T H E
J O U R N A L
O F
PHYSICAL CHEMISTRY Registered i n U . S . Patent Ofice
@ Copyright, 1964, by the American Chemical Society
~
VOLUME 68, NUMBER 1 JANUARY 15, 1964
The Conductance of Symmetrical Electrolytes.'" IV. Hydrodynamic and Osmotic Terms in the Relaxation Field
by Raymond M. Fuoss and Lars Onsager Contribution N o . 1742 from the Sterling Chemistry Laboratory, Yale University, New Haven, Connecticut (Received September 18, 1963)
Using (1) the differential equation which defines that part of the total potential which has hydrodynamic origin, ( 2 ) the corresponding Poisson equation, and (3) the appropriate boundary conditions, the term AX, in the relaxation field is calculated up to terms of order c')' in concentration in the conductance function A(c). The Boltzmann factor e' is kept explicit as exp( - Oe-"'/r) throughout the computation; the principal approximations made are to drop terms of order x a c3/'. The leading term contains the negative exponential integrals En(2xa)and E,(xa) and thus contributes c log c terms to A(c); these are identical with our previous result.lb The next term, linear in c, has a coefficient P 2 ( b ) , b = e2/aDkT, which contains most of the function K ( b ) which appeared in the electrostatic part AX, of the relaxation field. The kinetic term in A(c) is also computed to the same degree of approximation; combined with AX,, it gives the complete function K ( b ) , multiplied by a small coefficient.
-
In the first paper of this series, the equation of continuity for solutions of symmetrical electrolytes was reduced t o the form
v .esVp21
=
- V ( r )cos 8
(1)
where
consider species 1 as cations. Then for 1-1 electrolytes, el = + e and € 2 = - e, where e is the elementary charge, 4.8022 X 1O-Io e.s.u. The function V ( r )cos 8 is defined by the equation (1)
{ =
e2e-"'/rDkT -= De-"'/r
(2)
is the ratio of the electrostatic potential energy of two ions of opposite charge to the thermal energy k T , and fi?l is the potential of the total directed force acting on an ion of species 1 located by spherical coordinates ( r , 6) with respect to an ion of species 2 a t the origin. The dircction of the external field Xi is along the z-axis (5=r cos 8). The discussion is limited to symmetrical electrolytes, so that el = - e2; by convention, we shall
(a) Corrections to previous papers of this series: I, R. M.Fuoss and L. Onsager. J. Phus. Chmz.. 66, 1722 (1962): eq. 1.25, divide (VI VZ) by (WI a);eq. 1.27, divide right-hand side by (WI a);eq. 2.2, multiply right-hand side by cos 9 ; eq. 2.5, right-hand side is I- V ( r ) cos 81: eq. 2.7, right-hand side is [ - V ( r ) COB 01; eq. 2.31, replace 9 by 9 . 11. R. M .Fuoss and L. Orisager, ibid., 6 7 , 621 (1963): eq. 12, replace A by V a n d insert minus sign before V ( r ) ; eq. 15. insert minus sign before V ( r ) ; eq. 72, divide +.v'(b)by 3; e q . A4, second term on right-hand side is e*//h; p. 625, fifth line under (711, the exponent of (DT)is 3/2. 111, It. M. Fuoss arid L. Onsagor, i b i d , 67, 628 (1963): cq. 16, replace by f ; Table I, column hcadings are b; F ( b ) ; b; F ( 6 ) ; b; - F ( b ) . Symbols used in I. 11, and I11 will be uscd in general without redefinition here. Equations derived in these papers will be cited as I (1.25), etc. (b) R. M. Fuoss and L. Onsager, J. Phgs. Chem., 61, 668 (1957).
+
+
+
1
RAYMOND SI. Fuoss AND LARSONSAGER
2
+ v,) x
- V ( r )cos 9 =2ef(vzcos 6
(dT/dr)/(wl
+ a4
(3)
where v, and vr are the ficld and radial components of the velocity of an ion of species 1 in the vicinity of an ion of species 2, and q is the reciprocal of the friction coefficient of an ion of speciesj, so that
vjO= Xcpji
(4)
The potential pZ1was separated into two terms, hl’ and p21”, where ppl’ is the purelyelectrostatic part of the potential, satisfying by definition the homogeneous equation
V*erVpZ1’=
o
(5)
The solution of this equation was given in part 11. We now present a treatment of the equation
v.efVp2,”
- V ( r ) cos 9
=
(6)
which gives the part of thc relaxation field which derives from the motion produced a t the location of a given ion by the motion of its neighbors. This result, plus the osmotic term12calculated with the Boltzmann factor e‘ explicit, will be combined in the following paper with our previous results to give the following conductance equation
+
A = ’io - ~ c ’ ’ ~ 2 ~ c ’In
Ez’G(b)]c -
- [ E , ’ A ~ H (~) (2b - 3)A07~/b’ - 6E’c“’‘K(h) 7
(7)
wherea 7
=
Px/2 = 4.2016 X 1OB~’/’/(DT)’/’ (8) h = @ / a = t2/DkTa
(9)
and the functions H , G, and K are explicit functions of the Rjerrum parameter h. The coefficients
S
=
CY
+ Po&
(10)
and
IC‘
=
R,’i\o - Ez’
(11)
are determined by the valence type of the electrolyte, the temperature of thc system, and the dielectric constant and viscosity of the solvent, They are the same ones which appeared in our first solution of the p r ~ b l e m . ~I t will be noted that the functional form of h ( c ) is the same as before; what will he changed is the depndcnce on Q of thc coefficient of the term linear in concentration, which is now separated into four terms. ‘I’hcsr include our former * ] ( a ) ,plus new terms which aDDcar as a conse(Ii1cncc of retaining the Boltzmann factor cf explicitly in (1) instcad of approxThe Journal of Physical Chemistry
imating it by the first, few terms of its series. As point,ed out in part 11, one of the new terms predicts for lower dielectric constants a. decrease in conductance of the form which previously required the hypot,hesis of a thermodynamic equilibrium between free ions and ion pairs. The ternis calculated in this paper slightly decrease this “ion pair” term from AX+ 1. The Hydrodynamic T e r m in the Relaxation Field. We designate by AXDthat part of the relaxation field v2)/ which is produced by the velocity term (v, (w, w2) in the c q u h o n of continuity6; in principle, it, can he derived by the following sequence of operations. First, the corrcsponding part of the equation of continuity (6) is solved for pzl”, where
+
+
p21”
5
kTX21”
+
e%”
0.1)
and XU’’ and e”, respcct.ivcly, measure the changes in distribution function fitand potential produced by the velocity field. T h o rcsult is substituted in the corresponding I’oisson equation0
+,
AO” = - ( ~ ~ k T / 2 e ~ ) e ~ ~ ~ ~(1.2) ” =
+
- (x2/2e2)erpZ1”
x2&’1,h
(1.3)
Solution of (1.3) then gives e”, and the component in the direction of thc external electrical field of the force on the central ion due to the hydrodynamic motion of its neighbors is given by
AX, = -eV,O”(a) (1.4) i.e., by the x-component of the negative gradient of the potential e O”(r,tY) calculated a t r = a. The fact that only the value of (be”/bx) a t r = a is desired, rather than e” as an explicit, function of rand 9, pcrmit,s considerable simplificat,iori of the mathttmatical problcm. As we shall show, it is necessary to integrate neither the equation of continuity nor the Poisson equation in order to evaluate (1.4), because an adequate approximation can be expressed as a dcfiriitc integral in terms of V ( r )by using the propertics of ~ 2 1 ’ ’ and 0” as defined by their differential equations, (6) and ( 1 . 3 ) , and the boundary conditions which these functions must satisfy. The quantit,y V ( r ) cos tY is an ahhrcviat,ion for the scalar product v.Vr’ which appeared on expansion of the divergence of the product of distribution function and velocity. Wc shall use our previous value’ of velocity (2) (3)
It. 31. F u o s ~arid L Onsager, J . Phm. Chem.. 62, 1339 (1958). R. 31. Fuons and L. Onsaaw. Proe. S m t l . A r n d ’ S c i . 17. S..47, 818 (1961).
R . RI. Fuoss and L. Onsaper, J . Phus. Chem., 61, 668 (1957) ( 5 ) See eq. I (1.25). (6) See “9. I (1.19). (7) it. h i . Fiioss. J . Phyn. Chem., 63, 633 (1959).
(4)
v
=
i jl(r)
+ r1 j z ( r ) cos 9
(1.5)
where j l ( r ) and J ( r ) arc given explicitly by eq. 8 of ref. 7, and the constants I3 arid E which appear in them have the values given by ( 5 ) , (6), and (15) of t,he same paper. Since { is only a functioii of r ( i e . , is sphcrically symmetric)
rer = rl ef(d[,/tlr) =
er[jl(r)
+ fz(r)l(d{/dr) cos 9
(1.7)
k)e(;a~lsoi . r , = cos #. Substit~ihngexplicit values of the constants arid functions from ref. 7 , the result is
v .efvpzl’’ =
[Xcer/nqx2(wl
+ w 2 ) ] F ( r )(dildr) cos 9 (1.8)
F(r)
=
C1/r3 - C:ze-Yr(1
+ xr)/r3
Q”(r) (’os 9
c1
=
(1
+ xu + x2a2,/3)/(1 + xu)
(1.10)
arid
Cq = e““/(l
+ xu)
(1.11)
Introduction of a new dopendent variable 9 ( r ) simplifies the form of (1.8) considerably by absorhing into the coefficient the parts which arc indepcndcnt of the coordinates; it is defined by t,hc equation p Z l ” ( r , 8 ) = BXep(r)
COS
+
O / T ~ X ~ ( Wwq) ~
(1.12)
l‘he equation of continuity then reduces to
v(p cos 9)
+ V{.V(p
cos 29)
= - [ F ( r ) cos
alp ](d{idr)
( I . 13)
It will he noted that e’ divides out a t this stage; it will reappear in the Poisson equation. After making the vector operations indicated in (1.13), cos 9 can next be factored out, leaving an ordinary dil-ferential equation in one variable. Further simplification is obtained by approximating df;/dr by (-p;r2) in t h c gradient tcrm. (This neglects long range screening, hut givcs a closc approximation near the ion, r < I/x, where acmracy is riced(1d.) ‘I‘hctsc manipulations reduce the equation of continuity t,o thc form
+ ( 2 / r - P/r2)(dp/dr) 2cp(r)/r2 = P(r)e-“‘(l
+ xr)/r2
= - tl
f?”(r,tY)/Xa
(1.17)
AXJX
=
@’(a)
(1.18)
that is, if IVP evaluatr Q” for r = a , the problem is solved. I’sing the riew drpcwlent variable, the Poisson equation t)ccomes ( b a n
+ ( 2 , ’ ~ )dQ”/dr
- (2/r2
+ x2er/2)&”(r) =
befp(r)/2nq(wl
+
w2)
(1.19)
The hamnicr a d tongs approach to the problem would now tw to solve (1.14) for p, adjust the solution to ttic t)ouridary conditions, sutlstitute the result in (1. I:)), so11.e this equation i n t u r n arid adjust its solution to t t i c tmundary eoiiditions, differeritiate partially with rcsspcct to x , arid then set r (qual to a. Preliminary investigation, however, showed that A X , is i n geiiclral sinall cornpared to AXe arid therefore can have little riiirricrical influmce oti the total linear coefficient. An adrquatc upper estimate for AX, can bc found hy a much lcss laborious route than by the formal attack just outlined. If we drop the term x2pf,2 compared to 2 / r 2 in (1.19), the differential tquation which dcfiries Q”(r), we still have a good approximation to a description of the situation near the ion where r < 2 ~ - ” /x~; i r i (+feet, long range screening is being ntAglected again The net effect will be to suppress a damping function ~ - ’ ~ ( l y r ) iii thc integrand of the equation which determines AX,; as we shall show in the Appendix, such approximation introduces 110 catastrophir errors. The equation for &”(r) thus becomes
+
d*&”/dr2
+ (2,’r) (dQ”/dr)
- (2/r2)Q”(r) =
befv(r)/2sq(wl
+ w2j
(1.20)
S o w consider the identity
(1.14)
We next consider ttic I’oissori equation (1.3). Substitutioii of p ( r ) gives Ae” - x2ere”/’2 = - [pXefp(r) cos trl/2T?e(wl
(1.16)
so that the desired result can he stated in the compact form
(1.9)
with
-EIH”(u)/~
=
Again to simplify t h e coefficients, we introduce a new function Q ” ( r ) , tiefined by
d2G)”/dr2
where
d2p/dr2
AX,
(1.6)
arid t,hr scalar product, rcviiiccs to v.Vef
Now O”(r,9) must be of the form f i ” ( r ) cos 9 and by use of the houndary conditions* which the potcvitial must satisfy, we find
+
(121)
The quantity in brackets is recognized as the operator in
w2)
(l.l?i)
(8)
See cq. I1 (19-20).
I‘olumP 68. S7tmher I
January, 196’4
4
RAYMOND &I.Fuoss AND LARSONSAGER
the differential equation which defines cp(r) (and hence which determines ~ 2 1 " ) . By substituting (1.21) in (1,14),rearranging, and integrating, we obtain
sam
erp(r) d r
211
=
(2/r) d&"/dr - 2Q"/r2
=
(1.23)
and that by virtue of the boundary conditions on the potential a t r = a r(d&"/dr),
+
-
12n?l(w
___
~
12an WI
+4
Jam
+
WZ
Define I(%) as the integral in (1.33). Then
saa
C1
I(x)
erp(r) dr (1.27)
W Z ) ~
=
1
l
sam
+ (x2a2/3)
e'F(r)e-"'(l
+
(x2a2/2)Jam ET)
dr
=
a)
Il(x) = =
+
(r)
rl.Vpzl" = [PXe cos 29/nqx2(ol
+
w2)]
[dpldr] (1.31)
Substitution of these last two results (1.30) and (1.31) in the boundary condition (1.29) gives
+
(dp/dr)a = -xz/3@a(l xa) (1.32) Since we shall eventually drop terms of order c'", we may neglect xa compared to unity in the denomThe Journal of Physical Chemistry
+
er[l - e-"'(l
(1.36)
sa
eree"'(l
+
ZT)
+ xr)]e-"'(l + x r ) dr/r3 dr/ra -
2xr) dr/r3 - x 2 =
X Ecos 29/6~?7u(l xu) (1.30) cp
Using the
+ xr) dr/ra -
ere-"'(l
ereWzxr(l xr)2 dr/r3
sarn
(1.29)
Using our previous' value for V, we obtain
From the definition (1.12) of
(1.35)
(1.28) where
(T
(1.34)
+ xr) dr
erF(r)e-"'(l
dr
WZ)
=
=
is the integral which is to be evaluated. approximations for C1and Cz above I(x) = I1(x)
-
(1.33)
-
(1.26)
+ + Vpzl"]-rl = 0
(v-r&
+ x r ) dr
erF(r)e-"'(l
0
We thus have the estimate of &"(a) expressed in terms of the values of dp/dr a t the boundaries r = a and r = 00, and a definite integral over known functions. The boundary condition for pzl" a t r = a is9 [2v/(w1
l
The hydrodynamic part of the relaxation field is seen to be determined by a definite integral over known functions, one of which is the radial part of the inhomogeneous term in the differential equation which defines ~ 2 1 " . The presence of the screening function e-"' guarantees the convergence of the integral. We now proceed to the evaluation of this integral; it is clearly a function of concentration through x. Since our ultimate goal is a conductance function valid up to terms of order c3l2 x 3 , we shall develop the integral as a series and drop terms of order x 3 and higher. To this approximation, the constants C1 and Cz defined by (1.10) and (1.11) can be simplified as
(1.24)
Now combining (1.27) and (1.22), we obtain &"(a)
WZ)]
+
(1.25)
the integral of (1.20) becomes = - [b/27rq(w1
+
WZ)
= 0
&"(a) =
3&"(a)/a
-xzaeb/367rq(~14-
=
+ x2a2/3+ O(x3a3) Cz = 1 + x2a2/2 + O(x3a3)
&".(a)
=
(dQ"/dr),
AX,/X
[P/12m7(wl
-
that is, the integral on the left is given in terms of the integral of known functions and the values of d y l d r a t the boundaries. The integral on the left of (1.22) is also proportional to the result of multiplying (1.20) by dr and integrating between limits a and 00. Noting that d[2&"/r]/dr
inator here. At the upper limit, dp/dr vanishes, of course. Combining these results, we finally obtain
sam
e'e-2"'(1
ere-'"' dr/r
I z - I 3 - xz14
+
(1.37) (1.38)
The last two terms of (1.36) above can be considerably simplified, because they have coefficients x2. The integrals - are functions of x,which converge for x = 0; going to this limit, we see that the last two terms combine to give
- (xzu2/6)Jam e'/' dr/r3
=
-Ax2a2/6
(1.39)
where A is a constant which will be evaluated below. Hence (9)
See eq.
I (1.31).
CONDUCTANCE O F SYMMETRICAL
ELECTROLYTES
I(x) = II(NJ- Ax2a2/6
(1.40)
+ XKT)
ere-X"'(l
Jam
(1.41)
dr/r3
with equal to one or two. We shall therefore first evaluate this auxillary integral. As shown in part II,loer may be approximated by e-"e'/', where e-" is the square of the activity coefficient. Using this eZTl6(Xx)=
sam
ep/7e-hxrdr/r3
where
j,(b)
and only Il(x) remains to be calculated. I n (1.38),l2and 13 are both of the plattern
Iz(XH) =
5
+
+ j,(b)/2 + 3/41 + O(x3a) (1.54)
eZTI1(x)=x2[~,(2xa) - E,(xa)/2
(1.43)
Setting N equal to zero evaluates the first coefficient, which also appeared in (1.39)
I6(0)
=
Jam
= [eb@ -
1)
+
(1.45)
1 ] / ~ 2
Next, partial differentiation of (1.43) with respect to x gives
b16/dx = =
B
-X2x
sam
e'/'e-'"'
+ 2CxE,(Xxa)
drlr -
Cxe-x"a
e@/re)\Xr
+ 2Dx
(@Ir - l)e-h"r dr/r
drlr
(1.47)
+ E,(Xxa) (L .48) (:I.49)
--x2/2
just cancels the exponential integral on the right of (1.47). Thisreduces (1.46) and (1.47) to
$" (ep/' - 1)eB
Again setting x
=
hxr
dr/r
3K(b)
=
E,@) - (e*/b)(l
er = 1
=
Then N may be divided out of (1.50) and for D we finally find D
=
-),2e-XXa/4 - (A / 2)j,(b)
(1.52)
(1.58)
+ + {2/2
(2.1)
fII
COS$
dS
(2.2)
where
n=
- fizO)kT/ni
(2.3) and the integration is over the sphere of radius a which contains the ion. Since (fiz
=
2 r a 2 sin 6 d$
(2.4)
- fl2O
(2.5)
and fl2
(1.51)
+ l/b)
I n order to be consistent with the present calculations of relaxation and electrophoresis in which er is kept explicit, we must recalculate the virtual force A P on the ion produced by the local concentration gradients, using nln2erfor fl;, the unperturbed distribution fun& tion. As before, this force is given by the integral
0, we see that B must vanish B = O
(1.57)
8. The Osmotic Term.I2 I n our previous ca1cuk~tjon of this term, we used F12,the perturbation to the distribution, which was calculated on the basis of the approximation
d!? (1.50)
(1.56)
which is almost the pattern of K(b), where
=
+ X2xe-"""/2 + 2Dx
+ eb/6b2 - 1/6b2
2P2(b) = E,(b) - (eb/b)(l - 1/3b)
AP
c=
+ j,(b)/2 + 3/4 -
The part of P, (b) which is sensitive to b is
which, with
-X2x
+
[~~pe-~'/12~~& wa)]P~(b) (1.55)
eb/2b
(1.46)
The integral on the right of (1.46) contains an exponential integral
sam
=
Pl(b) = E,(2xa) - E,(xa)/2
(1.44)
ePIrdr/r3
Substitution of I.L = P/r converts this to an elementary form which integrates to
A
AX,/X where
+ Bx + Cx2E,(Xxa)+ Dx2 =
(1.53)
This is substituted in (1.36) and the result in (1.33), to give for the hydrodynamic term
and expanding through terms of order x 2
A
E,(b) - In b - I'
is the function introduced in the discussion of the eleotrostatic part or the relaxation field.]' If the above results are substituted in (1.37)) we find for 1,
Xx Jam e'/re-Xxr dr/r2 (1.42)
e2'16(Xx) = A
=
= Xl2flZ0
(10) See eq. I1 (55-57). (11) See eq. I1 (A.3). (12) R. A I . Fuoss and 1., Onsager, J . Phw. Chem., 6 2 , 1339 (19583, In eq. 14,AP should be replaced by A P / e l .
Volume 68,Number 1
January, 19tL$
the integral becomes AP
the radial component of the velocity vanishes.13 If we set R = a in (2.15) and use the definition of w in
[ 2 ~ k T ~ ~ f l z ~ (X~ ) / n l ]
=
~ 1 2 ( a , 6sin ) 6 cos 6 d 9
(2.6)
&IOz =
=
-
(tzX/kT) [&(a) -
-
U ] COS
6
37rva(w1
e28/h-T (2.7)
-
Since O(a) $'(a) is of order xu, we neglect it here compared to [R(a)- a ] . At r = a ir=
p - - 2 ~ eb
(2.8)
AP/Xt
=
(2.9)
At r
=
(x2a2Xeebe-"/6p) [R(a) - a ] (2.10)
=
+ + xa +
(I - ~ 1 ) R ( a = ) - T ] [ u - (p/x2a2){ 1 - e-Xa(l
+ e-'[a
xa) )I
- (~/x2a2)(1- e-"(l
x2a2) 11
(2.11)
After expansion and rearrangement, this gives
R(a) = ~ e - ~ b j / 4 ( 1 Tl)
+ O(xap)
(2.12)
TI
=
e-*(]
+ b + b2/2)
3K(b)
+ w 2 )1(2eb/3b
=
E,@) - (eb/b)(l
(2.13)
x2a2b2/24(1- Ti) -
2P3(b)
vo2
=
(Xt/47r~)(Zi?/Ba3
+ l/2a)
The Journal of Physical Chemistry
=
(3.1)
2En(2xa) - I.J,(xa) - In b -
+ 3/2
-
113
P (3.2)
In the next paper of this series, this result will be combined with the electrostatic relaxation field from part I1 and the electrophoresis from part 111 to give the conductance function (7) announced in the introduction.
Appendix In discussing the differential equation which defines Q"(r), an approximation was made which will now be investigated. The full differential equation is A(Q" cos 9)- r2eerQ"(r) cos 19
=
Aefq(r) cos 6 (Al)
where A is the coefficient shown in (1.19). This equation, whose operator is of course precisely the same one which appeared in treating AX,, has the soli~tions'~ for its homogeneous form
(2.13)
Here R [not the R ( a ) of ( 2 . 7 ) ! ]is the distance at, which
(2.19)
WZ)]
r
(2.14)
The first term will give a small linear term in the conductance equation; its multiplier e-" has been dropped because it would lead only to a term of order c"'. This multiplier will bc retained in the second term of (2.14), however, because this one (also small) will be combined with other terms in which the difference between e-" and unity can become significant We note that the second term contains a factor ueh b = p e b / b z , and that AX, contains a function which differs from K ( b ) , the essentially exponential function which appeared in AX,, by the term in e b / b 2 S o w the s-component of the velocity function used in setting up the differential equation (6) for pZl" reduces a t r = a a n d x = Oto
+ 1/61
+ AP/Xe = [ x 2 p / 1 2 ~ q (+~ 1 X [Ps(b) + 3e-2'K(b)/2] + p2x2/24(1- TI)
where x 2a2ebe - 2r/6b
(2 18)
2,
which appeared in the electrostatic term AX, of the relaxatioii field. 3. Combination o/' Hydrodynamic and Osmotic Terms. The contribution to the relaxation field from the velocity field is given by (1.55). To this we add the osmotic term just derived, with the coefficient of the second term converted to hydrodynamic symbols. The result is
Substitution in (2.10) then gives =
(2.17)
1
AX,/X
where
AP/Xt
=
It will be seen that the second term now naturally combines with the eb/fibZin the function I-'z(b) of (1.57) in AXL. Then this term, together with eb,/2b and the exponential integral from j p ( b ) , combine to give the same junction
a, R(a) can be evaluated easily by expanding
e - x a as a series and dropping higher terms
w2)
x2a2b2/24(1- T i ) [x2ppe- "/ 12aq (wl
Substituting in the expression (2.6) for A P and inserting the factor of from integration over 9,we obtain AP
+
Writing a 2 / bin the equivalent form p a / b 2 in the second term of (2.14) and using (2.17) to replace a in the result, we finally have for the osmotic term
as before; for symmetrical clrctrolytes n2 = 8aqt2//nkT = 8nnp
(2.16)
it is easily shown that
Using our results (and symbols) from part JI xi2
Xew
(13) (14)
Reference 4, eq. 5.21 and 5.22; ref. 7. eq. 10-18. See eq. I1 (39, 40).
CONDUCTANCE OF SYMMETRICAL ELECTROLYTES
= ebY'(l
fl(r)
+ yr)/y2r2
(A2)
fz(r) = eYt(l - y r ) / y 2 r 2
643)
and use of Green's function as before will lead to the result &"(a) =
B
s"
e-Yr(l
+ yr)era(r) dr
in analogy to I1 (47). If the term in dropped, the result would be &"(a)
=B
lam ercp(r) dr
L e . , the screening function e-"r(l Since e-Yr(l
y2
+
(A4) in (Al) is
=
y r ) disappears.
(A61
= 1
(A7)
=
J5"e-?'(l + y r ) d r ) dr
and
I(0) =
som
(x4r/24)En(xr)- e-2"r/4r3 (A12)
To this, we add C3/r2,the solution of the homogeneous equation L ( a ) = 0 which vanishes a t r = a ,and use the boundary condition (1.32) to evaluate Cs. To terms of order x 2
+ O(X'U)
C3 = -x/6
+ (2/r) dcpldr - 2p(r)/r2 = F(r)e-"'(l + x r ) / r 2
x/6r2
(-414)
in (A8) and performance of the indicated integrations gives
+
I ( % )= ~ " f i ( ~ , y ) E , ( p x ~(7x2/16)E,(~x~) ) x2f2(y,a)E,(xa) - ~ e - ~ " / 6 a f 3 ( y , ~ ) e - ~ ~ ~-/ 8 a ~
+
f4(y,u)e-8xa/8a2 (A15) where y2 = 92x2 = x2/2
(AW
p = l + q
(A17)
s = 2 + q
(AIS)
fi(x,y) = 3/16
-
x2/8y2
(A191
+ + ~ ~ a ~ (-420) / 3 ) 1 + xa(q - 5/3) - (x2a2/24p)(3/9 +
fdy,a) = (x2e-Y5/8y2)(1
ya
2 - Q) - x3a3/24 (A21) and f*(y,a) = 1
dr
(A13)
Substitution of
(A8)
(A9)
in order to study the long range effects of the damping function, e-yr. Since we are interested here in long range effects, we shall approximate the differential equation (1.14) for cp(r) hY d2p/dr2
+ x2r2/6 - x3r3/6)+
(e-"'/4r3)(1 - x r / 3
f3(y,a) =
Hence we shall consider the two integrals
I(?)
qP(r)=
= ( ~ p ( r )-
+ rr) 5 1
lim exp@e-"'/r)
(A10) by quadratures. The result is, after approximating CI and Czin F ( r ) by unity
(A51
for all values of y and r and is monotone decreasing, the integral (A5) is an upper bound for the desired integral (A4). We shall show, by investigation of Ow0 closely related integrals, that (A5) is an adequate approximation to (AB), and therefore our result, which was obtained by dropping the y 2 term in (Al), is likewise justified. The point a t issue really is whether neglect of the screening function (A6) introduces serious error ; convergence of the integral is already guaranteed by the e - x r terms in cp(r). The Boltzmann factor e{ is always greater than unity, but for large distances, T > 1,") rapidly converges to unity Iim er
7
i e . , the short range term ( - b ' / r 2 ) in the operator is dropped. The operator in (A10) can be rearranged to
which permits us to obtain a particular integral of
(A22)
In the limit of y (or q) going to zero, I ( ? ) reduces to I(0). No singularities appear. On expanding e - Y 5 inf2(y,u), the terms in y-2 from fl and fz are seen to cancel and are therefore harmless. The ( 3 / q ) in f3 is matched by a term of opposite sign, which appears when E,(pxa) is expanded as a Taylor's series in y
E,(pxa) (h10)
- xa(2 - 9 )
=
+ y2e-xa(l + xa)/2x2 + o(q3)
&(xu) - ye-y5/x
(~23)
The whole integral is of order x 2 In xa and higher, because the term (-xe-ya/6a) cancels out when the exponential functions e-'"' and e--sxa are expanded as series and the indicated multiplications are carried out. The final result after series expansions and the above cancellations is Volume 88,Number 1 Janua.ry, 1964
RAYMOND M. Fuoss AND LARSOKSAGER
8
I ( 7 ) = (x2/16)[4E,(xa) - E , ( ~ x u )7 E , ( s x ~ ) ]- (79 22q)x2u2/192 O(x3u3) (A24)
+
+
Expanding the exponential integrals, and regrouping terms, (A24) becomes
I(?)
=
x2[0.25In xu
+ 0.121041
(A26)
Since the equations are not to be used when the distance 1/x becomes less than about 5u (due t o failure of both model and mathematical approximations), 0.25 In xu in absolute value is a t least 0.40 and becomes numerically greater as concentration decreases. At xu = 0.2, I ( 7 ) = - 0 . 3 8 ~ and ~ Z(0) = - 0 . 4 7 ~ ~ the ; difference is about 25% and, as just mentioned, decreases as concentration decreases. Hence as asserted in section 2, suppression of the screening factor e-Y‘(l rr) in the integral produces no gross error. We shall next show that the net error in the coefficient of the linear term of A(c) is still smaller and becomes negligible as the dielectric constant decreases. Terms from AXo always appear in the conductance function with the coefficient Et2times a factor of order unity while terms from AXe have coefficient E’& times factors of order unity. Consider, for example, the coefficient of the term in In T
+
E‘
E‘]& - E’2
The Joairnal of Physical Chemistry
E’1 = p 2 ~ 2 / 2 = 4 ~r2/6
(A27)
(A28)
and
(A25)
while
I(0) = x2[0.25In xu - 0.068753
The Etl&part comes from AX,, the E‘z part from AXo. The values
E’z
= flx/30/16~‘/~
(A291
where the Onsager electrophoresis coefficient Po is defined as
po
=
5Ex/3PquC1’~
(-430)
are substituted in (A27). Here u (equals times the velocity of light) is the factor which converts e.s.u. potentials to volts and 5 is the Faraday equivalent. The result is E’lAo[l - ( 5 ~ / 2 ~ ~ ) ( l / A o ~(A31) ~)]
E’
Substituting numerical values for the universal constants
E’
=
E’i[l - 2.460 X lO-*/pAoq]
(A32)
Since = 560.4 X lO-*/Dzs
(A331
and the Walden product A,? usually varies between one half and unity, it is seen that the second term in the brackets (the hydrodynamic part) is small, and eventually negligible as D decreases, compared to the leading electrostatic term.
