RAYMOKD 31. Fuoss
1722
COMPARISON O F E X C H 4 N G E AND
IARS ONSAGER
Vol. 66
TABLE 111 HYDROLYSIS COXSTAXTS FOR CHLOROACETATE I O N
Reacting molecule
a
AND
.4ND CHLOROACETIC
ACID
k , (1. mole-' see.-')
Ich (sec.-l)
CHzClC003 83 X 10" exp( -27,900 & 600)/RY' CHLCICOOI-I" 2.11 X 1Ogexp(--24,800 f 500)lRT I?. A. Kenney and V. J. Johnston, J . Phys. Chem., 63, 1426 (1959).
acid and chloroacetate systems. In the chloroacetic acid system, activation energies for exchange and hydrolysis were experimentally indistinguishable; in the chloroacetate system they are significantly different. Pre-exponential factors for both exchange rereactions are within a factor of two of that predicted on the basis of gas phase collision theory. The difference in the activation energies for the exchanges (1900 f 800 cal. mole-l) is consistent with electrostatic effects. A value of 1900 cal. mole-1 is equivalent to the work done in water a t 80' in bringing together from an infinite distance tJvo unit charges of like sign to a distance of 2.7 A. apart.
2 26 X 10" exp( -26,400 f 400)/122' 0.91i X 1011exp(-24,500 =k 400)/RT
'l'hc slightly smaller negative entropy of activation for the reaction involving the two ions (AS* for exchange of the molecule is -10.6 e.u.) is not readily explainable. (The uncertainty in both entropy changes is approximately 1.G e.u.) The results are consistent with the assumption that chlorine exchange in both the chloroacetic acid-chloride and chloroacetate-chloride systems occurs by a simple bimolecular displacement type reaction. The results obtained in this work indicate that the hydrolysis and exchange reactions occur quite independently of each other. Acknowledgment.-This work has been supported by AEC contract AT(40-1)2826.
THE COXDUCTANCE OF SYMMETRICAL ELECTROLYTES. I. POTENTIAL OF TOTAL FORCE BYRAYMOND 31.Fuoss AND LARSONSAGER Contribution N o . 1691 from the Sterling Chemistry Laboratory, Yale University, New Haven, Connecticut Received April 18, 196%
By means of a multiplicative expansion of the distribution functions which describe local ionic concentrations, the 1932 Onsager-Fuoss equation of continuity can be integrated with explicit retention of the Boltzmann factor, instead of approximating the latter by a truncated power series. The result is expressed in terms of the potential ,uji of total force acting on a given ion: the present approximation to vpji includes the external field, the forces due to neighboring ions and to the asy.mmetry of the ionic atmospheres, and the virtual forces due to local concentration gradients. The differential equations whlch will lead to the forces from the velocity field also have been derived.
The Fuoss-Onsager conductance equation] A
= A,
- XC"~4- EC log c
+ JC
(1)
precisely reproduces experimental data for dilute solutions of symmetrical electrolytes in solvents of high dielectric When combined with the hypothesis of ion a~sociation~-~ to pairs, it leads to the more general conductance function A = A, -
+ ECYlog CY f JCY K A C Y ~ (*2 )~
which successfully describes observed values of conductance up to moderate concentrations (Ka = 0.2) in a wide variety of ~ y s t e m s ~ ocovering -~~ the (1) R. M. Fuoss and L. Onsager. J. Phys. Chem., 61, 668 (1957). (2) J. E. Lind, Jr., J. J. Zwolenik. and R. M. Fuoss, J. Am. Chem. Soc., 81, 1557 (1959). (3) J. E. Lind, Jr., and R. M. Fuoss, Md., 83, 1828 (1961). (4) J. E. Lind, Jr., and R. M. Fuoss, J. Phgs. Chem., 66, 999 (1961). (5) J. E. Lind, Jr., and R. M. Fuoss, ibid., 66, 1414 (1961). (6) R. L. Kay, J. Am. Chen. Soc., 81, 2099 (1960). (7) R. M. Fuoss, ibid., 79, 3301 (1957). (8) R. M. Fuoss and C. A. Kraus, ibid., 79, 3304 (1957). (9) R. M. Fuoss, ibid., 81, 2659 (1959). (IO) F. Aocascina, A. D'Aprano, and R. M. Fuoss, ibid., 81, 1058 (1959). Fuoss, 'I. ibid., 81, 1301 (11) F. Accasoina, S. Petrucoi, and R. & (1959).
range of dielectric constant down to about 12, below which ionic interactions higher than pairwise become significant. As frequently happens in the feedback between theory and experiment, experiments designed to test the generality of the theory underlying (2) revealed some limitations, which in turn have suggested further theoretical studies. The functional form of ( 2 ) appears to be correct, in that no systematic deviations between calculated and observed conductances appear within the claimed range of validity (0 KG < 0.2) of the equation. Problems have, however, arisen in the interpretation of the constants. The equation involves three arbitrary constants: do,J , and KA. From each of these, one can compute the centerto-center contact distance a between the charged spheres which are used to represent the
10, but run concave-down in the range below 10, and the computer has even delivered negative values of .KA for systems in solvents of high dielectric constant, Again, these values of K A are uncertain, because here the J-term now masks the association term, but the trend is clearly observable.*~” These two problenzs, variable a~ at low D and variable UK at high D,have one element in common. To first approximation, both the J - and KAlerms of ( 2 ) are linear in concentration. This suggests that thc solution of the problems might be found irr a re-examination of the approximations which lead to the linear terms in (1). Furthermore, tJhe present theory is mathematically inelegant, in that the ad hoc hypothesis of ion association must be invoked in order to convert (1) into ( 2 ) . Since ion-pairing is assumed t80be a consequence of electrostatic forces it should be possible to deduce a conductance equation in which KA does not appear as an arbitrary constant. The purpose of the present series of papers is to develop a more general theory of the conductance of dilute solutions of symmetrical electrolytes. We shall use the same model as beforelvz4to represent the system: charged spheres in a cont,inuum. The previous theory is based on the following approximationZ5for the Boltzmann factor (22) H. Sadek and R.. AM.Fuoss, J . Am. Cham. Soc., 81, 4507 (1959). (23) R. M. Fuoss And 1’. Acoascina, “La Conducibilith. Elettrolit>ica,” Edizioni dell’ Ateneo, Rome, 1959: “Electrolytic Conductance,” Interscience Publishers, New York, N. Y . , 1959, Chapter XVI. This reference will bc cited here frequently aa a 8oiirce of previously derived equations; the notation FA. 3.29, for example, means eq. 29 of Chapter 111. (24) R. M. E’uoss and L. Onsager, Proc. Natl. Acad. Sci. U.S.,41, 274, 1010 (1955). (25) FA. 12.23: FA.9.17.
us = l
r
+b +p/2
1723 (4)
where is the ratio of electrostatic potential energy of one ion in the field of another to the thermal energy kT. In the present treatment, e t will be retained explicitly in the integration of the equation of continuity. Anticipating the final result, it will be found that K A is replaced by an explicit function of a, which has as its asymptotic limit the value previously assigned to KA; in other words, the conductance function is reduced to one involving only two parameters, A. and a. I n this paper, we present the integration of the equation of continuity, in order to obtain the potential of the total force acting on a reference ion, and in subsequent papers, we shall use the result to obtain the various terms of a revised conductance equation. 1. Statement of Problem.-The starting point for the theoretical study is the Onsager-Fuoss equation of continuityz6 d i n (ftjvu)
+ divl (fjtvji) = 0
(1.1)
wherez7 fjt(r2J = n,nlZ(rd = n,nij(r12) = f d r d (1.2) are the symmetrical distribution functions used before and v , ~is the velocity of aj-ion in the vicinity of an ion of species i. The distribution functions and the potentials are related through the Poisson equations28
- (47r/D)2,€tf,i
(1.3) In order to save space, we shall take as given the material in chapters 8, 9, 10, and 13 of ref. 23. The velocities which appear in (1.1) are determined by the external field and the internal forces. The primary observable is specific conductance, which, as the ratio of current density to field strength, depends explicitly on ionic mobilities. Conscquently, the problem of calculating conductance as a function of concentration reduces in principle to the mathematical problem of solving the differcntialeq. 1.1 and 1.3, subject to the appropriate boundary conditi0ns.2~ The previous attack1Vz4began by an additive expansion of the distribution functions (fli = f”ji f’,,); here, in order to keep the Boltzmann factor in fojjz= n,n,et explicit instead of approximating it by a truncated series,25we now make a multiplicative expansion as @kj
=
+
fli(r21)
=
fO&) [1
+ Xjt(r2l)l
(1.4)
where foj6(r)
=
n,n,e x ~ ( - e ~ # ~ / k T=) njnier (1.5)
is the distribution function30for zero external field and xj, is the perturbation produced in the spherically symmetrical folz by the external field X . (261 L. Onsager and R.NI. Fuoss, J . Phys. Chem., 36, 2689 (1932); Equation 2.2.4; FA. 9.6. Further citations from this reference will be given as OF. 2.2.4. etc. (27) OF. 2.1.6: FA. 8 3 , 8 7, 8 10. (28) OF. 2.4.2.: FA. 8 8. (29) FA. X. (30) FA. 9.13 and 9.15.
RAYMOND M. Fuoss AND LARSONSAGER
1724
Vol. 66
We shall assume as before that the field X is beloJv The velocity is given by83 that corresponding t o a detectable \Ken effect31*32 and shall therefore drop all terms of quadratic or v d r ) = vi@) 3- wi[Kji(r) - 1cTv 1nfjt(r)] higher order in the field. (1.21) The potentials are expanded as and congruently for vij (r12). L4ssumingsuperposi$ 3 ( ~ ~ 1= ) P'k) -I- $'J(rzl) (1.6) tion of the asymmetric parts of the ionic atmoswhere $03(r) is the potential in the absence of an pheres, t,heforceE4Kjt(r) in turn is given by external field and $'3 is the perturbation, assumed Kji(r) = eiXi - ei~p,bz.'(a)- eiv2t,bj(r) (1.22) to be proportional to X. The latter is related to the corresponding term in the distribution function by A parenthetic comment on the second term on the the Poisson equations right is necessary. It represents the force on an i-ion due to its own at,mosphere and is small comnjA$jj' = - ( 4 7 / 0 ) 2 t e z f 0 3 t ~ 3 t (1.7) pared to the t,hird term which represents the force The asymmetry potentials are assumed propor- on the i-ion due to a neighboring j-ion and the tional to the chargcs by introduction of the func- atmosphere of the lat,ter. Formally, the second ~)], at r = a ; since tion O(r,O); specializing to the case of a simple term is [ - - ~ ~ v & ( r ~calculated electrolyte ( j = 1, i = 2 ; el = E = - - 2 ; nl = a central force can produce no motion, the term $io in $i drops out here, leaving [ - eiv2$%'( r) 1.: n = na) By the symmetry properties35 (1.14), this is [eiV~$i'(r) l a ; hence the result above where $l'(r) = E @ ( T , ~ ) (1.8) !bL'(r) = -eEZO(r,8) (1.9) I n terms of 8, the Poisson equations become n1elilO = L Z T ~ : ~ ~ X ~ ~(1.10) /D
(1.11)
n2f,Ae = 4ne1f'X21/D
on expanding the sums in (1.7) and using the symmetry properties
xd-r) A'(rlz)
-xZ&-) = xi&-> xdr) = 0
=
=
A'(-r>
=
(1.12) (1.13)
--$%'(r) (1.14)
P2$i'(r) la (1.23) The following steps combine the relevant functions to give the starting point for the mathematical analysis. The force vector (1-22)is substituted into (1.21) and the result into the equation of continuity (1.1). The expansions (1.4) a,nd (1.6) for distribution functions and potentials are next introduced, and the various product,s are expanded, dropping terms of order X 2 . All differentiationP are reduced to different,iations with respect to the coordinates of rZl: v2 = v = . - V I ; il, = il = A,. The final result of these manipulations is V~h'b)
~ . { e ~ [ ( ~l wf2w2)Xi l -
where
62(~zV$i')
rzl = r
=
-r12
(1.15)
Equations 1.10 and 1.11 can bc conibincd by multiplying the first by n2e2w2 and the second by n l ~ l and ~ l thcn adding. The result is simplified by introducing a constant y2defined as y2 =
(4a/DkI')
(nlf12W1
+
112€*2&1*)/
(w1
+
[elwIV$l'(a)
(1.16)
+ w2)kTVXei + - enwnV$,'(a) 3 +
(01
(VI
- VJ])
=
0
(1.24)
Oil introduciiig O(r,$) from (1.8) uid (1.9), (124) tmomes
v w2)
-
( E ~ w ~ v $~'
[ei(elXi - E*VO
-
+
+
1 c T ' ~ ~ a l e2~0(a) VI
- v2)]
=
0 (1.26)
which, for a binary electrolyte, becomes
It will be notled that, the Boltzmann factor eS appea~rsexplicitly as a multiplier after the divergence (1.17) y2 = p K 2 = K 2 / 2 operator. I n the earlier work, et was approximated where ~2 is the familiar Debye-I-Itickel parameter by the first fern t'erms of its series a t this stage of the analysis, and disappeared from subsequent equations. The validity of the previous result K' = (4T/DkT)(nlf1' f 7Z2E22) = for solvents of high dielectric constant and t,he 8nne2/DkT (1.18) eventual failure a t low dielectric const,ants both thus become clear. The t,erm in vO(a) is a conThe result of the above stcpb is stant which me shall t'emporarily absorb as a cor~1c2A.8= y2kTerx,l (1.19) rect.ion in the external field X (without change of notation) ; it will be calculat'ed later and inserted. Once x g 1 has been determined by integration of It has been show137 that vj is proportional to e i ; (l.l), integration oi (1.19) will give 0, and theiice theref ore the relaxation field AX
==
-Yr$I'(a)
=
-~l(dO,'d.~),, (1.210)
(d1) L Oniager J Chem Phys , 2, 599 (1934) ( 3 2 ) I1 C . Eokstrorn and C. Schmelzer, Chem. R e $ , 24, 367 (19.39).
(35) F A . 9 . 3 8 4 2 . (36) FA. 9.37. (37) FA. 13.124; R. A t . Fuoss, J . Phgs. Chem.. 63, 633 (1959).
CONDUCTANCE OF SYMMETRICAL ELECTROLYTES
Sept., 1962
1725
+
(2.5') p21 = PZlt PZl" (1.26) Bincc the velocity is divergence-free, and since 1 and define these terms as solutions of the equations depends only on r (Le,) is independent of COB 9. = V d VYLI[ = 0 (2.6) z / r ) , (1.25) rearranges to v.er~ p 2 1 " = V ( r ) (2.7) V . e r ( d i -- kTVxzlj = VI
- 2 ~
- vp =
2e'(v, cos 8
+ ur)(df/dr)
(1.27)
We now require the soIution of (1.27) and (1.19), subject to the boundary conditions (1.28)
0 ( ~ , 8 )= 0
[r(bO/br)- 0Ia U,1(..,6)
=
0
= ElXi
(r- Y 2 J a = 0
(1.29) (1.30)
(1.31)
where
tlXi - ~ ' 0 8- kTVX*i
Y21
(1.32)
2. Integration of the Equation of Continuity.First, me define a function 2 2 1 by 221
=
-x21
- (€2/JCT)0
(2.1)
which simplifies (1.27) to
+
~ - e r ( E ~ X i ~ T v Z , , ) = V(r)
(2.2)
where we have abbreviated the inhomogeneous term of (1.27) as V ( r ) . The Poisson equation becomes A0 - y2er0 = (y21cT/e2)erZ,l
(2.3)
Xexl, define pz1 by p n = - E1XX
- liT&l
(2.4)
-
0 (2.8) ?;ow (2.8) is a homogeneous equation of second order; we therefore need ti+o independent solutions, both of which satisfy the boundary conditions for r = a, and which reduce to the asymptotic value (- E ~ X Xfor) large distances. The complete solution then will be a linear combination of these two, with coefficients chosen so that the final expression for p'21 also conforms to the boundary conditions. The following procedure yields two independent approximate solutions which will give results valid through terms of order c in the final conductance equation. The largest part of ~ ' is the contribution from the external field, because the other term of (2.4) is an effect resulting from the external field as cause; we therefore approximate p'21 in the second term on the left of (2.8) by ElXX =
ElXT cos -9
(2.9) In we use the Debye-Hixkel limiting approximation for $30 -p21'
i=:
r,
l
= -
EI$ZO/JCT= -ele2e-K'/rDkT
p = 3/L)k2'
(2.5)
The symbol p2t is used in (2.4) as a reminder that the quantity there defined is the potential of the total force acting on an ion of charge el ; it includes the external potential Xz, the potential due to neighboring ions and to the asymmetry in the ionic atmosphere produced by the external field, and the potential of the virtual forces arising from local concentrat ion gradients which are expressed by xzl, the perturbation of the dis tributioii function. The Boltamann coefficient er of Vpzl in (2.5), which becomes large when pairwise distances become small or the dielectric constant becomes low l/rD), explicitly takes into account the large pairwise interactions a t short distances which were discarded as fluctuation terms by the previous series expansion of ef, and which had to be restored by use of the mass action hypothesis in order to attain a usable conductance function for systems in solvents of lower dielectric constant. Integration of (2.5) will be carried out in two steps. From our previous work,l we know that V ( T )gives rise to terms of order c log c and c in the final conductance equation. We first write p Z 1 as the sum of two terms
(r
+ Vr*Vpz'
Ap~i'
(2.10)
The abbreviation
so that ( 2 . 2 ) simplifies to ~ . ( e { ~= p ~V (~r ))
Here, we shall consider only (2.6), solution of which will give the leading cl/a term in the relaxation field, part of the c log c term, the pseudo-mass action term, and a relatirely small linear term. Treatment of (2.7) will be deferred until later. Expansion of (2.6) gives
(2.11)
sirnplifics (2.10) to {
=.
Be -"/r
(2.12)
E'or later use, we define a t this point the Bjerrum parameter b
b
=
c2/aDkT = @/a
(2.13)
Using the approximation (2.9) and the above symbols, (2.8) becomes
Apsl' = PelX b(e-"'/r)/bx (2.14) A particular integral of (2.14) is founda8to be (2.15) -- ( & ~ X / K[b(e-"/r)/bz] ?) which, when combined with the solutions of the homogeneous equation Ap'zl = 0 gives a complete solution pp =
+
+
-~iX[(Air Bl/r2)COS 19 p p ] (2.16) From the condition that p'21( ,8)= - clXr COS 0 , p~pl' =
(38) For tho method of obtaining particular integials of equation5 of the type (2.14), see equations (13)-(60) of Chapter XIV of ref. 23
and the relevant discussion.
2
~
RAYMOND M. Fuoes AND LARBON~AGER
1726
we see that Ai = 1, and requiring that pf2, remain finite for vanishing concentration, we find B, = -p/~"by series expansion of the exponential in f i p and inspection of the result BJr2
+ (P/K2r2)(I -
K~
r /2 f
K3T3/3..
.) #
03
(2.17) a t I( = 0, provided B1 has the above value. thus have one solution of (2.8) pclq1.1' =
- ~ 1 X [ r-
We
( , # / K ~ T ~ ) ~ ~ (cos T)]
9= --€lXRI(T) cos ir (2.18)
whcre Q ~ ( T )=
1
- e l K r ( l + KT)
where g2(T)
=
1 - e-"' (1 $- KT
pgl' = -F1X[dR1(T)
-
(2.21)
K'7"')
( , I ? j K 2 T 2 ) g ~ ( T ) ]COS
'd
=
--ElXH(T.)
cos d
(2.32)
pjj
=C
I'loutitlary c m c i i t icrii (131j l ~ ~ ~ c o n i c s
(2.33)
o
(2.34)
=
--EjXX
- €20
(2.35)
-E3XP(r) cos iy
(2.36)
- E ~ X [ N P I (+T ~) V C ' P ~ COS ( T )8]
(2.37)
wherc
+
(2.23)
o
By methods exactly parallel to those used to solve (2.6), the solution of (2.34) is found to be
Af?,(r) IZe-'Zi1(Ir) (2.24) 'rllc c~ollslants,Iand 1) rrlllSt 1)c c\~aluatednext.
*4+B= I
=
where
(2.23)
l$oundnry condition (1 3 0 ) immcdiatcly givcs
e2~0)
V.e-'VpJj =
w1ic.re
E(?)
(2.31)
Then pj3 is the solution of
(2.22)
Coml-)iniirg (2.18) and ( 2 . 2 2 ) , thc solution of (2.6) then is p21'
p
n2e-f
v.e-'(qXi -
=
= -tiXe-'[Ir
COS
&-'&(T)]
j-091
whence p21.2'
+
where the constant8 are given by (2.29) and (2.30) and the functions in the brackets by (2.181, (2.19), (2.20), and (2.21). For later treatment of the electrophoresis, e shall also need pJs, the potential of the total force on an ion of a given species when the reference ion at the origin is of the same species. Since xjj = 0, and
(2.19)
- (B/K2r2)gz(r)]COS 8 = -eXBz(r) cos ir (2.20)
= -EXIT
Summarizing the results, we find for ~ 2 1 '
in this case, eq. 1.27 becomes
To find a second solution, we try e-rv; again using (2.9) to give an inhomogeneous equation, and applying the above methods, we find 2)
Vol. 66
J'i('T') =
1'
Pa(?) =
T
+
(P/K2T2)g1(T)
(2.38)
(p/kZT2)g2(1')
(2.39)
ill = T2/(7'2 - 1)
(2.40)
x = -1,
I)
(2.41)
4-V/2)
(2.42)
(7'2
-
Lllld 'r'Z(b)
1-
e!' (1 -- b
,1'o suinmarize, intcgratioir of the q u a t i o n of continuity (1.1), with explicit retciition of the Boltzmanii factor er = exp(-cr;$JO/lcT), gives (2.31) which, after 3, tedious but clcrnentary calculation, and (2.37) for the purely electrostatic part of the leads to potential of total force acting on a given ion when an external field is applied t o an electrolytic soluA BTI(b) = 0 (2.27) tion, (2.31) valid for unlike charges on the specified ion and the reference ion, and (2.37) for the case of where like charges. The force V,L+'' due t o the velocity TI@) = e w b ( l b b 2 / 2 ) O(KU) (2.28) field remains to be computed bg integration of The terms in 1'2 indicated by O(Ka) are terms of (2.7). Then, given the potentials p J t , pJ3, the rcorder Ka and higher, which came from the series laxation field and the electrophoresis can be obexpansion of e-Ka in evaluating (2.26) ; they would tained. We may expect the leading terms, X d i 2 lead to terms of order c3/2 jn A and are therefore and Ec log c, of eq. 1 to remain unchanged, because dropped. Solution of (2.25) and (2.27) imme- those come from long range interactions of ions; we also expect the coefficient of the linear term J c diately gives t o appear with an entirely new functional dependence on a, part of which x d l be through eb due A = -2'1/(1 - TI) (2.29) to the presence OP ej- in the Poisson equation (1.19>, B = 1/(1 - 2'1) (2.30) because er becomes eb a t r = a.
(r Vp2r'ja
=
[r.(clH/dr)],
+
+ +
+
-I
0 (2.26)
