked by using standard pycnometry techniques and the agreement between the two sets was better than 5 parts in lo5 and we are therefore confident of our technique. The disparity between the three sets of density data is such that an uncertainty of at least *2% in the salt concentration is required to explain it. Therefore, we carried out perchlorate analysis, using standard methods.14 This agreed to better than * O S % with the value expected from the zinc analysis. It may be noted that no such check was carried out by the other two g r o ~ p s . However, ~,~~ the experimental uncertainties in the intradiffusion coefficients are large compared with the possible errors in the concentrations of zinc perchlorate arising from the density problem. Thus the accuracy of the intradiffusion data and of the distinct diffusion coefficients derived from them will be little affected by a possible error in concentration. The error in the od values (and I,/) is in any case dominated by the error in the calculation of the activity coefficient derivative. This unresolved discrepancy in density data deserves further attention, particularly with regard to sample contamination, and is to be investigated. Zinc Ion Hydration Numbers. In previous ~ o r k , intradif~.~ fusion data for water and some, or all, of the ionic species in divalent and trivalent metal salt solutions have been analyzed by using the hypothesis that water is present in solution in a number of different environments. By this method average hydration numbers for the cations were estimated. In order to estimate the extent of zinc hydration in perchlorate solutions, we use here the three-environment modeL4 (The preferred model, employing four environments, could not be used because of the unavailability of diffusion coefficients for the perchlorate ion.) The basis of the model is detailed in the papers by Easteal and co-w0rkers.4~~ The water present in the salt solution is postulated to exist in three different environments. First, a number, Npnm,are coordinated to the metal cation: the residence time for this proportion of the total water concentration, f,,is assumed to be much greater than the time for a single diffusional step. This portion is therefore assumed to diffuse with the cation as a single entity, with diffusion coefficient 0,. Another proportion of the water molecules,f,,,, which are attached to the cation in a secondary layer (N,% water molecules per cation), are taken as diffusing with D,,, which has an intermediate value between D, and Db, the diffusion coefficient of the bulk water, which is the remaining fraction, fb, of the water present. Any effect of hydration of the anion is implicitly included in the bulk diffusion term. The observed water diffusion coefficient, D,, is assumed to reflect the proportion of water molecules in the different environments and is given by Dw = Dfc + D s e c L e c + D b f b
(7)
For the purposes of the calculation, D,,, is taken to be the ar-
5114
The Journal of Physical Chemistry, Vol. 94, No. 12, 1990
TABLE IV: Values of Effective Hydration Numbers ( N , ) for Metal Cations Obtained from Diffusion Measurements for a = 0.40" salt .Y, rmsdb data source Zn(CIO,),
12
Zn(C10,)2 (0-1 m )
16
0.03, 0.01,
ZnCI,'
16 12 12
O.0l9 0.014
MgCi, NiCI,
present present ref 8 ref 4 ref 4
'The best fit of eq 7 was for values of a = 0.4 (It0.05) in all cases; the uncertainty in N , is f 2 ; variation in a by 0.05 does not appreciably change the goodness of the fit, and the tabulated values all refer to fits using (Y = 0.40 for consistency and ease of comparison. bRoot mean square deviation of the simulated D(H,O) curve from the experimental curve. < Approximate value obtained at low salt concentrations (
