The Journal of Physical Chemistry, Vol. 82, No. 8, 1978
Velocity Correlation Coefficients
959
nomenon of self-diffusion. However, the motions a t various times are not strictly uncorrelated and these correlations of the velocities cause the reduction of A** to a lower value, here represented by A*. References and Notes *Ot
0
1
1
3
.
47%
Figure 8. Equivalent conductivity (normalized to dimension cm2 Si),
A", and hypothetical equivalent conductivity if ion-ion velocity correlations were absent, A " " (see text), as a function of the salt concentration at 25 OC.
however, this is not an essential condition; what is more important is that the mean fluidity of the water outside the first hydration sphere is reduced. Since the degree of reduction of fluidity of free water continues steadily with increasing salt concentration, all ions present in the solution suffer a retardation in their translational motion which causes the almost linear decrease of Di (i = a, c) (see Figure 2) and therefore of A** which is shown in Figure 8. I t may be shown that the direct electrostatic ion-ion contribution to this retardation is very small.29 Of course, a t any instant the ions are fluctuating and so perform translational motion, which in turn leads to the phe-
(1) D. G. Miller, J. Phys. Chem., 70, 2639 (1966). (2) R. Haase and J. Richter, Z. Naturforsch. A , 22, 1761 (1967). (3) H. S. Dunsmore, S. K. Jalota, and R. Paterson, J . Chem. SOC.A , 1061 (1969). (4) D. W. McCall and D. C. Douglass, J. Phys. Chem., 71, 987 (1967). (5) D. C. Douglass and H. L. Frisch, J . Phys. Chem., 73, 3039 (1969). (6) For consistency with other papers in this series the term selfdiffusion is used here in the broad sense of single particle diffusion as descrlbed by the integral of the autocorrelation function. (7) H. G. Hertz, Ber. Bunsenges. Phys. Chem., 81, 656 (1977). (8) H. G. Hertz, K. R. Harris, R. Mills, and L. A. Woolf, Ber. Bunsenges. Phys. Chem., 81, 664 (1977). (9) See, for example, W. A. Steele in "Transport Phenomena in Fluids", H. J. M. Hanley, Ed., Marcel Dekker, New York, N.Y., 1969. (10) 0.D. Fitts, "Non-equilibrium Thermodynamics", McGraw-Hill, New York, N.Y., 1962. (1 1) R. Mills and L. A. Woolf, "The Diaphragm Cell", ANU Press, Canberra, 1968. (12) K. R. Harris, R. Mills, P. Back, and D. S. Webster, J. Magn. Reson., submitted for publication. (13) D. S. Webster and K. H. Marsden, Rev. Sci. Instrum., 45, 1232 (1974). (14) R. Mllls, J . Phys. Chem., 77, 685 (1973). (15) A. F. Collings and R. Mills, Trans. Faraday Soc., 66, 2761 (1970). (16) J. H. Wang, J. Am. Chem. SOC.,75, 1769 (1953). (17) J. M. Fortes, Thesis, Montpellier, 1972. (18) G. Jones and M. Dole, J. Am. Chem. SOC.,52, 2245 (1930). (19) T. Shedlovsky and A. S. Brown, J. Am. Chem. Soc., 56, 1066 (1934). (20) R. Haase and K. H. Ducker, Z. Phys. Chem. (Frankfur?am Main), 54, 319 (1967). (21) G. Jones and M. Dole, J. Am. Chem. Soc., 51, 1073 (1929). (22) J. Richter, Thesis Aachen, 1967. (23) V. Vitagliano and P. A. Lyons, J. Am. Chem. Soc., 78, 1549 (1956). (24) J. R. Hall, B. F. Wishaw, and R. H. Stokes, J . Am. Chem. Soc., 75, 1556 (1953). (25) P. A. Lyons and J: F. Riley, J. Am. Chem. SOC.,76, 5216 (1954). (26) J. TBmas and K. UjszBszy, Acta Chim. Acad. Sci. Hung., 49, 377 (1966). (27) J. R. Jones, D. L. 0. Rowlands, and C. B. Monk, Trans. Faraday Soc., 61, 1384 (1965). (28) B. Brun, Thesis, Montpellier, 1968. (29) H. G. Hertz, K. R. Harris, and R. Mills, J . Chim. Phys., in press. (30) H. G. Hertz and R. Mills, J . Chim. Phis., 73, 499 (1976). (31) The suggestion made by Frank and Wen * that the second coordination sphere of any ion, Le., zone B in their terminology, has higher fluidity than bulk water, to the authors' knowledge has not yet found experimental support. (32) H. S. Frank and W. Y. Wen, Discuss. Faraday SOC.,24, 133 (1957).
Velocity Correlation Coefficients in Multicomponent Electrolyte Solutions L. A. Woolf Diffusion Research Unit, Research School of Physical Sciences, Australian National University, Canberra, A.C. T. 2600, Australia (Received October 3, 1977)
Equations are obtained which enable experimental transport and thermodynamic data for multicomponent electrolyte solutions to be used to calculate coefficients for velocity correlations between like ions, unlike ions, and ion-solvent. Tracer diffusion data for Nat, Cl-, and HzO in the system HzO-NaC1-KC1 are used with Onsager ionic phenomenological coefficients5 to obtain velocity correlation coefficients at five compositions. Introduction The transport properties of multicomponent solutions are determined by the microscopic interactions between the particles constituting the solution. Traditionally, however, the transport properties have been described in terms of phenomenological equations and there have been few attempts to relate the transport coefficients appearing in those equations to the microscopic properties of the 0022-3654/78/2082-0959$0 1.OO/O
solution. In 1969 Douglass and Frischl used the time correlation function approach to nonequilibrium statistical mechanics to relate velocity correlation coefficients to diffusion coefficients and particle mobilities for twocomponent systems consisting of three constituents. (Velocity correlation coefficients, VCC's, are the time integrals of ensemble averages of microdynamical velocities; see eq 2.) Some three years earlier, Miller2 had 0 1978 American
Chemical Society
960
The Journal of Physical Chemistry, Vol. 82, No. 8, 1978
L. A. Woolf
applied the phenomenological approach to obtain expressions in terms of Onsager coefficients for the transport coefficients of the ionic species in two component electrolyte solutions. Recently Hertz3 and Woolf and Harris4 have by separate approaches combined and extended the work of Miller2 and Douglass and Frisch’ to obtain expressions for the transport coefficients in terms of velocity correlation coefficients and values of the velocity correlation coefficients for two-component systems of two ions in water. Woolf and Harris4 used the ionic formalism of Miller, based on the concept of individual ionic electrochemical potentials which are physically unrealizable and related measured transport properties through Onsager phenomenological coefficients to velocity correlation coefficients of the individual ionic and solvent constituents. Hertz3 used a direct route through linear response theory to obtain expressions for the measured transport properties in terms of velocity correlation coefficients. Because he considered the constituents of the solution as neutral molecules some of the VCC’s were defined by use of an arbitrary instantaneous velocity for the anion-cation combination regarded as a salt “molecule”. The two approaches do not produce identical values of corresponding VCC’s except for the solvent-solvent coefficients. In this paper the work of Woolf and Harris4 is extended to three-component systems of four constituents comprising two single electrolytes with a common ion in a neutral solvent, water. These systems were previously considered in terms of Onsager phenomenological coefficients by Miller5 and for convenience and brevity this paper closely follows his terminology and utilizes some of his results; equations from ref 5 are given the prefix M. It should be noted, however, that Miller has used the solvent-fixed reference frame in his ~ o r k . ~ Here, , ~ @both because of custom in the statistical mechanical approach to transport processes, and because of an interest in obtaining information about ion-solvent correlations, the mass-fixed frame of reference is used. Only for the system H2O-NaC1-KC1 is experimental data available over a moderately extensive range of compositions (five in all) and, therefore, it is this system to which the present theory is applied. To enable calculation of some of the velocity correlation coefficients, tracer diffusion coefficients have been obtained for Nat, C1-, and H 2 0 in the electrolyte mixtures up to a maximum concentration of 1.5 M of each electrolyte.
Theory Isothermal transport of matter in multicomponent systems may be described by the phenomenological equation for the mass-fixed reference frame7
Jj = -
5 ajjxj
( i = 1,. *
.,n)
(1)
j= 1
where J,is the flow of species i in mol cm-2,Qij the Onsager coefficient in mol2 J-I cm-l s-’, and X I is the thermodynamic force in J mol-’ cm-’. The Qi, are related to the velocities of the constituent particles by’
with
(3) Here k is Boltzmann’s constant (1.3805 X J K-’1, T the absolute temperature, V the volume of the system
containing 1 = Niparticles of component i of individual mass miand velocity uL. By recalling that concentrations ci in mol cm-3 are defined in terms of the molecular weight Mi = Nomi where No is Avogadro’s number by c j = Nimi/VMj
(4)
and defining the velocity cross-correlation coefficient as
(5)
(7)
where
The velocity autocorrelation coefficient fi,, is related to the tracer diffusion coefficient Di by the Kubo relation
3Di A relationship between the ha, tiap, and f i j (i # j ) can be obtained by defining the mass-fixed frame of reference bf fiaa =
n
2 M i a i j= 0
( j = 1,.
i= 1
. ., n)
Then consideration of the quantity 3kT C$J4$ij leads to the relation
mifiaa + V (
j#i
cjMjfij
+ ~ j M i f i a p=) 0
(12)
which enables eq 7 to be reformulated as
In a system comprised of two cations, 1and 2, a common anion, 3, and a solvent 4 there are a total of 14 velocity correlation coefficients: the four autocorrelation coefficients fi,, come directly from tracer diffusion coefficients (eq lo), the six cross correlation coefficients fi, (i # j ) from the Q j j , and the four correlation coefficients between different particles of the same species f i a p can be obtained through either eq 7 or 12. For systems of three ions in water, Miller5 has used diffusion data, transference numbers, equivalent conductances, and thermodynamic data to obtain the six independent Onsager coefficients, lij, for the solvent-fixed reference frame 3
(Ji)4= - 2 liiXj j= 1
( i = 1, 2, 3)
The relation between the 1,. and Qij is given by7 lij = s 2 i j - ( C j / C & 2 4j - ( C j / C 4 ) S 2 4 i + ( c i c j / c 4 2 ) ~ 4 (i, 4 j = 1, . . ., 4)
(14)
(15)
I t is convenient to rearrange eq 15 by eliminating the solvent terms (species 4) using eq 11. The resulting equations can then be solved by matrix inversion to obtain numerical values of the Qij from the 1,. Expressions for Conductance, Transference, and Diffusion. Miller2 has shown that the conductance is inde-
Velocity Correlation Coefficients
The Journal of Physical Chemistry, Vol. 82, No. 8, 1978 961
TABLE I: Tracer Diffusion Coefficients in H,O-NaCl-KC1 a t 2 5 'CQ D x 109/(m2 8-1) c/(mol dm-') NaCl 0 0.25 0.25 0 0.50 0.25 0.50 0.50 0 1.50 1.50 1.50 1.50 0.25
KC1 0.25 0 0.25 0.50 0 0.50 0.25 0.50 1.50 0 0.25 0.50 1.50 1.50
Na+ 1.322 1,290 1.318 1.319 1.276 1.320 1.294 1.327 1.300 1.195 1.208 1.165 1.167 1.266
Cl1.964 1.913 1.914 1.963 1.854 1.924 1.881 1.904 1.939 1.677 1.716 1.671 1.662
H2O 2.35 2.30 2.29 2.38 2.27 2.25 2.20 2.28 2.40 2.12 2.10 2.11 2.06 2.32
I
/
d
a Sources of two-component data: Na' in NaCl, Na+ in KC1, C1- in KCl (R. Mills, Rev. Pure Appl. Chem., 11, 78 (1961)); H,O in NaCl (ref 10);H,O in KCl, obtained by conversion of tritiated water data of R. Mills, J. Phys. Chem., 1 7 , 685 (1973), using DHa0 = 1.03DHT0.
pendent of the frame of reference. Equation M12 may therefore be written
where = hN/10005 (17) and zi is the valence of ion i, A is the equivalent conductance, 5 the value of the Faraday, and N the total solution normality in equivalents of neutral salts per dm3 is N = 10001z31c3= 1000(zlcl + zzcz). Using eq 6 , 7, and 10 with 16
CY
vi # jE= 1
3
z: j#i=1
ZiZjCjCjfij
Transference Number, ti. As ti is reference frame dependent, customarily being for the solvent fixed frame, and the VCC's are for the mass-fixed frame it is necessary to replace the l,'s in eq M14 using eq 15 CYtj = zj
z3: z j [ n i j - (Ci/C4)Q4j
M h
ad
0 0
??
- (cj/c4)n4i t
-
0 0 0 0 0
I
j= 1
Y
I L+ Yl
(cicj/c42)n44] i = 1, 2, 3
(19) Specializing to cation 1, using eq 6, 7 , and 10 and the condition of electrical neutrality Cb,cizi = 0 with eq 11
+
Ga G+ /I mmo -'?e . 00% y
Y
0
rl
a
0
I
1
r l a
V[C2Z2f12 czzzfz4
2 8
+ C323fl3 - (Clzlfi4 +
i
+ C3Z3f~)l
(20)
Diffusion Coefficients. Because experimental measurements of diffusion in systems of three or more ions are presently described in terms of diffusion coefficients D,, for neutral components, Miller has obtained expressions relating neutral salt phenomenological coefficients, LiJ's, to ionic $'s (eq M28). The route to the D,j in terms of the 1, is tedious (via solution of eq M20) and of no value in the present situation where the interest is in the averaging
(0
0
e
0
0
l
o
0
9 rl
u
*
wo
The Journal of Physical Chemistry, Vol. 82, No. 8, 1978
962
of velocity correlations of individual particles. It will be apparent, however, from examination of eq M28, M20 with our eq 14, 6, and 7 that expressions for the D, in terms of VCC's will be quite complex.
Experimental Section Tracer diffusion coefficients of Na+ and C1- in the system H,O-NaCl-KCl were measured a t several compositions of the salt mixtures using the disphragm cell m e t h ~ d The . ~ NaCl and KC1 were analytical reagent grade and were not further purified; the radioactive tracers (22Na and 36Cl)were obtained from the Radiochemical Centre, U.K. Counting procedures followed standard technique^.^ Intra- or self-diffusion coefficients of H 2 0 in the salt mixtures were obtained using an NMR spectrometer.lO The values of DNa+and Da- in Table I should be accurate to fl% and of DHzOto f 2 % . Results and Discussion It is convenient to use Hertz's definition of VCC's3 A
fij A
fij
(i, j = 1, . . ., 4;i # j ) (i = 1,. . ., 3) = (c,N,V/3)fjap =
(CiNov/3)fjj
(21)
and to extend the definition to
(1~0~/3)fiap whe;e I is the ionic strength (I = 1 / 2 ~ ~ = 1 c , z 1 Values 2). of f44
=
the flj and f,, are given in Table 11. They were calculated using the measured tracer diffusion coefficients and the I,, given by Miller;5 denjsities of the mixed salt solutions were obtained from the original papers.ll Molecular weights used in the calculations were Na, 22.99; K, 39.098; C1, 35.453; and H,O, 18.016. The values of f l j listed in Table I1 includeAvaluesfor the systems H20-NaCl and H,O-KCl. These f L j are given in parentheses and for the cations are a t the same individual cation concentrations as those shown in the table; for the anion and the water the f l j are for the concentration of either NaCl of KC1 corresponding to the total anion concentration in the system H,O-NaCl-KCl. Because tracer diffusion coefficients were not obtained for K+ in the system it-was not possible to obtain values of the VCC for K+-K+ (fiz). Examination of Table I1 shows that the K+-H20, Cl--H20, and H20-H20 VCC's in the mixed electrolyte are virtually unchanged from those of the single electrolyte systems. For the anion and water this is in agreement with the single electrolyte systems4 where the C1--H20 and H,O-H20 VCC's for H20-NaC1 and H,O-KCl were found to be almost independent of the nature of the cation. The K+-H20 VCC's (f24)are also unchanged from the H,O-KCl system which suggests that the ion-water interactions for K+ are not affected by the other cation or by the increased C1- concentration. This is in accord with the common
L. A. Woolf
interpretation of little, if any,permanent hydration of the K+ ion. The Na+-H20 VCC, f14, however, is somewhat less negative than in H20-NaC1 and this may indicate the decreased repulsion between oriented water molecules of the hydration sphere of the cation and free water molecules due to closer packing of ions and water molecules in the multicomponent solutions. Both the K+-Cl- and Nat'-C1- VCC's (f23 and f13) ?re considerably diminished compared to the respective single-electrolyte systems. This suggests that the freedom of movement of one cation species is assisted by the presence of a second cation species; a finding in agreement with what can be argued from simple considerations of electrical neutrality. Support is given to this interpretation by the more negative values of the VCC for Na+-Na+ (til) which are almost an order of magnitude greater than for the H20-NaC1 system. The VCC's for Na+-K+ (f12) are, as expected, negative and become more negative with increasing concentration as is also seen for the Cl--Cl- VCC (f3& This effect is most apparent for increasing Na+ concentration (compare second and third and third and fourth lines of Table 11) which may be related to the higher chargelsize ratio of Na'. Velocity correlation coefficients provide a fundamental bridge between direct experimental measurements of transport properties of solutions and calculation of such properties based on statistical mechanical theories. At present it is possible in principle to undertake such calculations by computer simulation of molecular dynamics. For electrolyte solutions of the concentrations considered here, however, a suitable form of the potential functions for ion-ion and ion-solvent interactions is not available. Further development of the VCC approach must therefore await either increased development of the statistical mechanics of electrolyte solutions or a much more extensive accumulation than exists at present of experimental measurements of diffusion, conductance, transference numbers, and activity coefficients in multicomponent electrolyte solutions.
References and Notes (1) (2) (3) (4)
(5) (6) (7)
(8) (91 ~, (10) (11)
D. C. Douglass and H. L. Frisch, J . Phys. Chem., 73, 3039 (1969). D. G. Miller, J . Phys. Chem., 70, 2639 (1966). H. G. Hertz, Ber. Bunsenges. Phys. Chem., 81, 656 (1977). L. A. Woolf and K. R. Harris, J. Chem. Soc., Faraday Trans. 1 , in press. D. G. Miller, J . Phys. Chem., 71, 616 (1967). D. G. Miller, J . Phys. Chem., 71, 3588 (1967). J. G. Kirkwood, R. L. Baldwin, P. J. Dunlop, L. J. Gosting, and G. Kegeles, J . Chem. Phys., 33, 1505 (1960). D. D. Fitts, "Non-Equilibrium Thermodynamics", McGraw-Hill, New York, N.Y., 1962. R. Mills and L. A. Woolf, "The Diaphraam . - Cell", ANU Press, Canberra, Australia, 1968. K. R. Harris, R. Mills, P. J. Back, and D. S. Webster, J. Magn. Reson., in press. I. J. O'Donnell and L. J. Gosting in "The Structure of Electrolytic Solutions", W. J. Hamer, Ed., Wiley, New York, N.Y., 1959; P. J. Dunlop, J . Phys. Chem., 63, 612 (1959).
