J. Phys. Chem. 1985, 89, 1486-1491
1486
Isothermal Dhslon in Aqueous Solutlons of Chlorine: A Ternary Process Derek G . Leaist Department of Chemistry, University of Western Ontario, London, Ontario, Canada N6A 5B7 (Received: May 1 , 1984; In Final Form: November 28, 1984)
Diffusion in aqueous solutions of chlorine is usually treated as a binary process and described in terms of Fick’s law, J = -DVc. In water, however, the dissolved chlorine component diffuses as free molecular chlorine as well as hydrochloric acid and molecular hypochlorous acid produced by hydrolysis: Clz + H 2 0 = H+ + C1- + HOC1. Because aqueous HC1 diffuses twice as rapidly as HOCl, binary solutions of chlorine undergoing diffusion separate into two ternary solutions: the solution ahead of the diffusion boundary contains the chlorine component plus additional HC1, while the region behind the boundary contains the chlorine component plus the corresponding amount of HOCl left behind by the coupled flow of the more rapidly diffusing HCI component. Consequently, diffusion in aqueous solutions of chlorine is a ternary process described by a flux JI = -DIIVcl- D12Vc2of the partially hydrolyzed chlorine component (1) and an additional flux J2 = -D2’VcI- D22Vc2 of the HCl component (2).A procedure for estimating the ternary transport coefficients is developed. In dilute solutions where hydrolysis is extensive, the simultaneous flow of the HC1 component is predicted to exceed the flux of the chlorine component. Diffusion coefficients of aqueous chlorine measured by Stokes’ magnetically stirred diaphragm cell method are reported. The data are in good agreement with predicted ternary behavior.
Introduction Diffusion of aqueous chlorine has received considerable att e n t i ~ n l -in ~ view of important applications to zinc-chlorine batteries, chloralkali production, and gas absorption operations. In water, the dissolved chlorine component diffuses as free molecular chlorine together with significant amounts of hydrochloric acid and molecular hypochlorous acid produced by the hydrolysis reactionq Clz
+ H 2 0 = H+ + C1- + HOC1
K(25 OC) = 3.94
X
mol2 L-2
In a solution saturated a t 25 O C under a chlorine pressure of 1 atm (approximately 0.09 M total C12) for example, 35% of the total chlorine is hydrolyzed. Hydrolysis is almost 90% complete at 0.01 M total Clz. In previous studies?+’ isothermal diffusion of aqueous chlorine was treated as a binary process and was described in terms of Fick‘s law, J = -DVc, with a single diffusion coefficient. Consider, however, that aqueous hydrochloric acidlo diffuses twice as rapidly as molecular hypochlorous acid.’,2 (This is due, at least in part, to the exceptional mobility of H+.) Consequently, a chlorine concentration gradient will generate a flow of HC1 in excess of the HCl transported equimolar to HOCl as the hydrolyzed portion of the total chlorine flow. On the basis of this argument, we propose that diffusion in aqueous solutions of chlorine is a ternary process characterized by an additional flow of hydrochloric acid. In this paper a theoretical justification of the ternary diffusion hypothesis is presented and a procedure for predicting the transport coefficients of aqueous chlorine is described. To check the ternary diffusion hypothesis, diffusion coefficients are determined for aqueous solutions of chlorine by Stokes’ magnetically stirred diaphragm cell method. The measured diffusion coefficients are compared with predicted values. Many other solutes undergo hydrolysis in aqueous solution; some important examples are salts containing phosphate, carbonate, iron, or aluminum ions. Like aqueous chlorine, ‘binary” solutions (1) Chao, M. S. J . Electrochem. SOC.1968, 115, 1172. (2) Kim, J. T.; Jorn€, J. J. Elecrrochem. SOC.1978, 125, 89. (3) Kim, J. T.; JornE., J. J . Electrochem. Soc. 1977, 124, 1473. (4) Himmelblau, D. M. Chem. Reu. 1964, 64, 527. (5) Kramers, H.; Douglas, R. A.; Ulmann, R. M. Chem. Eng. Sci. 1959, 10, 190. (6) Peaceman, 0. W. Sc.D. Thesis, M.I.T., Cambridge, MA, 1951. (7) Vivian, J. E.; Peaceman, 0. W. AIChE J . 1956, 2, 437. (8) Spalding, C. W. AIChE J . 1962, 8, 685. (9) Connick, R. E.; Chia, Y. T. J . Am. Chem. SOC.1959, 81, 1280. (10) Harpst, J. A.; Holt, E.; Lyons, P. A. J . Phys. Chem. 1965, 69, 2333.
0022-3654/85/2089-1486$01.50/0
of these solutes are predicted to have complex multicomponent diffusional properties.’
Theory The diffusional flux of the aqueous chlorine component consists of fluxes of four major chemical HOCl, H+,C1-, and molecular C12, in proportions that vary with concentration. (OCIand C13- are also present, but in negligible proportion.’) The rate of hydrolysis is so rapid13 compared to diffusion that chemical equilibrium exists at each point along the diffusion path. Since the electric current vanishes in pure diffusion, there is a second restriction, namely that fluxes of H+ and CI- be identical. With two constraints imposed on fluxes of four solute species, there are two linearly independent”J2 solute fluxes relative to the solvent. It follows that isothermal diffusion in aqueous solutions of chlorine is a ternary process. Why is aqueous chlorine a binary system from the point of view of equilibrium thermodynamics but a ternary system with regard to its diffusional properties? l 2 At thermodynamic equilibrium the composition of an aqueous solution of chlorine is completely specified by a single concentration variable. For example, if the total chlorine concentration [Cl,] [HOCl] is given, the concentration of the solute species is determined by the following conditions: electroneutrality, [H+] = [Cl-1; stoi~hiometry,’~ [HOCl] = [H’]; and chemical equilibrium, K = aHOClaHCl/aCI2. Once a gradient in chlorine concentration is prepared, the flux of more rapidly diffusing hydrochloric acid species will exceed the flux of slower hypochlorous acid molecules; the region ahead of the diffusion boundary will contain more hydrochloric acid than hypochlorous acid, and the region behind the boundary will be correspondingly richer in hypochlorous acid. Because the stoichiometry condition [HOCI] = [H+] no longer applies, the composition of the nonequilibrium system cannot be described by the total chlorine concentration alone; the concentration of a second solute component must also be specified. For convenience, we will take the excess hydrochloric acid as the second solute component. Transport Equations. According to nonequilibrium thermodynamics,15molar fluxes of neutral chlorine (1) and hydrochloric acid ( 2 ) components are related to gradients in component con-
+
( 1 1) Leaist, D. G . J . Chem. SOC.,Faraday Trans. 1 1982, 78, 3069. (12) Leaist, D. G . Can. J . Chem. 1983, 61, 1494. (13) Morris, J. C. J . Am. Chem. SOC.1946,68, 1692. (14) [H+], may exceed [HOCI] in real solutions owing to loss of trace amounts of slightly volatile HOCl and slightly volatile HCI to different extents, thus making the system more strictly a ternary one. ( 1 5 ) Fitts, D. D. ‘Nonequilibrium Thermodynamics”;McGraw-Hill: New York, 1962; Chapters 7 and 8.
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 8, 1985 1487
Isothermal Diffusion in Aqueous Solutions of Chlorine centrations and chemical potentials by the ternary flow equations
-J2
+DI~VC~ = D2lVcl + D ~ ~ V C ,
(1)
-J1
= LllVPl
+ L12VP2 = L21V~1+ L22V~2
(2)
-J1
= DiIVcl
and
-52
The ternary diffusion coefficients Dik are related to the Ljk phenomenological coefficients by
How large is J2, the coupled flux of hydrochloric acid generated by diffusion of aqueous chlorine? So that this question may be answered, we turn now to the problem of estimating values for the transport coefficients of the system chlorine (1) hydrochloric acid (2) water. Particular interest centers on D2', since this coefficient determines the flux of the HC1 component produced per unit concentration gradient in the chlorine component. We begin with species flow equations11*12v'6
+
+
cm2 s-' estimated by Kim and JornE2 $om studies of chlorine transport at rotating zinc hemispheres. Dl(Clz) was derived from measurements on chlorinated solutions that had been acidified with perchloric acid to suppress hydrolysis:0 while Dz(HOC1) was obtained from measurements made at low concentrations where hydrolysis is nearly complete. The values we have adopted for the diffusivities of C12 and HOCl are in good agreement with the respectivevalues (1.85 f 0.10) X 10" and (1.30 f 0.06) X cmz s-l obtained potentiometrically by Chao.' Hydrogen bonding probably accounts for the lower diffusion coefficient of HOCl relative to C12. Expressions for the transport coefficients of the components in terms of the diffusion coefficients of the constituent species are obtained from the following relations: (i) The flux of the chlorine component equals the flux of molecular chlorine plus the flux of hypochlorous acid Jl = jl
j3= j4
J2
+ P4)
(6)
Lik Coefficients. The utility of the flow equations (4) for the species lies in the fact that the lik coefficients are readily estimated from data available for the diffusion coefficients of the various chemical species. This information can be used in turn to predict the transport coefficients of the vomponents."v'2 If we make the reasonable approximatio_ns, as others"J8 have done, that (i) the off-diagonal elements of 1 are zero and (ii) the diffusion coefficients of the species are constant, we have iik
=13 -
(9)
(iii) The gradient in chemical potential of the chlorine component equals the gradient in chemical potential molecular chlorine VPl = VP,
(10)
(iv) The gradient in chemical potential of the hydrochloric acid component equals the sum of the gradients in the electrochemical potentials of H+ and C1vp2
= vp3
+ VF4
(11)
Upon substitution of eq 4-1 1 into eq 2 (or by matrix manipulation"), relatively simple expressions for the component transport coefficients are obtained. LI, = 7,'
+ i22
Lj2 = L2' = -122
..
(12)
(13)
1..
(5)
and by local chemical equilibrium VPl = V(P2 + P3
(8)
(ii) The flux of the hydrochloric acid component equals the total flux of HCl less HCl transported with HOCl as part of the chlorine component flux
(4) where jjis the molar flux of solute species i, jio are the species phenomenological coefficients, and Vji, denote gradients in electrochemical potentials of the species. We number the species as follows: 1, molecular C1,; 2, molecular HOC1; 3, H+; 4, C1-. Note that the concentration of the chlorine component (Le. total chlorine) equals the sum of the concentrations of C12and HOCl species, cl = El + E2. The concentration of the hydrochloric acid component equals the concentration of hydrochloric acid species in excess of hypochlorous acid species, c2 = E3 - E2. As mentioned earlier, the species fluxes are constrained by electroneutrality
+jz
= 6ik(EiBi/RT)
(7)
where 6ik is the Kr?necker delta, R is the gas constant, T i s the temperature, and Di and Ei refer to the diffusion coefficient and concentration of species i. Diffusion coefficients of H+ and C1- can be :valuated from limiting ionic conductances by use of the relation Di = Ri"&O/P. From the limiting valueslga X,O(H+)-= 349.81 and X40(Cl-) = 16.35 cmz ohm-' mol-', we calculate D3(H+) = 9.315 X and D4(C1-) = 2.033 X cm2 s-*. Diffusion coefficients of the molecular species are more difficult todetermine. We will use the values Dl(C12) = 1.96 X and D2(HOCl) = 1.49 X (16) Stockmayer, W. H. J . Chem. Phys. 1960,33, 1291. (17) Garland, C. W.; Tong, S.;Stockmayer, W. H. J . Phys. Chem. 1965,
In terms of the degree of hydrolysis of the chlorine component defined by [HOCl] = -E2 a= (15) [HOC11 + [Cl,] CI the concentrations of the chemical species are E, = [Cl,] = (1 - a)c1 22 = [HOCI] = ( Y C ~
+ c2 = acl + c2
E3 = [H+] = acl = [Cl-]
(16) (17) (18)
(19) Here c1 and c2 denote concentrations (in moles per unit volume) of chlorine and excess hydrochloric acid components. We have RTLll = (1 - a)c,Bl acIB2 (20) E4
+
RTLl2 = RTL21 = -(YCIB~
(21)
It develops that Lll/cl = [(l - a)b, + ab2]/RT, the thermodynamic mobility of the celorine compnent, is simply the number-weighted average of D l / R T and D2/RT, the mobilities of molecular chlorine and hypochlorous acid, respectively.
69, 1718.
(18) Creeth, J. M.; Stokes, R. H. J . Phys. Chem. 1960, 64, 946. (19) Robinson, R. A,; Stokes, R. H. 'Electrolyte Solutions", 2nd ed.; Academic Prcss: New York, 1962; (a) Appendix 6.1, (b) Appendix 8.9, (c) Chapter 1 1 , (d) p 259.
(20) Peace.man6*'has used the porous diaphragm cell technique to obtain b(C1,) = 1.48 X lo4 cm2 s-I at 25 OC. Because HC1 was used to suppress hydrolysis, the reported value may be low' owing to formation of the slower trichloride species: C12 + ClCI3-.
1488 The Journal of Physical Chemistry, Vol. 89, No. 8, 1985
Leaist
Dik coefficients. Finally, expressions for the ternary diffusion coefficients of CI2(1) + HCl(2) + H 2 0mixtures are obtained by multiplying the Li0coefficients by the derivatives &,/&k according to eq 3. The chemical potentials of the chlorine and hydrochloric acid components are given by p , = p l o + RT In [ ( l - a)cl] (23) =
p2
+ 2RTln
~ 2 ’
[(acI + CZ)Y+I
D22
1
(24)
Upon differentiation eq 23 and 24 together with the equilibrium condition
+ c2)2
a(ac,
K=
1-a
Yi2
we obtain
0OC
002
GO4
006,
C08
0°C
c , / m o l Ii?er-
Figure 1. Calculated ternary diffusion coefficients of aqueous chlorine (component 1) without added hydrochloric acid (component 2) at 25 O C .
Corrections for activity coefficients of HCI are included.
”[ I+,
1 acL2 = RT in which Z = a c , abbreviation for
1
d In Y i
a
aC2
+ c2 is the ionic strength and A is a convenient a
A=
(3 - 2 a ) a q
+ c2
(29)
We have assumed that the activity coefficients of molecular C12 and HOCl are unity. Values of y,, the mean ionic activity coefficient of hydrochloric acid on the molar concentration scale, are obtained from published activity datalgbfor binary solutions of aqueous HCl at the same ionic strength. For computational purposes, it proved convenient to express the activity coefficient terms as A I d In y,/dZ d ln Y i -(30) 1 + 2AZ(1 - a)cl d In yi/dZ dc, d In Y i -dcq
( A Z / a ) d In yi/dZ 1
+ 2AZ(1 - a)cI d In yi/dZ
(31)
since the derivative d In y,/dZ is readily obtained from the activitieslgb of binary HC1 H 2 0 mixtures. C12(cl) H 2 0 at 25 OC. In Figure 1 calculated values of the ternary diffusion coefficients of aqueous solutions of chlorine without added hydrochloric acid (Le. c2 = 0) are plotted against the total chlorine concentration, c,. In order to grasp the essential features of diffusion in this system, it is useful to have simplified expressions for the diffusion coefficients. Such expressions may be obtained by omitting the relatively small activity coefficient terms d In y,/dci from the analysis. We then obtain 3(1 - a)B,(ClZ)+ aB,(HOCI) Dll zz (32) 3 - 2a
+
+
2(1 -
D,2 z [b1(C12)- b2(HOCl)]
(33)
a D2l z -[D+O - b2(HOCl)] 3 - 2a
(34)
3 - 2a
Dt0 022
z
+ 2(1 - a)b2(HOCl) 3 - 2a
(35)
where Dio = 2$(H+)b4(Cl-)/[b3(H+) + b.,(CI-)] is the limiting Nernst diffusion coefficient of aqueous hydrochloric acid, 3.33 X cm2 s-I a t 25 OC. Dl ,, the main diffusion coefficient for the chlorine component, is a relatively simple weighted average of the diffusion coefficients of molecular chlorine and hypochlorous acid [but not a simple
number-weighted average such as Dll = ( 1 - a)b1(Cl2) + aD2(HOC1)]. In the liqit of infinite dilution where hydrolysis is complete Dll(a=l)= D2, the diffusion coefficient of the HOCl molecule. In a ternary system with c2 = 0, it would seem that the gradient in component 1 is incapable of producing a coupled flux of component 2 since none of the latter is present. For this reason it is usually assumed that D21 0 as cz 0. However, the system C12(1) HCl(2) H20provides a striking exception to this rule. In this case hydrolysis of aqueous chlorine generates equimolar amounts of HOCl and HCl. Since HCI diffuses more rapidly than HOC1, the flux of HCI in excess of HOCl appears as a flux of HCl component, even in solutions free of added HCl. Indeed, values of D2, shown in Figure 1 exceed Dll at low chlorine concentrations. In that region, the initial flux of the HCl component along the chlorine gradient will exceed the flux of chlorine, even though the system is free of added HCI! In a sense, the chlorine flux generates “its own” flux of the HC1 component. In the limit of infinite dilution, the value of DZl equals the difference in diffusivities of HCI and HOCl, D,O - D2(HOCl) = 1.84 X cm2 s-I. The cross-coefficient DI2 measures the flux of the chlorine component produced per unit gradient in HCl concentration. The calculated DI2values shown in Figure 1 are small and positive. This feature is most easily understood by considering transport along a gradient in HCI concentration in a solution with a uniform chlorine concentration. Because HCI suppresses hydrolysis, the region of the system with the higher HCI concentration contains more molecular chlorine and less HOCl than the region of the system with the lower HCI concentration. This means that the HCI flux will generate a cocurrent flux of molecular chlorine together with a countercurrent flux of hypochlorous acid. (Note in eq 33 that D12 is directly proportional to the difference in diffusivities between molecular chlorine and HOCl species.) Since the diffusion coefficient of molecular chlorine is slightly larger than the diffusion coefficient of hypochlorous acid, the net flux of the chlorine component is cocurrent to hydrochloric acid. Therefore, DI2is positive. At zero concentration where hydrolysis is complete, the fluxes of chlorine and hydrochloric acid components are chemically uncoupled and DI2vanishes in that limit. D22,the main diffusion coefficient of the HCI component, is given by Dio = 3.33 X cm2 in the limit of infinite dilution. At nonzero chlorine concentrations where hydrolysis is incomplete, D22falls well below this limiting value. There, part of the HC1 is transported as molecular chlorine with countercurrent flux of H O C l a transport mechanism that is slower than direct diffusion as C1- and highly mobile H+. The main source of uncertainty in the predicted Dik values resides in the values assumed for the diffusion coefficients of molecular chlorine and hypochlorous acid species. If we adopt (0.142) x cm2 s-I as an estimate of the probable errors in
+
+
- -
Isothermal Diffusion in Aqueous Solutions of Chlorine bl(C12) and b2(HOC1), the corresponding errors in the predicted 0.04 ternary diffusion coefficients could be as large as 0.4 X and 0.1 X cm2 s-l for Dll, DI2,D21,and x lo”, 0.2 x D22,respectively, over the range of concentrations shown in Figure 1. The assumption that the diffusivities of each of the species is constant is another source of uncertainty in the analysis. Because of the electrophoretic effect, the diffusivities of H+ and C1- vary slightly with concentration. Taking as a guide the electrophoretic corrections reportedlo for diffusion of binary HCl + H20mixtures, neglect of electrophoresis could produce errors in predicted D22 values as large as 0.03 X cmz s-l. The corresponding errors in the other diffusion coefficients would be even smaller since they depend less heavily on ionic mobilities.
Experimental Section In order to confirm experimentally that chlorine diffusion is a ternary process, attempts were made to measure diffusion of the solute by Harned’s conductometric method.21q22 In this procedure, solute fluxes are determined from electrical conductances measured along a column of diffusing electrolyte. Although the solubility of chlorine in water is relatively low, conductances of the solutions can be measured precisely. Therefore, Harned‘s technique appeared to be the method of choice for studies of aqueous chlorine. In practice, however, chlorine attacked both the epoxy cement used to assemble the diffusion cells and the grease on the sliding surfaces. Moreover, erratic conductance readings indicated that the columns of solution were unstable with respect to convection. (Coupled flows of hydrochloric acid driven ahead of the chlorine component may have produced density inversions and subsequent mixing.) We were unable to obtain reliable diffusion data using the conductometric method. The corrosive nature of the solutions and the possibility of convection prompted us to use the porous diaphragm cell method developed by Stokes?3 In this procedure, convection is eliminated by confining diffusion to the pores of a sintered glass disk interposed between two solution-filled compartments. The cells24 used for this work were constructed from Pyrex. Each compartment (nominal volume 30 mL) was fitted with a greaseless Teflon stopcock and magnetically stirred at 60 rpm. The diaphragms were Corning fineporosity disks (nominal pore diameter 4.5 X lo4 cm). The cells were calibrated2sa t frequent intervals with aqueous potassium chloride. Cell constants were reproducible to *0.3%. Reagents. Chlorine solutions were freshly prepared prior to each run from deionized, doubly distilled water and chlorine (Linde high-purity grade) that had been passed through a train of water-filled bubblers to remove traces of hydrochloric acid. Chlorine concentrations determined by thiosulfate titration26were reproducible to k0.0002 M. Reagent grade KCl was recrystallized from water and dried in a vacuum oven at 140 OC. All other materials were reagent grade and were used without further purification. KCl solutions from calibration runs were analyzed by potentiometric titration against silver nitrate. Procedure. At the start of a run, each diaphragm cell was filled with outgassed conductivity water. After sufficient water had been drawn through the diaphragm to remove any trapped air, the water in the bottom compartment was replaced by chlorine solution. The cell was thermostated at 25 f 0.01 O C and run for about 2 h to set up steady-state diffusion through the diaphragm. The top and bottom compartments were then refilled with water and fresh chlorine solution, respectively, and the cell was reset under the magnets. Diffusion was allowed to proceed for 2-7 days. During this time the cells were protected from light to avoid photochemical Harned, H. S.; Nuttal, R. L. J . Am. Chem. SOC.1949, 71, 736. Leaist, D. G.; Lyons, P. A. J . Phys. Chem. 1982.86, 564. Stokes, R. H. J . Am. Chem. SOC.1950, 72, 763, 2243. Kulkarni, M. V.; Allen, G. F.; Lyons, P. A. J . Phys. Chem. 1965,69, (25) Stokes, R. H. J . Am. Chem. SOC.1951, 73, 3527. (26) Sutton, F. “A Systematic Handbook of Volumetric Analysis”, 13th ed.;Butterworths Scientific Publications: London, 1955; p 313.
The Journal of Physical Chemistry, Vol. 89, No. 8, 1985 1489 decomposition of hypochlorous acid.3 At the end of a run, the contents of the cell compartments were drawn into separate gas-tight sampling syringes. Solutions from the top compartment were found to contain chlorine plus hydrochloric acid components, while solutions from the bottom compartment contained chlorine plus the corresponding amount of hypochlorous acid that had been left behind by diffusion of the hydrochloric acid component into the top compartment. Since the concentration of HOCl species exceeded the concentration of HCl species in the bottom compartment, the concentration of the hydrochloric acid component in that compartment was negative. Ten-milliliter portions of solution taken from the top (T) compartment were added to aqueous potassium iodide. Thiosulfate titration26 of iodine liberated by reaction of total chlorine with iodide (C12 2KI = 2KC1+ I2 and HOCl HC1+ 2KI = 2KC1 H 2 0 12) gave the concentration of the chlorine component in the top compartment, cIT. The concentration of the hydrochloric acid component, C2T, was determined by titration against standardized sodium hydroxide. The concentration of the chlorine component plus excess hypochlorous acid in the bottom compartment, cIB- c2B, was determined by iodometry as described above. Following Sutton,26 titration of KOH produced by reaction of the excess hypochlorous acid with potassium iodide (HOCl + 2KI = KOH + KCl + 12) gave C2B, and hence cIB. In order to reduce loss of volatile chlorine, each sample of solution was delivered by weight beneath the surface of the iodide solution directly from the sampling syringe. Weights of delivered solution were converted to volumetric units by assuming that the solutions had densities2’ identical with aqueous hydrochloric acid at a concentration equal to c1 + c2. Since the total solute concentrations were kept below 0.04 M (to further minimize volatilization of chlorine), this approximation led to negligible error. Duplicate determinations of c1 and c2agreed to within k0.0002 M. Within experimental error, the concentration of the hydrochloric acid component in the top compartment was identical with the concentration of excess hypochlorous acid in the bottom compartment: C2T + c2B, = 0. Data Analysis. The differential equations describing ternary diffusion in diaphragm cells have been integrated.28 In the experiments used here, the initial gradient is entirely in the chlorine component (1). We have
+
+
ACl(t) -ACl(0)
+
+
- (Dll - D2) exP(-PDlt)
- (Dll - 01) exp(-PD20
Dl - 0
2
(36)
where Aci(t) = cjB(t) - ctT(t)refer to differences in concentration across the diaphragm, fl is the cell constant, and Di are eigenvalues
(39)
of the diffusion coefficient matrix. A standard iterative least-squares p r o c e d ~ r e ~ was ~ - ~used l to fit the Aci(t)/Acl(0) vs. fit data to the equations (27) Harned, H. S.; Owen, B. B. “The Physical Chemistry of Electrolytic Solutions”, 2nd ed.;Reinhold New York, 1950; p 253. (28) Cussler, E. L.; Dunlop, P. J. J . Phys. Chem. 1966, 70, 1880. (29) Margenau, H.; Murphy, G. M. “The Mathematics of Physics and Chemistry”, 2nd ed.; Van Nostrand: Princeton, NJ, 1956; p 517. (30) Bevington, P. R. ”Data Reduction and Error Analysis for the Physical Sciences”; McGraw-Hill: New York, 1969; Chapter 11. (31) Leaist, D. G.; Lyons, P. A. J . Phys. Chem. 1981, 85, 1756.
1490 The Journal of Physical Chemistry, Vol. 89, No. 8, 1985
Leaist
in which a1 =
(Dll - DZ)/(DI - 02)
az
= DZI/(Dl - Dz)
(42) (43)
Values of the ternary diffusion coefficients were determined from the least-squares parameters a,, az, D l , and D2 by using the relations Dll
= alDl
+ ( 1 - al)DZ
/
I
02
00
I
I
I
I
1
06
04
,
I
08
10
I t /io5s cm-2
(44)
Figure 2. Concentration differences between top and bottom cell compartments plotted against @t, with c1 = 0.0126 M and c2 = 0: (O), measured values; (-), least-squares fit to eq 40 and 41; (---), predicted values using diffusion coefficients taken from Figure 1.
l0k
Results and Discussion Five or six diaphragm cell runs of different duration were carried out a t each of two mean chlorine concentrations: cl = 0.0126 and 0.0190 M. The results of the experiments are summarized in Tables I and 11. In Figures 2 and 3, measured differences in solute concentrations between the top and bottom compartments are plotted against fit. Since the solutions studied were free of added HCl, the concentrations of the HCl component (2) in the top and bottom compartments were both zero at the start of each run,hence Ac2(0) = 0. But as the chlorine component ( 1 ) diffused into the top compartment and Ac,(t) decayed monotonically to zero, the upward flow of chlorine produced a cocurrent flow of the HCl component. Because the concentration of the HCl component in the top compartment was larger than in the bottom compartment as diffusion proceeded, values of Acz(t>O) were negative. For values of @tgreater than about 0.4, the chlorine gradient driving the coupled flow of the HCl component into the top compartment had decayed to a sufficiently small value and the HCl component began to diffuse back into the lower compartment. For runs of the longest duration, it may be noted that the difference in concentration of the HCl component across the diaphragm is as large as that for the chlorine component, even though the initial difference for excess HCl was zero. This indicates that fluxes of the two components were similar in magnitude. Treatment of diffusion as a binary process would therefore be a poor approximation to the actual diffusional behavior of the system. Table I11 shows values of the least-squares parameters al, 02, D1, and D2 and their standard deviations30 determined by fitting eq 40 and 41 to the measured concentration differences. For comparison, values of the parameters predicted by theory (using Dikvalues taken from Figure 1) are also shown. The least-squares fits are plotted as solid curves in Figures 2 and 3; the dashed curves give concentration differences predicted by theory. Experimental values of the ternary diffusion coefficients of aqueous chlorine together with standard deviations30are recorded in Table 111. As anticipated, values of Dzl (the cross-diffusion coefficient that measures the flux of the HCl component produced per unit concentration gradient in the chlorine component) are large and positive. The major source of error in our diaphragm cell data is analytical error associated with titration of the (volatile) chlorine component. In addition, adsorption onto the sintered glass and in surface transport effects produce small errors (about 1%)19c923 the measured diffusion coefficients at the low concentrations used here. Because of these difficulties, we are unable to match the r ~a previous excellent accuracy achieved by Dunlop and C u s ~ I e in study of ternary diffusion with the magnetically stirred diaphragm cell technique. Nevertheless, our results demonstrate conclusively
'
1
,
!
' I
'
1
C.:O0190M
I
02
00
I
I
'
I
I
,
I
I
06
04
,
I 1
08
10
jtii05s c m 2
Figure 3. Same as in Figure 2 except c1 = 0.0190 M and c2 = 0. TABLE I: Diaphragm Cell Data" at 25 O C for 0.0126 M Total CI2 (1) without Added Hvdrochloric Acid (2)
@tb 0.277 0.373 0.471 0.650 0.735 1.012
Aci(t)/Aci (0) 0.571 0.442 0.365 0.280 0.218 0.118
"Ac,(O) = 0.0252 M, a = 0.82.
AC2(t)IACI(O)
-0.184 -0.236 -0.218 -0.184 -0.200 -0.155 units of los s cm-2.
TABLE 11: Diaphragm Cell Data" at 25 OC for 0.0190 M Total C12 (1) without Added Hydrochloric Acid (2) @tb 0.273 0.360 0.480 0.650 0.980
Ac~(~)/AcI(O) -0.128 -0.165 -0.222 -0.164 -0.102
Acl(t)lAcl(o) 0.616 0.490 0.385 0.241 0.180
"Ac,(O) = 0.0380 M, a = 0.72.
units of lo5 s cm-2.
TABLE III: Least-Squares Parameters Determined from Diaphragm Cell Ruos c1 = 0.0190 M CI = 0.0126 M uarameter 01
21b
D,b
observed' 0.58 (0.05) -1.71 (0.06) 1.79 (0.06) 2.55 (0.05)
predicted 0.92 -0.80 1.56 2.86
observed" 0.90 (0.17) -1.77 (0.17) 1.90 (0.10) 2.49 (0.06)
"Quantities in brackets are standard deviations. cm2 s-I.
predicted 0.88 -0.71 1.61 2.73 units of
Isothermal Diffusion in Aqueous Solutions of Chlorine TABLE IV: Comparison of Observed and Predicted Ternary Diffusion Coefficients" for Aqueous Chlorine (1) without Added Hydrochloric Acid (2) at 25 OC c1 = 0.0126 M
Dik Dll DI2 D21 D22
observedb 2.12 (0.06) 0.11 (0.10) 1.28 (0.12) 2.22 (0.05)
"In units of viations.
predicted 1.67 0.12 1.04 2.76
~1
= 0.0190 M
observedb 1.96 (0.14) 0.03 (0.05) 1.05 (0.23) 2.43 (0.12)
predicted 1.74 0.16 0.80 2.60
cm2 8 . bQuantities in brackets are standard de-
that diffusion of aqueous chlorine is a ternary process. Furthermore, the data are in satisfactory agreement with theory. Strictly speaking, the experimental diffusion coefficients listed in Table IV are not differential diffusion coefficients but rather integral values representing averages with respect to concentration of the differential values over the range of compositions from the mean composition of the top compartment to the mean composition of the bottom compartment. However, check calculations (see Appendix) indicated that differences between the integral and differential values could be safely neglected since they were well within our experimental error. Estimates of the diffusion coefficient of aqueous chlorine reported previouslyes range from 1.35 X to 1.90 X cm2 at 25 O C . Because simple binary diffusion was assumed, these values are apparent binary diffusion coefficients representing unspecified averages of the ternary diffusion coefficients of the system. Therefore, we are unable to make a detailed comparison of our results with previous measurements. The present study has shown that the transport coefficients of aqueous chlorine are sensitive to concentration. This has certainly contributed to the wide scatter of the r e p ~ r t e d diffusion ~-~ coefficients. Apparent chlorine diffusivities are generally lower than our values of D l l ,the true diffusivity of the chlorine component. This discrepancy is understood as follows: because diffusing chlorine creates a gradient with opposite sign in the concentration of the HC1 component, a countercurrent flux of chlorine equal to D12Vc2 is generated which lowers both the overall flow of the chlorine component and its apparent diffusivity. It is interesting to compare the ternary diffusion coefficients listed in Table IV with apparent diffusion coefficients obtained by assuming simple binary diffusion. At the end of each diaphragm cell run, the upper solution contains chlorine and additional HC1, while the lower solution contains chlorine and additional HOCl left behind by the more rapidly diffusing HCl. The extra HOCl in the lower solution is titrated by thiosulfate along with the chlorine component. If we were unaware of the ternary nature of the solution, more chlorine (Le. the extra HOCl) would appear to remain in the lower compartment than is actually there a t the end of each run. Consequently, the apparent binary diffusivity of chlorine measured by the diaphragm cell technique is lower than the ternary diffusivity D l l . For a specific example, consider the data for 0.0 126 M C12listed in Table I. The initial difference in concentration of the chlorine
The Journal of Physical Chemistry, Vol. 89, No. 8, 1985 1491 component across the diaphragm is Acl(0) = 0.0252 M. At P t = 0.650 X lo5 s cm-2, the difference in chlorine concentration has decayed to (0.280)(0.0252 M) = 0.00706 M. At this point the difference in concentration of the HC1 component is (-0.184)(0.0252) = -0.00464 M. This means that the upper compartment contains 0.002 32 M extra HCl, while the lower compartment contains 0.002 32 M extra HOCl. Thiosulfate titration of the extra HOCl would yield an apparent chlorine concentration difference of 0.00706 + 0.00232 = 0.00938 M at the end of the run. Consequently, the apparent binary diffusion calculated from the e x p r e ~ s i o n ~ ~ - ~ ~
is (0.650 X 10-5)-1In (0.0252/0.00938) = 1.52 X cm2 s-'. Note that the ternary chlorine diffusivity obtained at this composition ( D l l = 2.12 X cm2 s-l) is 40% larger. This example serves to illustrate that the diaphragm cell technique yields apparent binary diffusivities for aqueous chlorine that are much lower than ternary values. The experiments reported in this study confirm that diffusion of aqueous chlorine produces substantial flows of HC1, even in solutions free of added HCl. The system C12(l) HCl(2) + H20 demonstrates an interesting exception to the r ~ l e Dzl ~ ~ -0 ~as ~ c2 0.
+
-
-
Acknowledgment. Eng Hon Gan provided skillful technical assistance. We are grateful to P. A. Lyons for the loan of diffusion cells. This work was supported by the Natural Sciences and Engineering Research Council of Canada. Appendix
Integral diffusion coefficients (&) determined by the diaphragm cell method are averages of the differential values (Dik) over the composition range from the mean composition of the top compartment ( E l T , t2T)to the mean composition of the bottom compartment ( E l B , tZB).lgd Numerical integration of
with ci - ZIT EE-
EiB
-
using the predicted concentration dependence of the Dik indicated that under our experimental conditions the integral values were within f0.04 X cm2 s-l of the differential values evaluated at the mean cell composition (6 = 0.5). Registry No. Clz, 7782-50-5; HC1, 7647-01-0; HOCl, 7790-92-3.
(32) Gosting, L. J. In 'Advances in Protein Chemistry"; Academic Press: New York, 1956; Vol. 11, pp 160-182. (33) ODonnel, I. J.; Gosting, L. J. In "The Structure of Electrolytic Solutions"; Wiley: New York, 19S9; Chapter 11. (34) Kim, H.; Reinfelds, G.; Gosting, L. J. J . Phys. Chem. 1972, 76, 3419.
