CONDUCTANCE O F SYMMETRICAL ELECTROLYTES
2581
MerNBr, which is known to be associated in acetonitrile,l’ the coefficient of the linear term which is found is too small, as shown by the absurdly small a values (Table IV) calculated from the experimental values of L. There is no doubt that eq. 5 empirically represents the data. There is likewise certainty that the theoretical I; coefficient based on the sphere-in-continuum model d.oes not give even the right sign for the observed linear coefficient in the systems of high vis-
cosity. The experimental results therefore show that the hydrodynamic term A z ( a ) / v of eq. 6 must be large and negative for solvents of high macroscopic viscosities and clearly the quantity in the denominator must be smaller than the bulk viscosity. The theoretical problem to be considered is the motion of ions around segments of much larger molecules. (17) A. D’iiprano and R. M. Fuoss, J . Phys. Chem., 67, 1722 (1963).
The Conductance of Symmetrical Electrolytes. V.
The Conductance Equation’s2
by Raymond M. FUOSS,Lars Onsager, and James F. Skinner Contribution No. 1771 f r o m the Sterling Chemistry Laboratory, Yale University, New Haven, Connecticut (Received A p r i l 19, 1966)
Using the potential of total directed force on charged spheres moving in a continuum under an external field and correcting for electrophoresis, the conductance equation A = A. Xcl/’ E’c In (6El’c) Lc - Acf”Aois derived. Here S = aho Po, CY = 0.8204 106/(DT)”’, = 82.50/q(DT)’/’,E‘ = El’Ao - Ez’, El’ = 2.942 X 1012/(DT)3, E2‘ = 0.4333 X 108/q(DT)2, L = Li Lz(b),Li = 3.202Ei’Ao - 3.42032’ ape, ab = 16.708 X 10-4/DT, L2(b)= 2El’Aoh(b) 44E2’/3b - 2E‘ln b, h(b) = (2b2 2b - l ) / b 3 , fz = exp[-8.405 X 106c1/’r1/’/(DT)”/’], and A = K A = (4rNa3/3000)eb. This equation can be generalized to A = A0 - Sc’/’yl/’ E’cr In (6Ei’cy) Lcr - K ~ c r f ” Awhere , (1 y) = K ~ c y ~ f ”The . first equation reproduces conductance data for 1-1 electrolytes in solvents of higher dielectric constant. The second is required when the dielectric constant is small enough to stabilize ion pairs in contact. This result confirms our earlier (1957) conductance function and establishes the functional form directly from the equation of continuity, the equations of motion, and Poisson’s equation. Values of the contact distance calculated from the parameter b for a given electrolyte in a series of mixed solvents usually increase with decreasing dielectric constant. The source of this variation lies in inadequacy of the sphere-in-continuum model and not in mathematical approximations made in solving the differential equations.
+
+
+ + +
+ +
+
I n preceding papers of this ~ e r i e s the , ~ conductance of symmetrical electrolytes has been investigated theoretically, using the model of rigid charged spheres in a hydrodynamic and electrostatic continuum. This model was the basis of an earlier treatmentof the problem4e5which led to the conductance equation A
=
A0 - Xcl”
+ EClog c + J ( u ) c
(1)
+
+
Equation 1 finally gave the theoretical explanation for the observation that, in the usual working range of (1) Grateful appreciation is expressed to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. (2) Several sections of this paper were presented a t the Symposium on Electrolytes during the 145th National Meeting of the American Chemical Society, New York, N. Y., Sept. 10, 1963.
Volume 69, Number 8 August 1986
R. M. FUOSS, L. ONSAGER,AND J. F. SKINNER
2582
concentrations, conductance curves for solutions in solvents of high dielectric constant usually lie above the limiting tangent. Furthermore, the transcendental term required by the theory explained why empirical algebraic extrapolation functions never quite fitted precision da,ta. Use of the auxiliary function A‘
+ Scl” - Ec log c A’ = bo + J C
A(obsd.)
gave a function which permitted extrapolation to zero concentration on a linear scale. (Since S and E are theoretically known, given an approximate value of Ao, the right-hand side of (2) is determined. The approximation quickly converges on iteration.) Comparison with a wide variety of experimental data6 in solvents of high dielectric constant fully confirmed the functional form of (1). For solvents of lower dielectric constant (say less than about 30 for the alkali halides in dioxane-water mixtures, for example), the experimental A’-c plot become curved, showing that the function (1) no longer reproduces the observed conductances. The ad hoc hypothesis that pairs of oppositely charged ions in contact, in concentrations determined by the dielectric constant’ and the stoichiometric concentration, would not contribute to (d.c.) transport of charge led to the function8bg
+
A = Ao - S’C‘’~~~’~E c log ~ CY
+J
c -~ KACTf2n (4)
(This function obviously reduces to (1) as KA approaches zero and/or y approaches unity.) It was found that (4)reproduced experimental data for solutions in both aqueous and nonaqueous mixtures of solvents for all the 1-1 electrolytes for which precision data a t low concentrations ( K U < 0.2) were available. Reinvestigation3 of the problem was prompted by two motives : (a) dissatisfaction with the arbitrary insertion of the mass action hypothesis into the argument which produced (4) from (1) and (b) the observation that the a values calculated from the linear coefficients J ( a ) systematically increased for a given electrolyte BS the dielectric constant of the solvent mixture decreased. The first point is simply this: if ion association is the consequence’ of Coulomb forces only, then the corresponding decrease of conductance with increasing concentration should be predictable directly from the equation of continuity, the equations of motion, and the Poisson equations.’O I n effect, (4) was derived by the following sequence of arguments. First, assume that (1) gives the conductance of free ions, ignoring the possible presence and effects of ion The Journal of Physical Chemistry
pairs. Second, assume that the concentration of ion pairs is given by the mass action equation 1- y
=
KACy2f2
(5)
Third, use the Arrhenius hypothesis that = A/Ai
(6)
where Ai, the conductance of the free ions, is given by (1) when c is everywhere replaced by c y ; and, finally, combine these three equations. Regarding the second point, the dependence of a on D, at least two possible explanations suggest themselves : either the dependence is an artifact, a consequence of mathematical approximations made in deriving (1) from the fundamental diff erential equations, or else the model is an inadequate physical approximation to the real systems under observation. The results of our present calculations confirm our earlier more approximate treatment. We therefore conclude that a more realistic model, specifically one which includes short-range ion-solvent interactions, must be considered. Also, other sources of linear terms cannot be excluded. Two kinds of mathematical approximations were made in deriving (1); all terms in the development which would eventually lead to terms of the order c8” in A(c) were systematically dropped, and the Boltzmann factor er was approximated in the equation of continuity by the first three terms of its power series. Both in principle and in practice, the effect of the first kind of approximation can be eliminated by restricting the range of application of (1) t o sufficiently low concentrations, the latter being practically defined as concentrations from which the deviations between observed and calculated conductances are random and such that their root-mean-square value is within the estimated experimental error. The effect of the approximation by the truncated series, on the other hand, is built into the end result at the very outset of the calculation and can never be removed once it is in. A new statement of the problem in terms of the po(3) Equations from these papers will be cited as III(61), etc. I, J. Phys. Chem., 66, 1722 (1962); 11,ibid.,67, 621 (1963); 111, ibid.,67, 628 (1963); IV, ibid., 68, 1 (1964). Footnote 1 of IV lists corrections to 1-111. (4) R. M. Fuoss and L. Onsager, Proc. AVutZ.Acud. Sci. C’. S., 41, 274, 1010 (1955). (5) R. M. Fuoss and L. Onsager, J. Phys. Chem., 61, 668 (1957). (6) R. L. Kay, J . Am. Chem. Soc., 82, 2099 (1960). (7) R. M. Fuoss, ibid., 80, 5059 (1958). (8) R. M. Fuoss, ibid., 79, 3301 (1957). (9) R. M. Fuoss and C. A. Kraus, ibid., 79, 3304 (1957). (10) It. M. Fuoss and F. Accascina, “Electrolytic Conductance,’’ Interscience Publishers, New York, N. T.,1959, Chapters VIII-
XIII.
CONDUCTANCE OF SYMMETRICAL ELECTROLYTES
tential of the total directed force (part I) transforms the equation of continuity into a form I(2.5) which can be integrated with retention of the Boltsmann factor explicitly in its exponential form. We can therefore now consider the consequences of the earlier series approximation. When the relaxation field (part 11) is computed from the new equation, significant differences as contrasted with the earlier result6 appears. The relaxation!field AX, calculates to
AXJX
1-
+ Eic log c + Aic - A2cf2
(7) The square root and transcendental terms are exactly as before, but, where a linear term appeared before, two terms Alc and A2cf2now emerge. Of course, a t low concentrations in solvents of high dielectric constant, the activity coefficient is nearly unity, and the last two terms of (7) therefore closely approximate a linear term =
(YC~”
Alc - A2~f2 A ~ c
(8)
which is the result previously obtained when the Boltsmann factor was replaced by the initial terms of its series; in other words, (7) approaches our earlier result when the Boltzmann exponent [ is small, which is precisely where the series approximation is valid. The first of the two significant details in (7) is the appearance of the multiplier f 2 in the last term; such a multiplier would appear if an equation of the form (5) were involved in computing ionic concentrations from stoichiometric. Second, and even more revealing, is the form of the coefficient A,; it can be written A2
== A4[Ep(b) -
(eb/b)(l
+ l/b)l
(9)
where b
=
2/aDkT
(lo) The quantity in brackets in (9) is the functional part of the Bjerrum approximation’l to the association constant KA. When this result is combined with the electrophoresis correction (part 111) and the other higher terms (part IV), the conductance equation which results is A
=
A. - Sc”’
+ E%In r 2 + Lc - AAocf2
(11)
where L and A are constants, which will be discussed later, and r 2 =: 6 El‘c is proportional to concentration. For solutions of low concentrations in solvents of high dielectric constant, it is clear that (11) reduces immediately to (l), with J
=
E’ In (6E1’)
+ L - AAo
(12)
On the other hand, (11) is the limit of (4) as y approaches unity if J has the value given by (12) and A
2583
is identified with K A . The constant L separates into two terms, one of which is independent of the contact distance a; as will be shown later, a large part of J is independent of a, so a is much less precisely determined from the data than is the former total linear coefficient J . The purpose of this paper is to combine the relevant equations of parts I-IV in order to derive (11) and to compare the results with experiments and with the earlier conductance functions. Briefly summarized, we find that numerically the value of the contact distance calculated from J and from L is about the same, and it still shows a systematic dependence on dielectric constant. However, the contribution of the part of the linear term which depends on a to the total conductance is in general small, and any additional small linear term could easily account for the variation of a. On the other hand, the coefficient A closely approximates the former’ K A and eq. 4 in the form A = A0
- S C ~ / ’+ ~ ‘ E’cy / ~ In (6El’cy)
+
Lcy - K.4cyf2A (4a)
can be considered to be theoretically established. The principal contribution of this series3 of calculations is in fact to establish that retention of the Boltzmann factor without approximation leads directly to a term in cf2 into the conductance equation, which in turn heuristically justifies the use of (4a) as the more general function of which (11) is the limit. We shall refer to (4a) as the 1965 equation in order to distinguish it from earlier equations and shall use it in the analysis of conductance data. 1. The Conductance Function It is convenient to introduce a t this stage the dimensionless variable12 7
= (6El’c)’’’ = 816.11c1/’/(D~~)”/” (1.1)
which is the ratio of the Bjerrum distance p/2, near which most ion-pair distribution functions have a minimum,la-17 to the Debye-Huckel distance ( l / ~ ) . This variable is a rational reduced variable for the description of electrolytic solutions; two solutions a t different concentrations a t different temperatures in solvents of different dielectric constant have identical (11) See ref. 10, Chapter XVI. (12) R. M. Fuoss and L. Onsager, Proc. Nutl. Acad. Sci. U . S., 47, 818 (1961). (13) N. Bjerrum, Kgl. Danske Videnskab. Selskab, 7, No. 9 (1926). (14) R. M. Fuoss, Trans. Faraday SOC.,30, 967 (1934). (15) R. M. Fuoss, J . Am. Chem. SOC.,57, 2604 (1935). (16) H. Reiss, J . Chem. Phys., 25, 400 (1956). (17) J. C. Poirier and J. H. de Lap, ibid., 35, 213 (1961).
Volume 69, Number 8
August 1966
R. M. FUOSS,L. ONSAGER,AND J. F. SKINNER
2584
electrical properties at the same value of T . For example, the coefficient of the leading term in the relaxation effect simply becomes a numerical constant ac1/2
=
7/3(1
(1.2)
(1.3)
f = e-r
Alternatively, T is the electrostatic part of the chemical potential, expressed in units kT. The discussion which follows will be limited to the case of 1-1 electrolytes although the equations are valid for all symmetrical electrolytes. For nunierical work, computer programs for 1-1 electrolytes can be easily adapted to x-x electrolytes by replacing the dielectric constant D by D/x2 and the viscosity q by q / x . This is because charge-square always appears in the numerator when D appears in a denominator in electrostatic terms, while charge appears to the first power in the numerator of hydrodynamic terms, which always have viscosity in the denominator. The electrostatic part of the relaxation field is given by =
F(u)
+ +
+ Ax,/X
(1.4)
(1
- Tl)f(b,KU) = 0.4377Ti - 0.2612 - TiKU/2
+ a) - (T2/3)E,(SKu) T2e-2rK(b)+ ( ~ ~ / 3 ) ( Ib n+ r - 1) + (T2/3)f(b,KU) = -T/3(1
E,(SKCI)=
- q/(2
&(SKU)]
Ti
=
e-b(l
-
+ a) + Ti(p -
'/z)
+ b + b2/2)
(1.6) (1.7)
andig AXJX
=
- ( ~ ~ / 3 ) (2 8p/3) = - 0 . 1 1 4 4 ~ ~ (1.8)
The first term of F ( a ) in (1.5) is the classical leading term acl/' due to the relaxation field. The other terms are of higher order in concentration; some have their origin in the retention of the Boltzmann factor e' in explicit form in the integration of the equation of continuity. The negative exponential integral E,(sKu) contributes to the c log c term when it is expanded in the approximate formz0
En(z) =
-r - l n z + 2
+ O(z2)
The Journal of PhVsicaZ Chemistry
- In (2
+
(1.11)
+ q) - In 2 - In + In b +
(1.12)
= In (2T/b)
(1.13)
T
where the identity In
KU
has been used to replace KU by 7 ; this step yields a negative In b which cancels the positive one appearing in (1.5). Combining the various terms above, we obtain AX,/X = - ~ / 3 ( 1
+ q) 4- ( ~ ~ / 3In) T2e-2'K(b) + (T2/3)N'(b) 7
(1.14)
where
N'(b)
=
(1.4681 - 1.2916T1)/(1 12
- Ti) -
+ q + T1/2(1 - T i ) ] ~ a (1.15)
The last term in (1.15) leads to a small term in cs/'; (2 q) = 2.7071, and the second term in the brackets varies from 1.05 at b = 2 to 0.157 at b = 4 and thereafter decreases rapidly to zero. To simplify calculations, we shall therefore approximate the last term of (1.15) by ~ K U
+
"(b)
=
(1.4681 - 1.2916TJ/(l
- Ti)
- 3KU (1.16)
Next, the hydrodynamic and kinetic terms21 will be considered. Denoting their sum by AX,,k, we have AX,,k/x
+
[Kzfi/127V((J1 3e-2K(b)/2]
where
(1.9)
but the logarithmic parts of the exponential integrals q)/ in f(b,~ta) cancel except for a constant, In [(l (2 q ) ] . Expanding the integrals, we have
+
-r
SKU
(1.5) (1 - Tl)f(b,~a) = (T1/2)[En(pKa)
(1.10)
The relaxation field is clearly a function of b; for b larger than 3 or 4, the main dependence is through the term in K(b), the remaining terms not being very sensitive to b, while K(b) is essentially exponential in b. It will be more convenient to postpone further discussion of the dependence on b until the final equation is assembled. The expansion of E,(sKu)in (1.5) gives, to terms of order K ~ U ~
wherei8 F(U)
O(K2U2)
and on inserting numerical values for p , S, and q
+ a)
for all symmetrical electrolytes, and the limiting activity coefficient is given by
AX,/X
+
(1 - T d f ( b 4 = (T1/2)[In (sip) 2q - 1 - K U ] - q/(2 q)
(18) See II(A27). (19) See II(70). (20) See ref. 10, Figure 13.1.
(21) See IV(3.1) and (3.2).
(Jz)
+
1 [Pz(b)
+ /32~2/24(1- T1) (1.17)
CONDUCTANCE OF SYMMETRICAL ELECTROLYTES
2E,(2~a) -
2Pz(b)
E,(KU)
- hl b
+ 3/2
-
2585
-
1/3b2 (1.18) Expanding the exponential integrals as before, we find
- 1/3b2 - 2r
2P2(b) = -In
+ 3/2
The entire relaxation field is now obtained by taking the sum of (1.14) and (1.29); denoting the sum simply by AX, we have
+ 2(E’/Ao)c In r 6(E’/Ao)K(b)e-22‘~ + 2E1’N’(b)c 3.4677Ez’~/Ao+ El’c/(l - Ti) -
AX/X = -acl/’
- 3 In 2 + 3Ka (1.19)
=
-In r
- 1/3b2 - 1.7338 + 6r/b
2Ez’c/3Aob2
Next, the coefficient of the first term of AX,,k is converted into practical units. The reciprocal friction Coefficients wj are defined by =
voj
Xes. J J
(1.20)
where the field X is measured in electrostatic units. Single-ion conductances Xoj are related to the mobility uojthrough the Faraday constant F Xoj
=
FuOj
(1.21)
The factor lov9times C, the velocity of light, converts electrostatic units into volts; by combining these definitions, we find WI
+
a2
=
10-*CAo/Fe
(1.22) .
+
w2) = ~ o K / ~ , & C ~ / *
Note that the transcendental term is c In r rather than c In c; we have now transferred to this term part of the former Jc term. -41~0, the c”‘ term (which, it will be recalled, arises from numerical approximation of the exponential integrals) is most simply computed in terms of r. Note also that this term increases very rapidly with decreasing dielectric const b > 2 by the empirical formulas
H(b)
=
3.936
+ 329.5 exp(-3.344b1/’)
(1.41)
and
G(b) = 10.583 - 21.10b”’ These functions are shown in Figure 1; for b
Figure 1. Dependence of H( b) and G(b) on b. (Note location of zero ordinate.)
(1.42)
>
3,
H changes only slowly with b and asymptotically approaches a value of about 4 for large b (ie., becomes insensitive to the contact distance a). In general, the quantity El’Ao H is much larger than E2’G. These facts are of practical importance in numerical calculations, as will be shown later. For b greater than about 4, the K(b) term begins to dominate the linear terms. (To give a feeling for the magnitude of b, we mention that b = 140/D for a cor,tact distance d = 4.) The logarithm of the function
K(b) = [E,(b)
-
(e /b)(l
+ l/b)1/3
0
0 -
I
‘3
7
b
II
15
Figure 2. Comparison of approximate and explicit association functions and of b parameters derived therefrom.
(1.44)
which is the first term of the asymptotic expansion of K(b) and also the function of b which appears in the direct theoretical derivation of the association constant.’ It will be seen that (1 - 3/2b)K(b) and K’(b) approach each other as b increases. The curves cross a t b = 4. K’(b) has a minimum a t b = 3 while K(b) goes through zero a t 2.35. From the experimentally determined association constants, a value of b is computed, and then b determines the contact distance a through the definition (10). Also in Figure 2 (ordinate scale to the right) is shown the ratio b’/b, plotted against b’ for pairs of values obtained by solving K’(b) and (1 - 3/2b)K(b) for b for equal values of the functions. The greatest difference occurs around b The Journal of Physical Chemistry
Y
(1.43)
multiplied by (1 - 3/2b) is shown in Figure 2. (The multiplier is an estimate for the coefficient (1 Ez/EIAo)which implicitly is contained in the ion-pair term of (1.38); see eq. 2.10 in the next section.) Plotted on the same scale is the logarithm of the function
K’(b) = 2eb/3b3
2
= 6, where b’ is about 15% larger than b for the same (experimental) value of the association constant. Either function would of course give a self-consistent set of a values if used for data for a given set of electrolytes. Naturally, K cannot be interpreted as a factor in an association constant for b less than 2.35, where it goes through zero to negative values. It will be noted that K(b) has the same dependence on b as the Bjerrum function &(b). It appears that our derivation of K(b) from the differential equations and its interpretation as a factor in the association constant imply that association ceases from b < 2.35. Since such a critical value of b is physically unrealistic, we prefer to conclude that the derivation leads to a function which
CONDUCTANCE: OF SYMMETRICAL ELECTROLYTES
for larger values of b is a good approximation to the directly derivable’ function K’(b). For numerical calculation of K ( b ) , the following development is convenient. We have
+ Jb(eu
E,@) = .Ep(l)
- l)du/u
1
+ In b
(1.45)
by writing the definition of the positive exponential as the integral from minus infinity to unity plus the integral from 1 to b and adding and subtracting 1 in the latter integral.. The integral in (1.45) converges and can be expanded in an absolutely convergent series n h
m
m
where the second sum equals 1.31790. By series expansion and grouping terms (eb/b)(l
+ 1/15)
+ 2/b + l / b 2 +
3/2
=
21
+
2 2
(1.47)
2587
with free ions, relative concentrations being given by (5). We now see that, for the limiting case of small association constants, a t least, the hypothesis of ion-pair equilibrium no longer is needed as a special ad hoc assumption; retention of the explicit Boltzmann factor es in the distribution function automatically leads to a decrease in conductance (for b > 4) which arises from pairs in contact because it is a function of eb, which is the value of es when r = a. It is unnecessary to consider how the two ions approach to contact or how long they remain in contact; the equation simply states that a decrease in conductance is an inevitable consequence of Coulomb interaction when b exceeds about 4. It will, of course, be convenient to retain the vocabulary of the ion-pair model. Furthermore, we propose a stochastic extension of (1.38) to lower dielectric constants (where K(b) exceeds unity) by writing A
=
A. - Xc 1/2 y1/2
where
+ E‘cy In r 2 y + L c y AcyA exp(-2ry1’*)
m
21
= n=l
b”/(n
+ l)!
(1.48)
where
L = El’AoH(b) - EZ’G(b)
and m
2 2
b”/(n
= n=l
+ a)!
+ In b + b3/(n+ 1) [ ( n+ 3) !I
3K(b) = -0.!3228 - 2/b - l b 2 2b n=l
(1.50)
To simplify the equations in the following discussion, we shall temporarily neglect the r 3 term in (1.38) and abbreviate the coefficient of the linear term by L. Then the conductance equation becomes
+
A = A. - SC’;~ E’c In r 2
+ Lc 6E’K (b)ce - 2‘
(1.54)
and (1.49)
Substituting (1.45) and (1.47) in (1.43), inserting Ep(l) = 1.8951, and collecting terms, we obtain the rapidly convergent series m
(1.53)
(1.51)
If we neglect E&’ (which is relatively small) compared to El’ in E’, and replace K(b) by the leading term of its
A
-
which gives precisely the limiting form of eq. 4 obtained when .KA is small and y approaches 1. As already stated, (4) is the equation which results by assuming that (1) gives the conductance of the free ions and that nonconducting ion pairs are in equilibrium
6E’K(b) /A0
=
6E1’(l - EZ’/El’Ao)K(b)
(1.55a)
As stated in part 11, since an equation of the functional form (4) reproduces experimental data over a wide range of variables and since, in the limiting case of high dielectric constant, it reduces to (1.51) which is the approximate solution of a rigorous formulation of the conductance problem, we suggest that (1.53), which has the same dependence on c as does (4), is a close approximation to the exact solution of the problem.
2. Comparison with Experiment For convenience, we list below four versions, (l), (4), (11), and (1.53), of the conductance equation t,o which frequent reference will be made in this section
+ Ec log c + J(u)c - Scl”yl/’ + Ecy log cy + J c y A
asymptotic expansion, the last term of (1.51) becomes 6E’K(b)~e-~’ (4~~Na~/3000)e*A~ce27 (1-52)
=
A
= A0
=
A0
-
Xcl”
(2.1)
K ~ ~ y f ’ h(2.2)
+ + Lc - Aho@ A = A, - Xc‘/’yl/’ + E’cy In r 2 y + Lcy A
=
A. - Sc1l2 E‘c In r 2
Acyf”n
(2.3) (2.4)
Volume 69, Number 8 August 1966
R. M. FUOSS, L. ONSAGER,AND J. F. SKINNER
2588
Considered as functions of Concentration, (2.2) and (2.4) are identical; for y near unity, both approach (2.3), which for f 2 = 1 (or K A = 0) in turn approaches (2.1). We shall first consider (2.1) and (2.31, using data for potassium chloridez3and for cesium iodidez4in dioxanewater mixtures in the higher range of dielectric constants and for some quaternary ammonium salts in a c e t ~ n i t r i l e . ~The ~ results are summarized in Table I, where two entries are given for each value of D. The first one gives the constants obtained when (2.3), the three-parameter equation, was fitted to the data. The quantity 6h is the mean deviation between calculated and observed values for the five points of the run. The contact distances were computed by solving
L
- Ah0 =
El’AoH(b)- Ez’G(b) - 6E’K(b)
=
78,54 78.54 60.16 60.16 41.46 41.46
*
KCI
149,895 o,oo5 149.907 f 0.003 100.73 f 0.05 100.74 f 0.02 69.13 0.04 69.09 f 0.01
e2/bDKT = 560.4 X 10-s/bDzs
+ E’c In + ( L - AAo)c Sc’/’ + E‘c In r 2 + J’c
A
=
A. - Scl”
A
=
A.
72
(2.7b)
The coefficient J‘ is given in the L column, and the contact distances were obtained by solving J’ = .Fl’AoH(b) - Ez’G(b) - 6E’K(b)
(2.8)
for b and then calculating a by (2.6). It will be immediately noted that the lumped linear coefficient J’ The Journal of Physical Chemistry
6A
a
C~H~OZ-H~O
*
173,0 8,1 -o,20 o,08 193.9 f 0.5 .. 176 f 91 -0.4 f 1.4 202.4 f 2.9 ... 234 48 1.3 193.7 f 3.0 ...
o,oo2 2,85 0.003 2.80 0.017 0.012 3.19 3.27
o,oll 3.78 0,011 3.90
217 f 90 0.50 f 0.82 163.4 i 2.6 ... 192 f 83 -0.6 f 1.3 230 i 3 ... 261 f 22 -0.79 f0.61 289.8 f 1.6 ... 315 f 27 1.9 f 1.2 276 i 3 ...
0,018 0.016 0,017 0.015 0.005 0.005 0.007 0.008
2.41 2.56 3.49 3.37 4.36 4.28 5.09 5.21
Am3NBuBPL in MeCN
(2.6)
(2.7a)
A
CsI in C4H8o2-H20 78.54 154.20 f 0.05 78.54 154.17 f 0.02 60.18 99.44 i 0.05 60.18 99.46 i 0.02 40.57 66.88 f 0.01 40.57 66.90 i 0.01 29.79 56.46 i 0.03 29.79 56.42 f 0.01
36.01 116.33 f0.19 36.01 116.48 f 0.07
The pattern is the same for all the examples of Table I (as well as for many other sets of data which have been examined): a good fit of the function to the data (small 6h) but a large uncertainty in L and practically no certainty in the value of A. This simply means that fz is so near unity that, if we insist on three parameters, the mean-square computation gives us a wide band of L and A values because the two terms cannot be sharply separated. A large L paired with a large A (for positive A ) will fit the data just as well as with both constants small. It would require data of at least 0.001% precision in this range of concentration to resolve the two terms; such data simply do not exist. Also, it is clear that a calculated from L(a) by (1.54) will have a large uncertainty, while, with 100% uncertainty in A ( a ) , no meaningful value whatsoever can be obtained by solving (1.55) for a. I n this range of dielectric constants, the L term dominates the A term. The only way to obtain a value of a is to lump the parameters as was done in (2.5). If, however, we approximate fz by unity in (2.3), we obtain the results shown in the second entries for a given dielectric constant when the two-parameter equation (2.7) is fitted to the data.
-.
L
AO
(2.5)
forb, whence a
Table I: Constants of Conductance Equation
68642530 - 7 . 3 i 8 . 6 1138 f 33 ...
0,066 5.80 0.063 5.08
AmaNBuBr in MeCN 36.01 159.75 f 0.05 36.01 159.67 f 0.02
1084 f 201 836 i 13
2.9 f 2.2
...
0,018 4.20 0.020 4.42
Pr3NMeI in MeCN 36.01 178.30 f 0.07 36.01 178.25 f 0.02
310 f 278 130 f 11
1.9 i 2.7
...
0,016 3.72 0.014 3.82
is very much more precisely determined than was L; also 6hoand 6A in general decrease significantly. These smaller variations simply mean that the two-parameter equation (2.7b) fits the data for systems with small b values (less than about 4) better than the threeparameter equation (2.3). I n other words, for high dielectric constants and/or large ions, the 1957 equation (2.1) and the 1965 equation (2.3) in its approximate form (2.7) are indistinguishabb. The reason is obvious; for systems described by small b values, the series expansion on which (2.1) was based is a valid approximation. I n principle, (2.3) is theoretically more sound, but precision of present conductometric technique is inadequate to separate the constants L and Aho. (For the reason just given, our computer program contains and “IF” instruction; the data are first fitted to eq. 2.4. Then if A < 10, the computation is repeated, fitting the data to eq. 2.7b.) The practical difficulty in separating the terms of the theoretical equation is clearly shown if their relative numerical values are examined. The values shown in Table I1 for tetrapropylammonium tetraphenylboride (23) J. E. Lind, Jr., and R. M. Fuoss, J . Phys. Chem., 6 5 , 999 (1961). (24) J. E. Lind, Jr., and R. M. Fuoss, {bid., 65, 1414 (1961). (25) C. Treiner and R. M. Fuoss, Z . physik. Chem., Falkenhagen
Festband, in press.
CONDUCTANCE OF SYMMETRICAL ELECTROLYTES
in acetonitrilez6 are typical. For this system, A0 = 128.38 j z 0.01, L = 861 f 179, A = -5.0 f 2.0, and 6A = 0.004. Computed by (2.7b), the parameters are A, = 128.41 f 0.01 and J’ = 1304 i 11. It will be seen that the square root term accounts for most of the change of conductance with concentration.
2589
for our special case. The coefficient J‘, defined in (2.7), is identified in (1.38) as
J’
=
Ei’AoH(b) - Ez’G(b) - 6E’K(b) (2.11)
Using (1.28) and (2.lo), this becomes
J’ = Elfilo[H(b)- 3G(b)/2b 6(1 - 3/2b)K(b)] (2.12)
Table I1 : Conductance of Pr4NBPh4in MeCN 104~
9.449 7.297 5.548 3.766 2.096
S term
E term
L term A term
118.83 -10.01 119.91 -8.79 120.94 -7.67 -6.32 122 20 -4.71 123 72
-0.80 -0.65 -0.53 -0.39 -0.24
0.81 0.63 0.48 0.32 0.18
A
J‘ 6h
0.44 0.002 0 . 3 5 -0.003 0.28 0.000 0.20 0.004 0.12 -0.002
E
(2.13)
El‘AoF65(b)
The coefficient J ( a ) of (2.1) is givenz7by
+
+
+ +
J ( a ) = 2E1’[h(b) 0.9074 In K U C - ~ / ~ ] A O a/&, 11/?oUK/12C1/’- 2E2’(1.0170 In K U C - ~ / ’ ) (2.14)
+
Substituting The approxiniate cancellation of the E and L terms is typical for systems with small b (here equal to 2.81 with d = 5.54 for the three-parameter equation and b = 2.98 and 6 = 5.22 for the two-parameter). The latter characteristic, incidentally, explains why earlier purely empirical attempts to determine E, the coefficient of the :I log c term, gave such uncertain results. The A term amounts a t most to less than 0.5% of the total observed conductance. It is obvious, from inspection of the L and A columns, that it is very easy to rob Peter to pay Paul here because the A column also is almost linear in e. Before considering systems in the lower range of dielectric constants where (2.1) and (2.7) no longer fit the data, we shall compare the linear coefficients of the two equations, J and J’. It will be recalled that the former appears in the 1955-1957 derivation where the series approximation for eb was made, while the latter appears in the present development in which e‘ was retained explicitly. These coefficients depend on dielectric constant in two vvays: explicitly through the coefficients El and Ez and implicitly through the appearance of functions of which in turn varies inversely as D. I n order to separate these two kinds of dependence, we shall consider the special case of equal mobilities (wl = w ) and further assume Stokes’ value (6~7a)-’ for w . The general case will not differ greatly from this special case which, of course, is much simpler to treat. Substituting 67r7wa = 1 in (1.26) and using (10) to eliminate a gives
In ~ac-’/’ = 0.5 In 24E1’ - In b
which follows directly from the definitions (10) and (1.27) of b and El’, and combining terms, (2.14) becomes
J(u)
=
E’ In 6E1’
+ E I ’ A o F ~ ~ ( ~ ) (2.16)
where F67(b)
E
3.202
+ 1.213/b + 26/b2 - 2/b3 2(1 - 3/2b) In b (2.17)
The term E’ In 6E1’ in (2.16) is simply the quantity which converts the transcendental term in (2.1) from the c-scale to the 7-scale and numerically makes up a large part of J which i s independent of b. The consequence is that an illusory precision has been ascribed to the values of a derived from J ; actually, the percentage uncertainty in a is considerably larger than that for J . Some numerical examples will be given later; here we shall continue the discussion of functional dependence. If both the 1957 and 1965 equations are written with the transcendental term in the T scale, (2.1) and (2.7) become, respectively
- Scl/’
+ E’c In + El’AoF~7(b).c (2.18)
- Scl/’
+ E’c In + El’AOF65(b).~ (2.19)
€1,
b2~2/4b= &?&‘C/Ao (2.9) Then substitution of (1.27) gives a relation between the coefficientsin 15
(2.15)
A
= A0
T~
and A =
A0
T~
The two equations now are identical except for the coefficients of the linear terms. The direct dependence of the latter on dielectric constant is immediately obvious: both go inversely as D3,as seen by inspection of the definition of El’ (1.27). The behavior of the b(26) D. S. Berm and R.M. Fuoss, J.Am. Chem. SOC.,82,5585 (1960).
Ez’/Ei’AO
=
3/2b
(2.10)
(27) See ref. 10,XV(34), (35), (36),and (38).
Volume 69, Number 8
August 1965
R. M. FUOSS, L. ONSAGER,AND J. F. SKINNER
2590
i.e., a quantity clearly deriving from contact pairs. The logical extension of (2.3) to solvents of lower dielectric constants therefore is (2.4), where A is now to be identified with an association constant K A , preferably in the explicit form7 K A = (4~rNa~/3000)e~ = 2.523 X 10-3d3eb (2.20) = 2.523 X
Figure 3. Comparison of 1957 and 1965 functions in the linear coefficient of the conductance equation. (Note location of zero ordinate.)
dependent part is shown in Figure 3, where F65 and FS7 are plotted against b. The D scale shown at the top of the figure is for 12 = 4.00. For the approximate range 80 > D > 40, the 1957 function is a very good approximation for the explicit 1965 function; presumably, the agreement is good also for D > 80 because the series approximation of the Boltzmann factor improves as D increases. But for D < 40, the two functions diverge rapidly as the exponentiallike function K(b) in F66 takes control. Since F6, is much simpler than F66, we shall eventually replace F65by F57as an adequate approximation. This study of the two equations suggests the method of analyzing conductance data for solvents of lower dielectric constant where ion association cannot be neglected. The explicit equation (2.3), of which (2.7) is a limiting form, fits the data over a somewhat wider range toward lower dielectric constants than (2.l), but, below D = 25, it too starts to show systematic deviations. In the range where the two functions of Figure 3 closely approximate each other, the contribution of the K(b) term, shown by the dotted line, .is practically negligible. (It is therefore not surprising that the A values for the three-parameter equation in Table I were so uncertain.) In other words, the linear coefficients of the 1957 and of the 1965 equations are nearly identical if the terms deriving from the negative exponential integrals are expressed as Ec log r 2 , in the range where the concentration of ion pairs is negligible. The significant difference between the two derivations is the appearance of the AAocf2 term of (2.3), which, as has already been emphasized, is a term obtained by setting T = a in an integral, The
JOUTnd
of Physical Chemistry
lOZ1fla(eb/b3)
rather than in the approximate Bjerrum form which is proportional to the function K(b) defined in (1.43). The coefficient L of (2.4) is, as we have just seen, closely approximated by a much simpler function similar to the special function F57. After considering some examples a t lower dielectric constant by the explicit functions H(b), G(b),and K(b) which appear implicitly in (2.4), we shall derive a general formula to replace As the dielectric constant is decreased, the coefficient A increases, both theoretically and experimentally. For a short range, (2.3) instead of (2.7b) may be used; the A and L terms can be separated. However, much above b values of about 4, the deviations between calculated and observed values of conductance become systematic and steadily greater as b increases. The A term now describes a loss in conductance due to ions in contact, i.e., in nonconducting pairs; it is evident that other higher order effects of ion pairs must also be considered in the range of lower dielectric constants. Contact pairs should not be counted as atmosphere ions in calculating the relaxation field nor the electrophoresis, nor in any of the other long-range effects. By writing cy for the fraction of solute present as free ions, (2.3) is transformed into (2.4), which has the same functional form as (2.2). In Table I11 are summarized the results of the calculations for potassium chloridezaand for cesium iodide24in the mixtures of lower dielectric constant and for tetramethylammonium tetraphenylboride in acetonitrile-carbon tetrachloride mixtures.z6 Since 6A is uniformly small and of the order of the experimental error, we conclude that (2.4) reproduces the data as far as concentration dependence of A is concerned and that it may be used to extrapolate for A,,. Now consider the constants A and L. Recall that, for small b, they could not both be determined with precision; in fact, for b around 2, A is completely uncertain. However, as seen in Table 111, as b increases (D decreases), the constants become separable in the intermediate range, and, considering their contributions to total conductance (see Table IV later), are fixed with fair precision by the data. Then as b increases further, the A term becomes so large that the L
CONDUCTANCE OF SYMMETRICAL ELECTROLYTES
2591
~~
Table III : Constants of Conductance Equation D
A
L
AI,
Table IV : Conductance of Me4NBPb A term
141.51 -11.00 142.59 -9.81 143.85 -8.46 145.34 -6.97 147.30 -4.95
AcetoniMe -1.10 1.69 -0.92 1.35 -0.73 1.00 -0.54 0.68 -0.31 0.34
-0.41 -0.34 -0.26 -0.19 -0.10
0.006 -0.006 -0.019 0.028 -0.010
6.812 5.268 3.910 2.511 1.532
84.87 -11.19 86.53 -10.12 88.30 -8.74 90.53 -7.01 92.56 -5.49
Mixture 1 -2.91 1.80 -2.49 1.40 -2.05 1.04 -1.51 0.67 -1.06 0.41
-1.85 -1.57 -1.28 -0.92 -0.62
0,002 -0.006 0.010 -0.007 0.002
6.017 4.764 3.797 2.680 1.530
46.57 48.77 50.90 54.10 58.90
Mixture 2 -3.92 -3.67 -3.38 -2.91 -2.17
0.23 -11.51 0.19 -10.81 0.15 -10.04 0.11 -8.77 0.07 -6.69
-0.009 0.012 0.005 -0.016 0.005
d
104~
0.007 0.005 0.009 0.010 0.017
4.00 4.38 4.86 5.47 5.61
10.308 8.198 7.094 4.129 2.083
8 term
A
KC1 in C4H80rH20
:*
30.26 25.84 19.32 15.37 12.74
56.45 0.03 370 52.32 2c 0.02 379 46.27 :+ 0.04 373 42.51 :+ 0.08 220 39.55 :+ 0.11 -433
2c 33
24.44
52.26 :+ 0.03
18.68 15.29 12.81
48.22 :& 0.07 349 2c 67 45.47 :+ 0.11 44 f 85 41.82 :+ 0.03 -869 f 22
f 23 f 42 f 61 f 108
16.6 f 1.4 32.4 f 1.3 159 f 4 460 i 12 1702 f 32
CSI in C4HsOrH20 430 2c 37
19.6 f 2.2 88 313 788
f7 f 14 f6
0.009 5.28 0.017 5.89 0.021 6.10 0.005 6.59
MeaNBPb in MeCN-CC14 36.01 152.32 :k 0.07 1645 f 1018 3.6 i 9.6 22.32 114.85 :k 0.07 2886 f 596 25.8 f 10.0 17.45 99.32 :k 0.04 2698 i 259 58 f 6 108 365 & 7 13.06 83.63 i 0.05 1997 483 f 271 1431 f 27 10.68 74.46 :k 0.10
*
E term
L term
6A
0.026 0.021 0.010 0.009 0.016
4.74 6.15 7.26 7.71 7.81
term becomes relatively invisible, as shown by the increasing value of 6L/L. The precision in A increases as b increases. We note that L goes through a maximum and eventually becomes negative. According to theory, L should asymptotically approach 4ElAo for large b which is always positive. We shall comment later on this discrepancy between theory and experiment. Finally, a systematically increases as D decreases, both for the aqueous and the nonaqueous systems, just as it did in the earlier analysis. We shall also return to t,his point later. I n order to demonstrate the shift in significance of the various terms in the conductance equation as dielectric constant is decreased, the data of Table IV are presented. They give the conductance of tetramethylammonium tetraphenylboride in acetonitrile and in acetonitrile-carbon tetrachloride mixtures of dielectric constants 17.45 and 10.68. The parameters are: acetonitrile, A. = 152.32 I 0.07, L = 1645 f 1018, A = 3.6 f 9.6, b = 3.28, 12 = 4.74; mixture 1, D = 17.45, A0 = 99.32 f 0.04, L = 2698 f 259, A = 58.4 f 6.3, b = 4.42, d = 7.26; mixture 2, D = 10.68, A0 = 74.46 f 0.10, L = 483 f 271, A = 1431 f 27, b = 6.72, it = 7.81. The square root term is alway,s a large one, and the E term is never negligible. The interesting detail is the relative values of the L and A terms as b increases: the shift from L dominant to A dominant as the dielectric constant decreases is evident. The pattern shown by the specific examples of Table I V is general and leads to a significant conclusion (which is intuitively correct, of course): no useful information can be obtained from the ion-pair term a t high dielectric constants nor from the linear term at low.
-12.69 -11.41 -10.29 -8.78 -6.77
6A
So far, the B r 3 term (-ca/') of (1.38) has been neglected. Here, B is defined by the equation B
=
(2b - 3)Ao/b2
Its origin is in the next term after In ( ~ K uin) the expansion of the negative exponential functions which appear in the relaxation and velocity fields. Its effect was investigated for a number of systems, by the following method. First, the data were analyzed by means of eq. 2.7 for A < 10 and by (2.4) for A > 10. Using the values of b so obtained, we calculated Ana,, = (2b - 3 ) A o ~ ~ y ' / ~ / b ~
(2.21)
and h(cor.)
=
A(obsd.)
+ AA,,
(2.22)
Then the values of A(cor.), which are the observed conductances corrected to first approximation for the ca/' term, were fed back into the computer, along with the original concentrations, and the values of A,, L , and A which would reproduce A(cor.) as a function of concentration were obtained. The net effects were as follows: (1) 6A was slightly decreased, indicating a better fit to the data; (2) L became a monotoneincreasing function of D-l (instead of going through a maximum) but did not increase as fast as the theoretical 4El'Ao; (3) the values of A were increased by about 50% for the systems with dielectric constant below about 20; and (4) the spread of it values for a given system was decreased, as shown in Table V for some tetraphenylborides in acetonitrile-carbon tetraVolume 69,Number 8 August 1966
R. M. FUOSS, L. ONSAGER,AND J. F. SKINNER
2592
chloride mixtures.26 The columns headed d correspond to values obtained neglecting Br3; the values obtained by using (2.22) are in the columns headed
eq. 2.17, consider the general case (2.14). Again substituting the identity (2.15),we find
J(a)
8.a/?‘
=
E’ In 6E1’
+ L1 + L2(b)
(2.23)
where
Li
Table V : Effect of Br3 Term on a Values D
d
%
D
5.83 6.05 7.08 7.18 7.37
36.01 19.46 16.13 12.31 9.80
Pr4NBPh4 5.54 8.67 9.16 10.36 10.11
36.01 19.87 17.18 13.92 11.16
BudNBPh4 6.30 6.27 8.57 8.28 9.09 8.65 9.66 8.97 10.43 9.38
Mel NBPh4 36.01 22.32 17.45 13.06 10.68 36.01 20.60 17.05 14.21 12.14
4.74 6.15 7.26 7.71 7.81
Et4Pu’BPhd 5.78 5.76 7.30 7.07 3.02 7.52 7.78 7.27 3.89 7.87
6
&a/?
5.51 8.41 8.36 8.90 8.61
While inclusion of the c”? term is indeed an improvement, the benefits are marginal. Considering the extra work involved in retaining it (usually much more than double the computation time, due to iteration until b converges to a constant value), we propose to neglect it in the analysis of conductance data and allow A and L to absorb the errors. There already has been neglected a multitude of ca” terms of unknown magnitude and sign; in order to be consistent, it now seems preferable to discard them all and restrict the range of application of the equations to concentrations where the unknown cS/?terms are negligible. The criterion is ah,the mean deviation between calculated and observed conductances. When this becomes significantly greater than the known experimental error, the functional form of (2.7) or of (2.4) no longer fits the data. The presumption is that higher terms than those retained have become significant. We return to a consideration of the coefficient of the linear term. According to (1.54),it should approach approximately 4El’Ao as b increases because H(b) levels off a t about 4, while E2’G(b)becomes very small compared to El’&. Experimentally, as was shown in Table 111, L goes through a maximum and becomes negative for small dielectric constants. Possibly part of the terms which we have collected in L should be in A , or perhaps the approximations that resulted in K(b) instead of K’(b) also affect L. In any case, the linear coefficient derived from J ( a ) for the case of unassociated electrolytes does show the right behavior qualitatively. Instead of the special case treated by The Journal of Physical Chemistry
=
3.202Ei’Ao - 3.42032’ f P.o
(2.24)
and
+
L z @ ) = 2E1’A0h(b) 44E2’/3b - 2E’ In b
(2.25)
with h(b)
=
+
( 2 ~2b -
(2.26)
i)/b3
Both L1 and L2 depend on dielectric constant through El’ and E2‘; the only part of J , however, which depends on b is the term Lz(b). It is from this part of the coefficient of the linear termt hat ion size ( a J ) is determined. It is easy to see that
L’ = L1
+ L2(b)
(2.27)
will go through a maximum and become negative. For lower dielectric constants, the terms in E2’ become small and h(b) 2/b; the dominant part of L’ is given by
-
L’
-
2El’Ao(l.601 - In b)
(2.28)
which goes through zero a t b = 4.96 and is negative for larger values. In Figure 4, the solid curves are L’ for the dioxane-water mixtures which were used as solvents in the work on potassium calculated for d = 3, 4, and 5 . The curve through the circles shows the experimental L values of Tables I and I11 for potassium chloride. The general patterns of the calculated L’ curves and of the observed L curve are the same: flat over the range of high dielectric constants, a maximum, and then a rapid drop (El’ varies
OIo-
500
0
I
- 500
0
0
20
40
D
60
Figure 4. Comparison of empirical and theoretical linear coefficients for various contact distances.
80
CONDUCTANCE OF SYMMETRICAL ELECTROLYTES
as D - 3 ) to negative values a t low dielectric constants. The value 8. = 3 fits the data quite well in the high range of D; then as D decreases, larger and larger values of 8. are weded to match the experimental L values. The conductance equation is a two-parameter equation because both L and K A depend on b; i.e., A = A(c; ho,b). A program was set up to solve it for these two parameters, but the standard deviation for ah, the difference between calculated and observed values, was usually significantly greater than the standard deviation for results calculated by the three-parameter equation A =: h ( c ; no,L , KA). This means that different values of b (or &, of course) may be required to give the coefficients of the c term and of the cf2 term which will best fit the data. Over the intermediate range of D, the agreement was good, as shown by Table VI, where d(L) is the value derived from the linear coefficient and ~ ( K Ais) the value from the ionpair term. The last column gives the percentage contribution to the total conductance of the linear term Lc (or Lcr) a t the highest concentration of the corresponding run; it is obvious that data of very high precision would be needed in order to determine L. (At the higher dielectric constants, the data permit determination of only hoand L, of course.) Table VI:
Variation of Contact Distance
KC1 in C4H802-H20 78.54 60.16 41.46 30.26 25.84 19.32 15.37 12.74
3.20 3.24 2.91 3.94 3.98 4.41 4.70 5.21
... ... ... 3.99 4.22 4.41 5.01 5.18
1.37 1.54 1.71 2.72 2.73 2.30 1.24 -.2.08
CSI in CaHsOrHzO 78.54 60.18 40.57 29.79 24.44 18.68 15.29 12.81
3.49 3.65 3.73 3.46 4.18 4.45 4.76 4.84
... ...
... ... ...
5.78 5.58 6.02
1.40 1.83 2.36 2.34 2.94 2.19 0.25 -4.50
The increase of a with decreasing D can be interpreted in several ways. One obvious way is to assume increasing solvation of the ions as the fraction of nonpolar component in the mixed solvent is increased.
2593
Alternatively, one could postulate2*,29 that the more polar component concentrates in the volume around the ions, so that the effective dielectric constant to be used in computing interionic forces is larger than the macroscopic value. I n this way, one could retain a fixed ion size for a given system. Enough specificion-solvent interactions have been observed to suggest that both solvation and local dielectric disproportionation could be involved. Certainly, the next problem for theoretical study is the effects of short-range forces. Finally, we realize that there are other sources of linear terms in the conductance equation besides those summarized by (2.23) and the short-range effects just mentioned. The solution viscosity contains a term proportional to concentration which carries over into the conductance. The volume of the ions, as suggested by Stokes,30decreases the space available in the solution for ionic motion and so decreases conductance. The Debye-Falkenhagen cl” term in the dielectric constant will produce a term linear in c in the conductance through the appearance of D in the coefficient S. The fact that an ion pair tends to orient parallel to the field means that it will be struck more often by an ion of the sign which will increase conductance when the resulting three-ion cluster dissociates; the triple-ion clusters themselves give an increase in conductance proportional in first approximation to concentration. There undoubtedly are other effects which are also absorbed in the experimental coefficient. All of these will cause the calculated a to vary. It must, of course, be kept in mind that the real solution is not made up of charged spheres in a continum; the most theory can achieve is to predict a one to one correspondence between the physical system which is too complex to be treated and a simplified model which is amenable to mathematical description. Including the effects of short-range forces and so on is in last analysis a refinement of the model.
3. Conclusion and Summary The conductance equation (2.3) which was derived from the consideration of the potential of total directed force is an improvement over the earlier equation (2.1) as far as functional form is concerned. In attempting to include the effects of both long- and short-range interionic forces in one comprehensive treatment, however, comparison with experiment shows that precision in detail in the coefficients has been lost. The coef(28) J. B. Hyne, J . Am. Chem. SOC., 85,304 (1963). (29) Y. H. Inami, H. K. Bodenseh, and J. B. Ramsey, ibid., 83,4745 (1961). (30) B. J. Steel, J. M. Stokes, and R. H. Stokes, J . Phus. Chem., 6 2 , 1514 (1958).
Volume 69, rumbe?’ 8
August 1966
R. M. FUOSS, L. ONSAGER, AND J. F. SKINNER
2594
ficients L and A are both approximations, the first valid for high dielectric constants and the second for low, to the quantities ( J - E log 6E1’) and KA, both of which can best be computed ~eparately.~JThe longrange interactions between ions not in contact are best described by (2.1); over the range of dielectric constant where the approximations made in its derivation (e’ = 1 rz/2), both (2.1) and (2.3) agree numerically, as shown in Figure 3. The short-range interactions which create ion pairs are best described by (2.20) ; for smaller values of dielectric constant, the coefficient 6E’K(b) approaches KA, and the values of a obtained from equating the experimental value of the coefficient of the cfz term to either differ at most by about 15% in the range where ion pairs might be expected to have physical stability, i e . , where their potential energy 2 / a D is significantly greater than kT. To summarize, observed conductance curves can be described by the equation
+r +
A =
A0
+
+
- Sc”’yr‘/’ E c log ~ (6El’c~) L c ~ K A C ~ ~(3.1) ~ A
where L is given by
L
=
L1
+ Lz(b)
(3.2)
and K A by (2.20). The two terms of L are given by (2.24) and (2.25). In order to obtain the constants Bo, L , and K A from the data, the following sequence is convenient for programming. A preliminary value of A0 is obtained by graphical extrapolation, and estimates (which need not be a t all accurate) of L and K A are made. As zeroth approximation, y is set equal to the conductance ratio A/& in the square root term, giving as a first approximation 71
=
A/[Ao - S ( C A / A O ) ~ / ~ ]
(3.3)
Since the linear and logarithmic terms usually just about cancel each other, (3.3) is already a fair approximation. (If the estimated A. is too small, the denominator of 3.3 can become negative, and then (3.4), which calls for the square root of cy1, would set
The Journal of Physical Chemistry
the computer off on an infinite loop. To avoid this trap, an IF instruction terminates the computation, should y1 come out negative.) Then as next approximation y2 = A/[&
+
- sc1/2ylr’/2 Ecyi log
(6&’~1)
+
LCYll
(3.4)
This process is iterated until the condition lyn
- yn-il
< 0.00005
(3.5)
is satisfied. Then, if the converged value of y is less than unity, the data are treated by the conventional least-squares program to obtain A,, L , and KA. From both L and KA, values of b are obtained by solving (2.28) and (2.20). To avoid possible loops (and wasted machine time), the solution of Lz(b)for b is terminated if b becomes less than 1 (which would correspond to absurdly large d values). The function eb/b3 has a minimum at b = 3; to avoid presenting the machine with the dilemma, of a double-valued function and the probable attempt to divide by zero, the calculation of 6 from K A is stopped if b becomes less than 3.05. The value a t the minimum of eb/b3is e3/27 = 0.75. If the corresponding experimental value from K A is less than 0.75, the machine is instructed not to attempt a solution. Finally, the calculation is terminated if b exceeds 25 (which would correspond to absurdly small ti values). For the range of dielectric constants where both L and K A can be determined with relatively good precision, the values of d from LZ and from K A agree within about 10%; for high dielectric constants, the value from K A becomes unreliable, while for low dielectric constants, the value from Lz becomes uncertain, as might be expected. If (3.4) converges to a value greater than unity, y is set equal to 1, and the solution of the three-parameter equation is continued. Then the data are automatically processed by the twoparameter equation A
= A0
+
+
- SC”’ EC log (6E1’~) LC
(3.6)
Also, if K A from the three-parameter equation (3.1) is less than 10, the data are analyzed by means of (3.6).
