March, 1963
(if n, = 0.0086 mole ~ m . -for ~ (FeCh),) n*h
=
62 1
CONDUCTANCE OF SYMMET~EICAL ELECTROLYTES
6 X
mole cm.-2 (if X gS 5 X V*
h
=
0.8 X 1011 sec.-l
(the estimated Debye temperature -480OK. gives a mean frequency of 0.8 X 10'8 sec.-*). Here SI,* and AS: are both positive (rather than negative as found for iodine) and AH: and AS: are very large. The activated state for vaporization-condensation may be presumed to be FeC& molecules or perhaps pairs of FeClp, i.e., incipient molecules. The normally large condensation entropy loss as gas molecules move to the activated state may be offset by virtual diasociation of FezCle into FeC& monomers. I n the gas phase ASo for this dissociation is 29.3 e.u.6 An entropy change approaching this magnitude plus the loss on moving to the surface then could give a net small positive AS,". A similar qualitative argument may be presented for a positive AB:. On vaporization, the six iron-chlorine bonds formed by each iron atom in the semi-ionic sandwich layer type solid are replaced by four interactions per iron atom in the gas dimer, or three if the activated state is a monomer; one would expect a large enthalpy and entropy increase in such a process. The estimated Debye frequency is about one hundred times larger than v*/h (using the molecular weight of FezCle). However, in addition to any surface reduction of the Debye temperature, one also would expect probability factors to lower the effective frequency. If the ions in the solid1 move more or less independently, a cooperative motion of the iron ion and its neighboring chloride ions is required for release of FeCL(g). If the ifon ion and three chloride ions must move together toward the surface while the other three neighboring chloride ions move away, a probability factor between (6) W. Kangro ahd H. Bernstorff, 2. anorg. allgem. Chena., 263, 316 (1950); H. Schdfer, ibid., '269, 53 (1949).
(l/2)' 20 S 1/6, if any three C1-, and 1/128, if only specified ones, might be expected; direct release of FezClawould be even less probable. a,,for the process 2FeCls(s) + 2 FeClz(s) Clz(g) is very small and could only be roughly estimated as with no temperature dependence apparent. With equilibrium data,' AHo = 26,000 cal., ASo = 39 e.u., and Pe = 4 X 10-6atm. at 408'K., one may tentatively suggest the following
+
AH(2) = 0 As@) = 11.2 AF(2) = -4600
AHc* = 0 As,* = -27.5 AF,* = 11,200
AH,* = 26,000 cal. mole-l AS,# = 0.3 cal. deg.-l mole AF,* = 26,000 cal. mole-l n*h = 3.2 X 10-24 mole cm.-2 (h 3 X lo-* cm.) v*/X 1.6 X loll sec.
-
v*/h is somewhat closer to the Debye frequency than in the case of Fe2C16. The vibration probability factor should be larger here; only two chlorine atoms must move together, while the iron ion with which they are associated in the solid moves away. AH,* appears to account for the full Vaporization enthalpy and no appreciable change in standard entropy is associated with vaporization activation. The entire entropy of condensation is accounted for by AS,* together with the standard state change term. The activated state would appear to correspond to an energetically unchanged chlorine gas molecule which finds an appropriate site on the FeClz-FeCla surface; however, the sites are few and far between and virtually no translational motion is permitted once the molecule enters the condensation site. Acknowledgment.-This work was supported by the National Science Foundation. (7) H. Schafer and E. Oehler, ibid., 271, 205 (1953).
THE CONDUCTANCE OF SYMMETRICAL ELECTROLYTES. 11. THE RELAXATION FIELD BY RAYMOND M. Fuoss AND LARSONSAGER Contribution No. 1715 from the Sterling Chemistry Laboratory, Yale Uniuersity, New Haven, Connecticut Received August lY,1962 The Poisson equation for the asymmetry potentials of a symmetrical electrolyte in the conductance process has been integrated by the use of the corresponding Green's function in order to obtain the purely electrostatic terms of the relaxation field. The Boltsmann factor in the distribution function was retained explicitly as an exponential throughout the calculation, instead of approximating i t as a truncated series as has been customary in previous derivations. The consequence of this refinement in mathematical methods is the appearance in the relaxation field of a term which will lead to a decrease in conductance with increasing concentration or decreasing dielectric constant. The decrease is proportional to the product of concentration and the square of the mean activity coefficient. It depends on dielectric constant through a function which has as its asymptotic limit eh/b3 ( b = ~a,/uDkT),which is the form of the theoretical association constant for contact pairs. This result means that the ad hoc hypothesis of ion pairing controlled by a mass action equilibrium is no longer needed to obtain a satisfactory conductance function; the former mass action term is derivable from the Poisson equation. It was missed in earlier theoretical work by too drastic approximation of the Boltsmann factor.
The relaxation field AX which retards the motion of ions moving in an external electrical field is given b.y the component in the field direction of the negative
gradient of the potential p' which the asymmetry in the distribution of the other ions produces a t the location of a given ion
RAYMOND 111. Fuoss AKD LARSONSAGER
822
AX = -v,+’(a) = -a+‘/ax
(1) The potential is obtained by solving the corresponding Poisson equation n,A+‘,
e
- ( 4 ~D)ZJ‘jiei /
fdr)
foji
expC-4Oj/kT)
=
(4)
by the first few terms of its series. For low concentrations in solvents of high dielectric constants, this ation is valid (as confirmed by the excellent t) in those cases between theory and experiments, but when the dielectric constmit becomes small, the exponent
Ei+J/JCT (5) becomes sigliLificantly larger than unity, especially for ions in contact, and then the trunaated series no longer leads to a conductance equation A = A (c) of the correct functional farm. (Theory, concave-up ; observation eoncave-down or inflection range, on a A - ellZscale.) An equation which accurately reproduces the observed conductance in these cases can be obtained by g fiast, that ion pBirs contribute nothing tance, and second, that their concentratioii ulated by a classical mass action equilibrium between free ions and psirs. The earlier forms of this theory encountered difficulties in the pairs; these finally were resolved by defining pairs to mean pairs in contact1 but even so, the ad hoc hypothesis of mass action equilibrium had t o be retained. If the decrease of conductance ascribed t o association were due solely to the effects of electrostatic forces between the ions, one should in principle be able to predict such a decrease from the solution of the Poisson equation, without recourse to an additional hypothesis. Since the theoretical association constant was found t o be proportional to eb, where [ = -
b = e2/aDkT (6) and since e b is the value which e l approaches at low concentrations for ions in contact, we began to suspect the series approximation of the Boltzmann factor in the equation of continuity as the source of the failure of the 1957 equation2foE “associated” elecbrolytes. I n the previous paper3 of this series. a new approach to the problem was made in which a multiplicative separation
fdr)
f”ik) [1 + x,i(r) I
(7) of the distribution function permitted the explicit =
8(r,6) = +l’(r)/el = -+z’(~)/Ez
(1) R. M. Fuoss, J . A m . Chsm. Soo., 80, 5059 (1959). ( 2 ) R. 31.Fuoss and L. Onsager, J . Phys. Chem., 61, 668 (1957). (3) R. M.Fuoss and L. Onsrtger, zbzM., 66, 1722 (1962). Symbols defined in part I will be used in part I1 without redefinition.
(8)
simplified the Poisson equation to the form (9)
~ 1 6 2 b . 0 = y2/CTerXzl
where the coefficient
+
y2 = q a ~ = z (4n/DkT)(n1e12w1
fO&) + f’&)
(3) which eventually required approximating the Boltzmann factor in the distribution function =
retention of the exponential function throughout the calculation. Introduction of a function 8(r,6) for the coordinate dependence of the asymmetry potentials for binary electrolytes
(2)
but in order to solve (?), the pertusbt+tionflji in the distribution must, of course, first be known, It can be obtained by solving the equation of continuity, which relates the distribution functions fji and the velocities of the ions. Previous solutions of the problem have always started with an additive separation
Vol. 67
np€2”2)/(wl
+ w)
(10)
reduces to K 2 / 2 for this case, where nl = n2, €1 = - E Z . Then the total electrochemical potential for an ioii of species 1 in the vicinity of a refereme ion of species 2
+
- ~ i X x kTxz1 $. $8 (11) reduced the equation of continuity to the elegant form pa
V(T) (12) where V ( r ) depends on the local velocity field. The latter is k n ~ w nto~give terms of order c log c and c in A(c). I n order to focus aktention on purely electrostatic terms (rather than hydrodynamic), p21 was separated into two terms V.(d
P21
=
Ap21)
=
P‘21
+
(13)
P121
where p l Z l was defined as the solution of the homogeneous equation
o
(14)
~ . e ~ ~ = p ’ z ~
leaving ptpzl,the term of hydrodynamic origin, as the solution of
v .er~,.4”21=
(15)
~ ( r )
Treatment of (15) will be given in a later paper. The solution of (14) was found to an approximation valid up to terms of order c3I2in A(c) ~ ’ 2 1=
- E ~ X R ( TCOS ) 6
(16)
where R(r) is given explicitly by eq. (2.31) of part I. If 8(r,&) is separated in a fashion analogous t o that for the total potential 8 = 8’
and likewise ~ A8’
+ 8“
(17)
the Poisson equation becomes
2 1 ,
- y2er8’ = (y2Xer/el)[R(r)-
7-1
cos 6
(18)
The purpose of the present paper is t o present the solution of (18), which through (1) and (8) will give the electrostatic part A X , of the rel’axation field. Before beginning the mathematical development, it will be convenient to define several new functions. Since B’(r,eS) is proportional to the asymmetry potentials t)’,, it must satisfy the boundary condition [T(dO’/&)
- e’],
= 0
(19)
and since it necessarily is of the form 8’(r,zX) = H ( r ) cos 6 = z H / r (4) R. 11. Fuass, zbzd., 73,633 (19%).
(20)
CONDUCTANCE OF SYMMETRICAL ELECTROLYTES
March, 1963
[r(dH/dr) - HI,
(rH’
=
- H)a = 0
(21)
The part of the relaxation field AX, corresponding to 8’ is given by
AX, = - e l ( ~ e ’ / a ~ ) a =
-E~[H/T - ( ~ ‘ / r ~ ) ( r H-’ H ) ] ,
(22)
(23)
By virtue of (21), this reduces a t once to AX,
=
-elH(a)/a
(24)
This means that we need only the value of H ( r ) at r = a in order to obtain AX,. I n order to simplify coefficients, we introduce Q(r),defined by
-€IH(r)/Xu which leads to the compact statement
(25)
A X J X = &(a) Here, Q(r) satisfies the differential equation
(26)
&(T)
(r2&’)’ -- (2
=
+ y2r2eS)&(r)= - (yzr2ees/u) [R(r)- r ]
+dr)
(28)
Substitute in (27) and define F(r) as the solution of
(r2F’)’ - (2
+ +)F(r)
=
- (y2r2eS/a)[R(r) - rl
(29)
whence g(r) is the solution of (r2g’)’
- (2
+ y2r2)g(r)
=
y2r2(ef - l)F(r)
4- y2r2(er- l)g(r)
(30)
It is clear that the first inhomogeneous term of (30) is of order K ~ Fand , since F gives as its leading term in A(c) the square root term
(NK), we conclude that g must give ternis of higher order than K ~ . Investigation shows that the leading term of g is of the order ~~a~In KU. The second inhomogeneous term on the right side of (30) is clearly of still higher order. Therefore, in order to obtain A(c) through terms of order c, we may neglect the contribution of g(r) to (26) and need only the solution of (29) which will give the relaxation field
=
0
(33)
The differential equation which determines F ( r ) can be written symbolically as
L(F) = - 4 k )
(34)
where L represents the operator on the left in (29) and -$(r) the inhomogeneous term on the right. The most direct method of evaluating F ( a ) is by use of the Green’s function5 G(r,x), by means of which the solution of a differential equation of the form of (29) can be expressed as a definite integral
F(r)
=
G(r,z)4(x)dx
Jm
a
=
1
G I ( ~ , 4 4 (dx 4
(35)
+ lm G11(~,2)$(z)dz(36)
where Gl(r,x) is the value of G(r,x) for r 3 z and G;1 for r 6 x . Tho first derivative of G(r,x) has a discontinuity a t r = 2; elsewhere G and its first two derivatives are continuous functions of r for a fixed value of X . Then
F(a)
(27)
which results from substituting (25) in (20) and the result in (18) and then expanding the Laplacian operator and dividing out cos 29. I n (27), primes denote differentiation with respect to r. Now (27) obviously is a difficult equation to treat, on account of the appearance of e’ as a multiplier of the unknown function &(T) on the left. By expanding essentially in terms of (eS - l), however, an approximation to AX,, valid through terms of order K ~ ,can be obtained. Let
Q(r) = F(r)
dF/& - F ) a
(r
the boundary condition simplifies to
623
= [GII(T,z)~(z) dx
(37)
Green’s function is constructed from the solutions of the homogeneous equation
L(F) = 0
(38)
which are
+ yr)/yZr2
(39)
f 2 ( r ) = eyT(l - yr)/y2rz
(40)
f1(r)
=
e-?? (1
and Green’s function is defined as G ( ~ , z= ) A(x)fi(r)
+ B(x)fz(r) *
[f1(4f2(r) - fz(x)f1(r)1/2p(x)W(4 (41) where p ( x ) equals 22 for our operator L and W ( R is ) the Wroeskian
W ( Z )= fifi’ - fi‘fi = -2/yx2 (42) I n (41), the positive sign holds for a 6 x 6 r and the negative for r x 00. The functions A ( x ) and B(x) are determined by the condition that G(r,x) must satisfy the boundary conditions (32) and (33). From the first
<
