J. Phys. Chem. 1981, 85,1137-1146
spectrometers. The Harvard spectrometer is supported by NSF Grant GP-37066X, the Kansas spectrometer by NSF Grant MPS 74-22178. Calculations were carried out at the University of Connecticut Computer Center.
1137
Supplementary Material Available: Tables 111-VI containing assigned microwave bands of S-ethyl thioesters (5 pages). Ordering information is available on any current masthead page.
Explicit Relations of Velocity Correlation Coefficients to Onsager b’s, to Experimental Quantities, and to Infinite Dilution Limiting Laws for Binary Electrolyte Solutions Donald G. Mlller‘ Diffusion Research Unit, Research School of Physical Sciences, Australian National University, Canberra, A.C. T. 2600, Australia (Received: October 2, 1980)
The relation of velocity correlation coefficients v i , ) to the Onsager formalism of irreversible thermodynamics is considered for a binary electrolyte. Explicit expressions are derived by relating Woolf-Harris f i j (1) to solvent-fixed Onsager lij, (2) to experimental quantities, and (3) to dilute solution limiting laws. The f i j equations are more complex than 1, eq.uations. Since velocity correlation integrals are the theoretical basis for f j j , limiting law expressions are used to investigate the behavior of these integrals at infinite dilution. All ion-ion integrals become infinite there. This implies that direct computer simulations may not be possible, and that some other correlation function approach may be more suitable for the electrolyte problem.
I. Introduction Linear transport processes can be described macroscopically in more than one way. The oldest established framework is the irreversible thermodynamics of Onsager.2-6 I t has been successfully applied to vector transport properties in electrolyte solutions by Onsagerzs7and o t h e r ~ , ~and - l ~ was recently reviewed.20 The resulting transport coefficients are the Onsager I , or friction coefficients Rip Our interest will be in the solvent-fixed lip9 An alternative method is based on linear response theory and involves time correlation functions.21 This approach (1)Visiting Fellow, A.N.U. Permanent address: L-202, Earth Sciences, Lawrence Livermore National Laboratory, Livermore, CA 94550. Portions of this work were done under the auspices of the U.S.Department of Energy by Lawrence Livermore National Laboratory under contract No. W-7405-ENG-48. Reprints available at LLNL. (2) L. Onsager, Phys. Rev., 37, 405 (1931); 38, 2265 (1931). (3) S. R. DeGroot a n d P. Mazur, “Non-Equilibrium Thermodynamics”, Interscience, New York, 1962. (4) R. Haase, “Thermodynamics of Irreversible Processes”, Addison Wesley, Reading, MA, 1969. (5) J. Meixner and H. G. Reik, “Thermodynamik der Irreversiblen Prozesse”, in “Handbuch der Physik”, Vol. 111/2, S. Flugge, Ed., Springer-Verlag, Berlin, 1959. (6) D. G. Miller, Chem. Rev., 60, 15 (1960). (7) L. Onsager and R. M. Fuoss, J . Phys. Chem., 36, 2689 (1932). (8) D. G. Miller, J. Phys. Chem., 64, 1598 (1960). (9) D. G. Miller, J. Phys. Chem., 70, 2639 (1966). (IO) D. G. Miller, J . Phys. Chem., 71, 616 (1967). (11) D. G. Miller, J . Phys. Chem., 71, 3588 (1967). (12) R. Haase and J. Richter, 2. Naturforsch. A , 22, 1761 (1967). (13) R. W. Laity, J. Chem. Phys., 30, 682 (1959). (14) R. W. Laity, J . Phys. Chem., 67, 671 (1963). (15) D. G. Miller and M. J. Pikal, J. Solution Chem., 1, 111 (1972). (16) H. S. Dunsmore, S. K. Jalota, and R. Paterson, J. Chem. SOC.A, 1061 (1969). (17) S. K. Jalota and R. Paterson, J. Chem. SOC.,Faraday Trans. I , 69, 1510 (1973). (18) A. Agnew and R. Paterson, J . Chem. SOC., Faraday Trans. 1,74, 2885 (1978). (19) A. Agnew and R. Paterson, J . Chem. SOC.,Faraday Trans. I, 74, 2986 (1978). (20) D. G. Miller, Faraday Discuss. Chem. SOC.,64, 295 (1977).
0022-3654/81/2085-1137$01.25/0
was initiated by McCall and Douglass22and Douglass and Fris~h.2~ A much more complete application to electrolyte solutions has been carried out by Hertz and collaborat o r ~ , and ~ ~ from - ~ ~a different viewpoint by Woolf and Harris.ni28 The results are velocity correlation coefficients (vcc) for various solution species, denoted by f i p Hertz et al.2“-26have gone directly from linear response theory to microscopic vcc’s, which in turn can be based on experimental quantities. The Hertz f i j coefficients reflect ion-ion, salt-salt, sal-water, and water-water interactions, but no ion-water interactions. Woolf et al.27v28have extended the work of Douglass and F r i s ~ hwhich , ~ ~ was based on using ions as species and on connecting the macroscopic mass-fiied Onsager coefficients (denoted by Qij) to microscopic velocity correlation functions. Woolf et al. defined their vcc in terms of these Q . . and used experimental quantities which were transformea into the relevant frame of reference by standard proced u r e ~ . The ~ ~ Woolf f i j coefficients reflect ion-ion, ionsolvent, and solvent-solvent interactions, but no salt-salt or salt-water interactions. These two vcc descriptions should be equivalent, since they both are macroscopic and describe the same phenomena. This equivalence can be tested since there is a common subset of ion-ion f i j among the different larger (21) W. A. Steele in “Transport Phenomena in Fluids”, H. J. M. Hanley, Ed., Marcel Dekker, New York, 1969, p 209. (22) D. W. M c C d and D. C. Douglass, J. Phys. Chem., 71,987 (1967). (23) D. C. Douglass and H. L. Frisch, J. Phys. Chem., 73, 3039 (1969). (24) H. G. Hertz, Ber. Bunsenges. Phys. Chem., 81, 656 (1977). (25) H. G. Hertz, K. R. Harris, R. Mills, and L. A. Woolf, Ber. Bunsenges. Phys. Chem., 81, 664 (1977). (26) H. G. Hertz and R. Mills, J. Phys. Chem., 82, 952 (1978). (27) L. A. Woolf and K. R. Harris, J. Chem. SOC.,Faraday Trans. I , 74, 933 (1978). In Tables 2-7, all fij* should be labeled f i , and in Table 2, f,, should be fc,. In eq 22, the denominator No should be replaced by CBNO
(28) L. A. Woolf, J. Phys. Chem., 82, 959 (1978). (29) J. G. Kirkwood, R. L. Baldwin, P. J. Dunlop, L. J. Gosting, and G. Kegeles, J. Chem. Phys., 33, 1505 (1960). Their equations are all in
mass units instead of moles, but the results needed here will also be valid in mole units as well.
0 1981 American Chemical Society
1138
The Journal of Physical Chem;stty, Vol. 85, No. 9, 1981
sets of dependent f . ? sof each description. Specifically, numerical results an3 expressions in terms of experimental quantities should be identical for the common subsets. At present there remain some unresolved discrepancies, and this issue will be briefly discussed later. We are interested in explicitly relating the macroscopic Onsager formalism for solvent-fixed quantities to the macroscopic velocity correlation coefficient formalism. This is most easily done following the path of DouglassFrisch and Woolf-Harris. Since the f i j are defiied in terms of the Qy and since Onsager coefficients can be transformed from one reference frame to another by standard metho d ~the , fij-lij ~ relationship is essentially a reference frame transformation within the Onsager formalism. Woolf and Harrisz7only partially carried out this fj1--lij transformation. Their principal goal was to provide numerical values of the velocity correlation coefficients f i j from experimental data for binary systems. These f i , are defined in terms of the six mass-fixed Onsager coefficients Qij, of which three are independent. Moreover, there are three solvent-fixed Onsager coefficients 1 , which can be related to the three independent Qij. These three 1.. had already been calculated from experimental data anJ tabulated>J2J6J7 Therefore from the expression for 1, in terms of Q i . (obtained from the frame of reference conversion), Wooif and Harris obtained the three independent ai.by numerical matrix inversion. The other three dependent Qij were calculated by their defining equations. The six f i j were then obtained from the six Q i j and the three required self-diffusion coefficients. However, explicit relations for the f i j in terms of either 1, or experimental quantities were not obtained. The purposes of this paper are to extend the WoolfHarris results to get explicit equations for the f i j in terms of the !ij, and in turn for f i j in terms of experimental quantities. The latter will make it possible to calculate the fija directly from tabulated experimental data. Such experimental f i j for real systems can then be compared against theoretical or computer models. In addition, the f i j will be expressed in terms of a Debye-Huckel limiting law model, since no other model theories are presently available explicitly for the f i j The limiting law used is due to Pikal,3O but ultimately goes back to Onsager and Fuoss.’ All results will apply to any valence-type binary electrolyte. Finally, we will present an explicit expression for D in terms of f L j to add to the earlier Woolf-Harris equations for conductance and transference number.27 We will find the f i j are complex functions of the lip They are also somewhat more complex than the 1, in terms of the following experimental quantities: transference numbers ti,conductance A, diffusion coefficient D, and selfdiffusion coefficients D;.The limiting law equations are equally complicated, without any cancellation of electrophoretic or relaxation terms. The complexities are primarily due to mass weighting. We will also find that the velocity correlation integrals for ion-ion correlations diverge at infinite dilution. Before turning to derivations, some comments may be in order about the relative advantages and disadvantages of the 1, and f ; j approaches for a binary system. (Similar considerations will apply to multicomponent systems.) Only three independent 1, are required to describe vector transport in an isothermal binary system, and the equations relating the 1, to standard experiments and vice versa are relatively ~ i m p l e An . ~ extended Debye-Huckel theory also exists.30 On the other hand, because the lij are solvent fixed, there are no explicit coefficients for ion-solvent or (30) M. J. Pikal, J. Phys. Chem., 75, 3124 (1971).
Miller
solvent-solvent interactions. Indeed, the extended Pikal theory shows those interactions to be part of all three lij terms.30 There is also no direct way to bring in the selfdiffusion coefficients of the ions and solvent in the binary description. However, ion self-diffusion coefficients can be included by expanding the binary to a 3- or 4-component system which includes tagged ions of one or both types (i.e., isotopic diffusion)?i32 The results are relatively simple, and connect the self-diffusion coefficient Di of ion i to lii and an isotope coupling constant lii*. The selfdiffusion coefficient of the water cannot be included in this solvent-fixed formalism. The f i j (and Q i j ) are related to velocity correlation integrals, which in principle could be obtained by molecular dynamics computer calculations. Moreover, the autocorrelation function is a natural outgrowth of this formalism and is essentially the self-diffusion coefficient. Thus the f i j bring in the Di and the self-diffusion coefficient of the water D, in a natural way. On the other hand, the f i j are not symmetric. There are six of them, and their expressions in terms of experimental quantities are more complicated than those for the Zij, In addition, the large variation of the f i j , due largely to their mass-fixed nature, skews any molecular interpretation, because mass interactions are small relative to other particle-particle interactions. Moreover, limiting law expressions do not show any simplicity. Finally, although f i j and Q i j are well behaved at infinite dilution, the more basic velocity correlation integrals become infinite there. This divergence may reduce the physical insight to be gained, and may well make direct computer simulations impossible. As far as a macroscopic description for electrolyte transport processes is concerned, the choice is of course a matter of taste or convenience. The results of this investigation should be of value to those interested in vcc’s or Qi., but the hoped-for simplicity and explication were not iound. 11. Preliminary Considerations Consider a binary (strong or weak) electrolyte in a
neutral solvent which ionizes as CrlAr, = rlCpl + r2AZ2
( 1) and where 0, 1, and 2 refer to the solvent, cation C, and anion A, respectively. Then rlzl + r2z2= 0 (2) ci = ric i = 1, 2 (3) N = rlzlc (4) r = r1 r2 (5)
+
c1q2 + c2zz2
I=(
)=
rlzlc(zl - z2) 2
-- N(zl - zz) 2
rzlz2c
--
2
(6)
where C is the cation; A the anion; ri the stoichiometric coefficients; zi the signed valence in equiv mol-I; ci the concentration of species i in mol dms; c the concentration of electrolyte-as-a-whole in mol dm-3; N the normality (equivalent concentration) in equiv dm-3; r the sum of the stoichiometric coefficients; and I the ionic strength in mol dm-3. The concentration of the solvent co in mol dm-3 will be used later. (31) S. Liukonnen, Acta Polytech. Scand., Chern. Incl. Metall. Ser., No. 113, 1 (1973).
(32) J. Anderson and R. Paterson, J. Chem. SOC.,Faraday Trans. 1, 71, 1335 (1975).
The Journal of Physical Chemistry, Vol. 85, No. 9, 1981 1139
Velocity Correlation Coefficients
The soluent-fixed Onsager coefficients 1, are introduced by (7)
where (JJ0is the flow of ion i referred to the soluent-fixed reference frame in mol cm-2 s-l, and
The integrand of the second integral is called the like-ion cross-correlation function. The remaining symbols used in eq 14-16 are defined in ref 27. The fij are defined by “normalizing” the integrals in eq 14-16 with respect to concentrations, as given in eq WH25, WH26, and WH45. When these are substituted into eq 14-16, we obtain the following expressions for the fij in terms of the Qij, with due regard to units: 1OOORTQij fij = i, j = 0, 1, 2 (17) Cj
where Xj is the thermodynamic force, pj the chemical part of the electrochemical potential in J mol-l, F the Faraday in C equiv-l, q5 the electrical potential in V, and x the distance in cm. The 1, have the units mol2 J-I cm-l s-l. The mass-fixed Onsager coefficients Qij are introduced by
where (JJM is the flow of ion i referred to the mass-fixed reference frame. The Qij have the same units as the 1,. Note particularly that for this system of coefficients Qij, the i and j include the solvent as well as the ions. Consequently, the ai, are a dependent set. This contrasts with the 1, system which has only ion terms and has an independent and minimal set of Coefficients. Since the mass-fixed reference frame is defined by
where Mi is the molecular weight in g mol-l, it can be shown29that 2
CM.Q11 = 0 1
i=O
j = 0, 1, 2
(11)
Both the li.and Oil satisfy the Onsager reciprocal relations ( O R R ) ~ 1,. 41 = 1.. 11 (12) Q U. . = QI.1 .
(13)
Consequently, only three of the four 1, are independent, and only six of the nine Qij are different. These six can be reduced to three independent ones by means of eq 11. Two such independent sets of three are all, QZ2, QI2 and QlO,
Q20, F12.
According to linear response theory, the Qijare directly related to velocity correlation integrals. These can be obtained from eq WH7-11 and WH14-16, where the prefix WH indicates equations taken from Woolf and Harris.27 The results are EiEjV Qij
=
$, (uia(O).uja(t)) dt m
i #j
(14)
Q.. =
(15) where ciis the concentration in mol ~ m - V~ the , volume in cm3, and k the Boltzmann constant in J deg-’. The integrand of the first integral in eq 15 is the autocorrelation function and is related to the self-diffusion coefficient Di by the relation
Di = ‘/31m(uia(o).uia(t))dt
(16)
fii
=
lOOOR TQii
-Di
i = 1,2
(18)
Ci
where Diis in cm2s-l, co is the concentration of the solvent in mol dm-3, R the gas constant (8.3144 J mol-’), and T the absolute temperature in degrees K. The fij have the units cm2 s-l, and 1000 is the conversion factor between cm3 and dm3. The ORR, eq 13, makes the Qij symmetric. The fij are not symmetric, but do satisfy the relation fij/ci = fji/cj
(20)
Therefore the nine fij reduce to six different ones. These six are related by the three eq 11, reducing the six to three independent fip However, the three Di represent additional required quantities. Consequently, the fi; formalism requires six actual coefficients for a binary system. Upon eliminating fil, fol, and fo2from eq 11 by use of eq 20, these relations, not given explicitly in ref 27, can be written as ClMl(D1 + fll) czMz(D2 + fzz)
+ c2Md12 + C d M O f l O = 0 + czMj12 + c&of20 = 0
(21) (22)
A symmetric form of cross-correlation coefficient (denoted by fij) can easily be defined by simply replacing c ’ in eq 17 by (c;c~)’/~.The use of a symmetric fij is suggested both for convenience and because it seems more reasonable that the correlation of particle A’s velocity with particle B’s velocity should be the same as the correlation of B’s velocity with A’s velocity. Finally, we note the notational differences between this work and Woolf et al.27,28The solvent is denoted here by 0 rather than 3; the concentration is in the customary mol dm-3 (mol L-I) rather than mol ~ m - and ~ ; the fij are in cm2 s-l as in ref 28, rather than m2 s-l as in ref 27. 111. lij in Terms of
Qij and $j The relation of 1, to Qjj is easily determined by the methods of ref 29, using eq 10 and the Gibbs-Duhem equation. The resultz7in our notation is
Equation 24 directly connects the 1, to mass-fixed Onsager coefficients, and in turn to velocity correlation functions by eq 14 and 15. To get the li; in terms of fij, eq 17-19 solved for Qtj are substituted into eq 24. Since the tabulated quantities are liJ/N, eq 3 and 4 are also used to get the desired results (equivalent to eq WH21, 22):
Miller
The Journal of Physical Chemistry, Vol. 85, No. 9, 1981
1140
112 _ AT I.
d l 2
103rlzlRT
- rd10 - rJzo
+ rlr2 - + foo (26)
The units of lij/N are mol2 dm3 J-l cm-’ s-l equiv-l. Comparison of the relative sizes of individual fij and Di terms calculated from experimental data for sample systems from 0.01 to 3N27*33 yields the following. For Zii/N, the largest contribution at all concentrations is from Di, with the negative fii (related to the like-ion cross correlation coefficient) becoming increasingly important as c increases. The fi0 term is next most important. The Do and fo0terms mostly cancel each other at all concentrations. For 112, the flz are dominant at lower concentrations, but flo and fiO dominate at the highest concentrations. Again the Do and fooquantities, while individually not so small at higher concentrations, nearly cancel each other at all concentrations. Equation 25 for lii/Nis interesting in that ii, io, and 00 interactions are present, but 12 or j 0 (I’ # i) are not. Such 12 and j 0 interactions would be expected intuitively from possible ion association for some salts, and are specifically predicted by Pikal’s theory30 for the lit The appearance of solvent-solvent terms could be considered implausible in solvent-fixed coefficients. However, the full set of f i . is a dependent set, so that the 00 term can be eliminated from all the 1,/N by means of eq 23. Elimination of the 00 terms yields
_ 112 N -
[
1O3r1zlRT r2f12 - r2( 1 +
%)flO
- rl(
1+
eliminate all fio contributions. However, it is not possible, for example, to get Zll/N in terms just of 11, 12, and 10 terms. Comparison of the relative sizes of terms of eq 28 and 29 up to 3N yields the following results. The Zii are completely dominated by the Di + fii term at all concentrations. The other terms are small, with the f12term larger at lower concentrations. The 112 term is dominated by f 1 2 at low concentration, but the io terms increase rapidly and are larger at the highest concentration. The difference in relative term sizes between set (25,26) and set (28,29) is interesting, since the (Q + fii) term dominates lii much more in eq 29 than it does in eq 25. The 12 terms are essentially the same. Consequently, the non-uniqueness of these expressions for 1, in terms of f i j makes it difficult to see whether f i j are really a more physical or basic description of transport. The limiting law expressionsto be presented in section VI1 will help us compare.
IV. Q i j in Terms of Zij Explicit expressions for the f i j in terms of the lij are best obtained through the Qi,, followed by use of eq 17-19. In order to solve eq 24 for the Q j j , these must be reduced to a minimum set. This is done by means of eq 11. An appropriate such set consists of &, Q22, and Q1p Before proceeding it is convenient to define densities of ions i, solvent, and solution as pi = ~iMi i = 0, 1, 2 (30) P = PO + PI + PZ = C&O + CiM1 + c2M2 (31) These densities have the units g dm-3 (g L-l) owing to our choice of concentration units. The use of eq 11 yields the following expressions for Q1o, Q20, and Roo:
Substituting these into eq 24 yields (28)
Po2h1
Equation 27 is also interesting in that 12 terms still do not appear explicitly, although the solvent-unlike ion term has appeared. However, it is possible to introduce 12 terms into liJN by eliminating the 2fi0 term in eq 27 by the use of eq 21 and 22. The result is
PO2122
*C)Of M 2O 0]
2~2M1M2 COMdMi
fl2 -
CiMJ10
1
~iMd20
-- COMO
CdMO
= 1,
(29)
Equations 28 and 29 are a set which contain all the expected terms in each lij/N, since, for example, a 20 contribution to lii/Ncould be expected as an indirect impact on a solvent-fixed coefficient from ion 2 disturbing the solvent structure. I t is also possible to rewrite eq 28 and 29 in terms only of 11, 22, and 12 terms, by using eq 21 and 22 to totally
+ P1)2Qll + c12M22Q22+ 2ClMZ(PO + Pl)Q12 (34)
= C22Ml2Q11 + (Po
+ P 2 ) 2 Q 2 2 + 2CZMl(PO + P2)Q12 (35)
=
PO2112
CZMl(P0
+ P1)Qll + ClMZ(P0 + P z ) Q 2 2 + (Po2 + POP1 + POP2 + 2PlPZ)QlZ
(36) Since the 1, and Qij. in eq 34-36 are both minimal sets, the Qij can be solved in terms of the 1, by Cramer’s rule, etc., with the following explicit results: Qi1
=
[(PO
+ PZ)+ ~L ~ i ~ M 2 ~-l z2~1M2(~0 2 +PZ)~~ZI/P~ (37)
a22
= [C22M12111+ (Po + P1)2h2 - 2CZMAPO
+ P1)k!1/P2 (38)
Ql2
= [-CzM1(Po
+ P&ll
- ClMZ(P0 + P l h + (Po2 + POP1 + POP2 + 2P1PZ)l121/P2
Equations 32 and 33 yield the after some cancellation, are 010
(33) The D iwere obtained from L.A. Woolf, private communication. References for these data are in ref 27, but the actual values used were not tabulated there.
= (Po
Qio
(39) and Qw. The results,
=
-cowl(Po
+ P&ll
- C1MZ24,2 + %(Po
+ Pz
-
Pl)llZl/P2I
(40)
Velocity Correlation Coefficients a20
= -col[-c2M12111
The Journal of Physical Chemlstry, Vol. 85, No. 9, 1981 1141
+ M Z ( P 0 + Pl)l22 + Ml(P0 + P1 P2)h1/P21 (41)
= ~o~([M?111 + Mz2122 + 2MiM21123/~~l (42)
Equations 37-42 could also be written in terms of weight fractions wi Wi
= Pi/P
with some multiplication and division of molecular weights. In addition the units of c, cancel out because the numerator and denominator have c to the same power. Consequently, c can be in any consistent set of units. Equations 37-42 are the first set of desired results. They are the expressions needed to obtain the fij, and show explicitly the mass weighting factors not observed in a purely numerical matrix inversion. Because the are also directly connected to velocity correlation integrals through eq 14 and 15, these equations are useful in obtaining experimental values of those integrals from tabulated 1,.
V. 4j in Terms of lij/N Given the Qij in terms of l,, we can obtain the fij explicitly in terms of lij using eq 17-19. Since 1, are typically tabulated as lij/N, the results will be expressed in this form by means of appropriate stoichiometric factors. The expressions are given by eq 43-48. These equations also
Equations 43-48 are our second set of desired results and allow the direct calculation of fij from tabulated Iij data*12J"19*34without the need of a numerical matrix inversion. They also show the explicit weighting factors due to the mass-fixed nature of the fij. Finally, they will be used to obtain the fv in terms of experimental quantities and in terms of limitmg law expressionsbecause the Zij have already been given explicitly in terms of those quantities.&gBO
ej
VI. in Terms of Experimental Quantities The necessary experimental quantities to obtain solvent-fixed li.in a binary electrolyte solution are tl, A, DB, and 1+ c(d yldc), where tl is the Hittorf (solvent-fixed) transference number of ion 1, A is the equivalent conductance, Ds is the solvent-fixed mutual diffusion coefficient, and y is the activity coefficient for concentrations in mol dm-3. The appropriate expression is eq 41 of ref 9, which in the above notation is
_lij N-
titjA
rirjDB
+ 1 0 3 F z ~ i ~ j103RTrrlzl(l + c d In y/dc)
i, j
= 1, 2
(49) The quantity t2is obtained from the expression tl tz = 1
+
(50) For practical calculations, it is convenient to use the commonly tabulated volume-fixed mutual diffusion coefficient D' and molality activity coefficients y. These may be used in eq 49 because it can be showng that D' (1+ m d In r/dm)
-
+
DB
(1 c d In y/dc)
(51)
where m is the molality in mol (kg of H20)-l. It is also convenient to define the quantity fl2
=
103rlz1RT .
r2 122
P1)E
a =r(1 + c dDSIn y/dc)
111
-czM1(Po + P 2 ) E - ClMZ(P0 +
+ (Po2 + POP1 + POP2 + 2PlPdG 112
I1 /P2
(45)
have c's to the same power in both numerator and denominator of each term, so that any consistent set of c units can be used.
(52)
to abbreviate the desired equations. We obtain our third set of desired resulh by substitution of eq 49 in eq 43-48 and use of eq 52:
~~~
~~
~
~~
~~
(34) A. J. McQuillan, J. Chem. SOC., Faraday Trans. 1,70,1558 (1974).
1142
Miller
The Journal of Physical Chemistry, Vol. 85, No. 9, 1981
foo =
RT(piti - ~ 2 t 2 ) ~ A( P I + P ~ ) ~c B + P2 - -Do (58) VlFP2 CO
These equations are simpler than those for f i j in terms of lij, but clearly are also mass weighted. They allow the direct calculation of the f i j from experimentaldata, without having to go through the 1,. The expressions for f 1 2 and the first two terms of fll and f i 2 have a certain similarity in form to eq 49. This can be emphasized by considering mass-fixed experimental quantities as follows. Woolf and Harris2' derived an expression for the mass-fixed transference number for the cation, using standard transformations.29 Their result, eq WH43, leads immediately to
+ P2 t l m = Pot1 -
(59)
P t2m =
Pot2 + P1 P
where tim is our notation for mass-fixed transference numbers, and we recall that ti are the solvent-fixed Hittorf numbers. From eq 59 and 60, we obtain Po
( P A r n - P2t29 = --(Pltl P
- Pd2)
(61)
Fick's law for electrolyte diffusion can be expressed by either
or
where Dm is the mass-fixed diffusion coefficient. The standard transformation procedure,29using eq 10 for the salt35and recalling that (Jo)ois 0, gives
Consequently, we have Po -DB
D m
(65)
P
and analogously a m
Po = -a,
(66)
P
which shows the similarity of eq 67-69 to eq 49. The relative numerical contributions of various terms to the f i j have the following general features. For the f i i , the A and D terms are both positive and more or less the same size. However, they add up to less than Di, so that the f i i are negative. The A and D terms in f 1 2 are nearly the same, but the A term is negative and the D term is positive, hence they nearly cancel. The sum is typically positive at lower concentrations, but for some salts the cancellation leads to negative values at higher concentrations. For tio,the A term can be negative depending on relative values of the ti but is almost always smder than the D term. For foe, the D and A terms are very small compared to Di so that fo0 is negative,
VII. Limiting Law Expressions for 4, Some 15 years ago Friedman used the correlation function approach to calculate the conductance of a dilute electrolyte and recovered the Debye-Huckel limiting laws for conductance. However, equations for all the velocity correlation integrals (eq 14 and 15) were not given explicitly in practical terms. There has also been a computer simulation of the diffusion coefficient by use of correlation functions and Brownian dynamics,41but again values for eq 14 and 15 were not presented. Consequently, there is presently no convenient model theory for the Qij or f i j , There is, however, an extended Debye-Huckel theory for the 1 , due to PikaL30 Therefore our results for f i j in terms of lij, eq 43-48, can be used to get f i j in terms of this Debye-Huckel model. However, Pikal's full extended theory is rather complex. Since most of the conceptual features of this Debye-Huckel model are exhibited by Pikal's "simplified limiting law" (SLL), we shall use it to obtain a dilute solution model theory for the fij. Pikal's expressions for the lij are in terms of lii/ci and lij/ (cicj)lI2for a multicomponent system. These SLL equations can be specialized to a binary system and written in simple form in terms of l i j / N after some manipulation of valence and stoichiometric factors. The results are
Since A and the self-diffusion coefficients Di do not depend on the reference frame,g we can rewrite eq 53-58 in terms of our desired mass-fixed quantities as
RT(tlm)2A rlpoa)m fll
=
f22
=
f12
z P
+--
D1
(67)
D2
(68)
+-rlpoBm P
(69)
P
RT(t2m)2A r2poBm (-z2)F
=
+--
-RTtlmtzmA Z l F
P
(35)The flow of salt is J, = Jl/rl = J z / r z ,since in diffusion electroneutrality requires that both ions move together.
where (36)H.L. Friedman, Physica, 30, 509 (1964). (37)H.L.Friedman, Physica, 30,537 (1964). (38)H.L. Friedman, J. Chem. Phys., 42,450 (1965). (39)H.L. Friedman, J. Chem. Phys., 42,459 (1965). (40)H.L. Friedman, J. Chem. Phys., 42, 462 (1965). (41)P.Turq, F.Lantelme, and H. L. Friedman, J. Chem. Phys., 66, 3039 (1977).
Velocity Correlation Coefficients
The Journal of Physical Chemistry, Vol. 85, No. 9, 198 1
k=-- 0.10740 x 10-l2 103172 ka P=r1r2 kbrlr2
(75)
1O3r1zlRTlk( P I ~ ~+ ?[ ( P O + P I ) -~ p2IA2' f22
=
r2P2
z22
(77) rr1z1 A: are the infinite dilution (limiting) ionic conductances, Ao the infinite dilution equivalent conductance, where 9=-
Ao = A?
+ Azo
(78)
and a and b are numerical factors which at 298 K are32 a = 0.22962 (79) b = 30.2475 (80) In obtaining these formulas, the following expressions for the ionic strength fractions si (pi in Pikal's notation) were used: 7 2 ~ 1 ~ 1 ~z1 r2 s1 = = -- Z z1 - z2 r
f12
(Po
>-
+ P1)Z2)2]l r2
=
1
1O3r1z1RT( k [ P ~ ( P O+ ~2)Al'+ P ~ P + O PI)AZO r2p2
P2qrlr2
2122
(Po
+P2k1
P2Z2]
r2
(85)
[
PlZl -
+
+ PJZ2
(Po
r2
71
(81)
r1
-- Z =z1 - z2 r f/2C2ZZ2
sz =
qr22( PlZl r1
1143
-22
The p terms are the relaxation contributions, and the q terms are the electrophoretic contributions. The units of k are (equiv2dm3 ~ m - ~of) a, are (mol3I2 dm3/2equiv-2), and of b are cm2C2 dm3l2equiv4 J-l s-l). These units are based on the customary use of mol dm-3 for the ionic strength (Le., the valence factors z;2 are assumed to be without units in eq 6, 81, and 82). The numerical values of a and b are obtained from eq 18 and 19 of ref 30; using the electronic charge 6 in esu, D = 78.54 at 25 "C, Boltzmann's constant k in erg deg-', F in C equiv-l, the velocity of light C in cm s-l, 7 = 8.949 X P at 25 OC, and T = 298.15 K. Our a is Pikal's Aij/lzizjl, and our b is Pikal's Bo/2. Anderson and Patersod2derived a SLL expression for Di for the 1-1 binary electrolyte case. They described the ternary system consisting of the binary plus a tagged ion in terms of lij, analyzed this system for isotopic diffusion, and introduced Pikal's SLL equations for the tagged and untagged ions. Their procedure, when applied to the general valence type binary, yields the result
1
Z1f2qr12 -
1'12 Pr22[-PlA?
+ -JL
r1
--
~
I ) 187)
+ P2A2°1[PlA10+ (Po + PSA2Ol + A0
(89)
Note that electrophoretic contributions have cancelled in the self-diffusion of the ions. Note also that there is no Debye-Huckel model for solvent self-diffusion, so that Do cannot be eliminated in fm Substitution of eq 73,74, and 83 into eq 43-48 together with some tedious algebra yields the desired limiting law expressions for the f i j : fll
=
1 0 ~ r ~ z ~ R[bo T ~+ ~P ( ~ - )p21Al0 ~ +~ rlP2
Z12
>-
2 ~ 8 2 '
Equations 84-89 are the fourth set of desired results. Equations 84-89 have no cancellation anywhere of relaxation or electrophoretic terms. Consequently, none of the f i . has any basic simplicity in terms of the classic Debye-kuckel model. This in turn indicates that the mass-fixed Qij and velocity correlation integrals lack basic simplicity in this model as well. The 1, have both relaxation and electrophoretic terms too, but their expressions (eq 73-74) are much simpler than eq 84-89. We now consider the relative contributions of various terms in the dilute solutions (0 II I0.1) to which these equations apply approximately. The contributions of Di to fll and f a are -p2A:/z;2 in the k terms and - r r ~ 2 ( A ~ ) 2 / A 0 in the p terms. The overall relative contributions of the k , p , and q terms are as follows. For fii, both the non Di and Di parts are dominated by the k term. The Di parts are increasingly larger but not much larger than the non Di parts so that sum of the k parts is small and negative, and smaller than the p and q terms. The q term is usually
1144
The Journal of Physical Chemistry, Vol. 85, No. 9, 1981
Miller
the largest and negative, so that f,, is negative overall. For the p and q terms are of about the same size (q larger), positive, and larger than the negative k term (z2 is negative). Thus f12 is positive. For f L 0 , the k term dominates and is negative because po is much larger than p,. Hence the fc0 are negative. For the f o O term, the noterm is 1000 times the k term, which is the largest of the others. Since the Do term is negative, so is foe. The limiting behavior of the f,, at infinite dilution can also be obtained from eq 84-89, keeping in mind that if c 0, then p, 0 (i = 1, 2), I 0, and p po. The infinite dilution values are denoted by f , O. The f l l and fi2 terms go to 0 at infinite dilution because the p and q terms vanish with PI2, and because, as c 0, p2 goes to zero in the k term and p 2 A ; - p2A; cancels. If p is expanded as in eq 31, it will be seen that pZA; will cancel in the k term and that c can be factored out. However, the p and q terms each have a po which survives and hence c cannot be factored out of them. Furthermore, I1I2/c is proportional to 1/c1f2,which goes to infinity as c 0. Consequently, f,,/c --m . On the other hand, f,,/11/2 is finite at infinite dilution. Here the k term will have a factor c1f2 so will vanish, but po survives in the p and q terms. Retaining these terms and using the definitions of eq 75-77, we obtain
TABLE I: Comparisons of Limiting Laws and Experiment
f12,
-
-
-
-
-
-
-
BaC1, NaCl (c = I = 0.01)
losf,,
expt SLL * dil
105f1,
expt SLL * dil
losflo
(c = 0.01,
I = 0.03)
expt SLL dil
-0.37 -0.029 -0.052 0.057 0.058 0.059 -0.00032 -0.00032 -0.00031
0.080 0.097 0.130 -0.0011 -0.0007 -0.0012
1O5foO
expt SLL * dil
-0.00042 -0.00042 -0.00040
-0.00040 -0.00040 -0.00040
1050,
expt SLL * dil
1.33 1.31 1.33
10'21, ,IN
expt SLL expt SLL
5.22 5.17
0.84 0.76 0.85 1.56 0.90
7.97 7.92
7.84 7.66
expt SLL
0.23 0.24
0.33 0.52
10 l1 122 IN 10111,2/N
-0.068 -0.312 -0.433
have a factor c, but the Do term remains. The desired expression is where values of a and b at 25 "C are given in eq 79-80. The f l z go to 0 at infinite dilution because the k term has a factor c and the p and q terms have a factor PI2 both of which go to 0. For f12/c,the k term will be finite. However, the p and g terms have p
