RAYMOSD M. Fuoss AXD LARSONSAGER
628
c=
-X
(A35) the logarithmic singularity is eliminated from both sides of (A34) and the constant C is determined. After cancelling these terms, K may be set equal to zero in the result
second differentiation with respect to K and a repetition of the compensating process shows that
F
= P2A/2
B=-X-
PA - Xj,(b) (A36) where j p ( b ) is the integral defined by (A3). Then a
= @(A
+ 1)2/2
@37)
Finally, letting K go to zero again, after cancelling expohential integrals on the two sides of the equation, D is evaluated
D
whence
Vol. 67
+ P(1 + 2X)jp(b>/2 +
(Xz/2)J(0)
+ X2a/2 4-3p(1 + X)2/4
(A38)
where J ( 0 ) is the integral defined by (A4).
THE CONDUCTANCE OF SYMMETRICAL ELECTROLYTES. 111. ELECTROPHORESIS BY RAYMOND M. Fuoss AND LARSONSAGER Contribution XO.172'0 from the Sterling Chemistry Laboratory, Yale University, New Haven, Connecticut Received October 18, 1962 The electrophoretic velocity in a dilute solution of a symmetrical electrolyte has been computed, with the following improvements over earlier treatments of the problem: (1)the volume force is calculated as the gradient of the potential of the total force acting on an ion instead of being approximated merely by the force due to the external field; (2) the Boltzmann factor is retained explicitly, instead of being approximated by a truncated series; and (3) the Oseen equation8 of motion (rather than the Stokes) are used. The result gives the Onsager 1926 limiting value ( - e j ~ X / S ? n ) as ) the leading term; to next approximation, this is opposed by a term proportional to concentration, which depends on b = 6 / a D k T in a non-exponential fashion. For example, for b = 1.5(D = loo), F ( b ) = 2.31 and for b = 15 (D i=: lo), F ( b ) = -0.77. The coefficient goes through zero near b = 5.
Forces acting on ions in soIution are transmitted to the solvent through which they move; the resulting local hydrodynamic currents in the latter are the source of the electrophoretic correction to the mobility. For a solution containing nt ions of species i per unit volume, the volume force acting on the ions is z2ntkl, where kt is the force acting on one i-ion. I n the steady state reached in the conductance process, the net force per unit volume of solution vanishes (there is no bulk flow) ; therefore Z( ntkt
+ noko = 0
(1) where ko is the average force acting on one of the solvent molecules and no is the number of the latter per unit volume. I n the vicinity of a specified reference ion of species j, however, the concentration of i-ions is njt and the net force acting on a n element of volume dV becomes
d F = ( 2 ,n,tkjt
+ n&ddV
(2) where kit is the local force acting on an i-ion, given a j-ion a t the origin of coordinates. Combining (1) and (2), the elementary volume force becomes
dF
= 2,
(njikjc - nfkt)dV
(3) In previous calc~lations1-~of the electrophoretic velocity from ( 3 ) , the force kjt was approximated by its leading term Xati, that due to the external field X . This approximation is sufficient to give the limiting square root term, but neglects contributions to the total force produced by the atmospheric fields. As will (1) L. Onssger, Physib. Z..87, 388 (lW6). (2) L. Onsager and R. M. Fuoss, J . Phys. Chem., 36, 2689 (1932). (3) D. J. Karl and J. L. Dye, ibid., 66, 477 (1962).
be shown here, these lead to terms of order c in the conductance function h ( c ) , and clearly should be considered in any attempt to deal consistently with terms of order c. Their previous neglect is undoubtedly part of the reason for the observed variation4 of the size parameter aJ derived from J , the coefficient of the linear term in A(c). The purpose of this paper is t o present a higher approximation obtained by using the gradient of the potential of the total force5 in the calculation of the velocity instead of merely the leading term X d The Boltamann function will be kept explicit I n our recent computations of the local fields and charge distributions in the ionic atmospheres, me found it convenient to consider for some purposes not just the electrical forces (due to the external field and to local charges) Ej,et but rather the total potential (Eli. at - kT grad In fit). According to the ecpations of motion for an incompressible fluid, the formal forces representing the concentration gradients will not affect the resulting velocity field: it is exactly compensated by a redefinition of the pressure to include the osmotic pressure. We retained the hypothesis that the atmospheric charge densities of neighboring ions may be superimposed. To that approximation (at least), the local electric field will be a gradient field (derivable from a potential), and the virtual osmotic force field (-kT grad In f) is inherently a gradient. field. I n the first paper5of this series, it was shown that the total potentials are
.
(4) J. E. Lind, Jr., and R. M. Fuoss, ibid., 66, 999, 1414 (1961); 66, 1727 (1962). ( 5 ) R. M. Fuoss and L. Onsager, ibid., 66, 1722 (1962). Symbols de. fined in parts I and I1 will be used here without redefinition.
CONDUCTASCE OF SYMMETRICAL ELECTROLYTES
March, 1963 pji
= -
ptt =
ElXR(T) cos 6
(4)
- €tXP(T)cos 8
depending on whether the reference ion has a different charge or a like charge with respect to the i-ion located by coordinates r , 6. The functioiis R(r) and P(r) are given by eq. 2.31 and 2.37 of ref. 5. Since the potential depends on both r and 6 (Le., on x, y, and x ) , the simple Stokes equation dv = dF/Gaqr
(6)
which is valid when the force d F has only an x-component, may no longer be used to calculate the component of local velocity in the field direction. Instead, we must use the more general Oseen equation6
+
dv = [dF (dF r)r/r2]/8nqr (7) which gives the velocity dv produced by a volume force dF, acting at the origin, at a point located by the vector r in a medium whose viscosity is 7 . We are interested only in the component of2 of velocity in the field direction produced by the force; it is obtained by projecting vj in the field direction, Le., by forming the scalar product of vi and i, the unit vector in the field direction dujz = [idFj
+ (rl.dFj) cos 9 ] / 8 n y r
629
the exponent {, we use the Debye-Huckel first approximation for $, giving =:
E2e--Xr/?DkT= pe-"'/r
(1G)
where p is a constant defined in (16). The partial integration leads to the result
ulr
= - (nejxpj377) Jam
e-rP(r)]dy/r
e-''
(1
+ xr) [e"(?> +
+ (ne,Xa/377) [ebR(a)- e-bP(a)] (17)
where b = 2/aDlcT
(181)
The result (17) follows from (15) by inspection: the limits of e l and e.-r for r = m are unity and the limits of both P ( r ) and R(r) for large r are simply r; hence the integrated term vanishes a t the upper limit. At the lower limit, where r = a, e*r becomes e i b plus terms of order xu; since the P-term has a coefficient (-e-r), differentiation gives it the same sign as the Rterm. There now remains the explicit evaluation of the integral in (17) and the function in brackets. The functions P(r) and R(r) are given explicitly by
(8)
where rI is the unit vector in the radial direction. Let dV = 2n-r2 sin ZY. d 6 dr be an annular element of volume located by r, 9.. Then from (3) dFj/dV
(9)
= -&nj,Vpjt
where kjt is replaced by Vpjt and I:,ntkt in (3) vanishes by the condition of electroneutrality since kt = X d . For a binary electrolyte, nl = nz = n and nll = ne-r,
n12
=
ner
(10)
where =
14 l k T
(11)
and $ is o€ course the electrostatic potential. Substituting (10) in (9), and recalling that €1 = - e 2 = E, we obtain dFl/dV
E
-nllVp11 -
n12Vp12
=
(12)
+ er\7(R cos 6)]
(13)
+ rl(dR/dr - R/r)cos 6
(14)
nrX[e-sV(P cos 6) where V ( R cos 6)
=
i(R/r)
and similarly for V (I' cos 6). I n order to obtain the electroplioretic velocity, the scalar products i.dF1 and rl.dF1 are formed, using (13) and (14), where i.rl = cos 6. The results are substituted in (8) and the integrations over 6 (0 5 9 5 n) and r are perflormed. The result, after collecting terms and rearranging, is viz = - (n+Y/37)
j:m[er d(rR)/dr e-I d(rP)/dr] dr
(15)
The form of (15) suggests a partial integration. I n (6) C. W. Oseen, "Hydrodynamik," Leipeig, 1927.
Akademische VerlagsKesellschaft,
Since the operations a t this stage of the derivation involve r as the independent variable, the functions P and R above are written as P(r) and R(r)for compactness. We must, however, bear in mind that both functions also depend on concentration via x 2 and on the contact distance a = P/b. The integrated term of (17) is easily evaluated, because at r = a, the exponential functions in gt(a) and g2(a)may be expanded as series, giving
+ O(x3a3) gz(a) = -x2a2/2 + O(x3a3) gl(a) = x2a2/2
(33:) (34)
Higher terms than the one of order x 2 a2 are not needed explicitly here, because the coefficient (nejXa/3q) is already proportional to concentration and terms of order x3a3 in gl(a) and gz(a) would therefore give
RAYMOND M. Fuoss AND LARSONSAGER
680
terms of order xa in &(a) and P ( a ) , whence terms of order e'" in A(c). To our present approximation, we may neglect them. The integrated term then becomes ( E I X T ' / ~ W P ) [(I ( ~ /-~ )Ti)-1 -
Jn(x) = (Pz/2)
7
(37)
= pX/2
Next we consider the integrals in (17). These are similar to the integrals which appeared in the calculation of the relaxation field' and will be evaluated by the methods described in the Appendix of part I1 of this series. Let I l represent one of the integrals in (17) I1
=
sarn
e-sP(r)e-"r(l
+ xr) dr/r
(38)
Jl(0)
=
sam
+ B, + CE,(2xa) + O ( x )
A/x
(39)
Since I I has a coefficient n ,- xz in the electrophoretic velocity, it is clear that part of the square root term comes from A / x , while B1 contributes to linear terms in A(c). The transcendental term in (39) does not, however, lead to an additional c log c term in A(c), because it will be found that
Ip
=
+ Bz - CE,(2xa)
A/x
(40) and the exponential integrals therefore cancel in the sum (11 12). On substituting the explicit value of P ( r ) , given by (19)-(26), the following integral appears as one of the terms of II
+
(44)
- 1 + p / r ) dr
(e-"'
(45)
(A more rigorous argument which justifies replacing by Jl(0) here is given in the Appendix.) By the methods used in part 11, Jl(0) is found to have the value = pJ,(b) = ~ [ j , ( b )- 1
Jl(O)
+ (1 - e - 9 / b I
(46)
where
+ In b + r
j,(b) = E,@)
(47)
E,@) is the negative exponential integral and I? is Euler's constant, 0.5772157. . . . The other two terms of I(x) are familiar integrals
+ pe-""/b
J z ( x ) = 2e-""/x JZ(x)
I1 =
+ xr) dr/r2
J1(x)
It diverges for x = 0; as we shall see shortly, it is of the form
e-3Kr (1
which clearly converges to Jll(0) = P2/2 a in the limit x = 0. The higher terms likewise are well behaved. Hence we need only J1(0)in order to obtain the desired conductance function
(Tz - 1)-'] = ( E , X T ~ / ~ X ~ P ) (35) FZ(~) where we have substituted n = xz/8np (36) in the coefficient of the integral of (li'), aiid made use of the abbreviation5
Vol. 67
=
E,(2xa)
(48)
4-ewZxa/2
(49) Pie now return to the calculation of I,. After using the device of compensating terms just described, and using the condition that the sum of M and AT must be unity, 1, can be put' into the form
+ M/3J,(b) + (NP/x2)14(x) + (14fP/x2)15(x) (50)
11= Iz(x) - / 3 1 3 ( ~ )
The int'egral I*(%),given by
I&)
e-xr (1
= Jam
+ xr)[l - e-nr (1 + x r ) ] dr/r3 (51) (52)
= s a r n f 4 ( x r )dr
is an elementary integral, and is found to have the explicit value
It clearly diverges for H = 0, but by adding and subtracting compensating terms in the integrand, we shall see that I(x) can be put into the form I(x)
=
al/x
+ + a2
a3
In xa f
a 4 lii ~
xa (42)
When inserted in (17) to give the velocity, the coefficient n reduces the al/x term above to the leading square root term in A(c); the a3 111 xu term is, as already mentioned, cancelled by an opposing term from Iz; and the a4 term is of higher order than xz c and is therefore dropped. First, write I ( H )as
-
1(x)
=
lam (e-r - 1 + {)e-"' (1 + Jl(4
If the exponential term is seen to give 17)
+JZ(4
XT)
-
dr
+
PJB(4
= (e-""/2a2)(1 - e-"")
I4(x)
(l/z
-
e-"")
+ (xe-""/a) X
+ x2E,(2xa) - (x2/2)E,(xa)
(53)
On approximating the exponential e-"" by the first few terms of its series, we obtain
14(x)
=
3$/4
+ z2E,(2xa) -
+
(x2i2)~n(xa) 0 ( ~ 3 a ) (54) The integral I5(x) is explicitly
15(x) =
lam
(55)
e-'f4(xr) dr
where fk(xn) is the same screening function which appeared in I 4 ( x ) . Using the methods described in the Appendix of part 11,we find (43)
in J 1 ( x )is expanded, the leading
R. IlI. Fuoss and L. Onsager, J . Phys. C h e n . , 67, 621 (1963).
16(x)
= x2L3/4 -
('/Z)jn(b) -
('/Z)Bn(Xa)
+
Ex(2xa) 1 (56)
Combining the five terms which make up 11, we obtain the sum
March, 1963
CONDUCTANCE OF SYMMETRICAL ELECTROLYTES
6311
TABLE I FUNCTIOK F ( b ) OF b = e2/aDkT h
1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
I n a similar fashion, unlike ions at origin and in d f ) can be evaluated; the result is
12
=
2e-Xa/~-t- pe-""/b (P/2)E,(xa)
+ pe-2xa/2 - 3p/4 +
+ PA[(l/zV,@)+ 1 -
(eh - l ) / b I
(58) On adding, considerable simplification by cancellation appears (11
+ Id/P
=
+
+
(4e-Xa/P~)(1 ~ a / 2 )
+
MF,(b) AFP(b) (59) where F,(b) and F,(b) are abbreviations for the bracketed quantities in the precediing two equations. Again expanding e-"" and retaining only terms of order "W, and abbreviating the last two terms OE (59) which depend on b by Fl(b),the integratedl terms reduce to (11
+ Iz)/P
4/Px - 2/b
+ Fi(bl + O(xa)
(60) The results (35) and (60) are now substituted in the expression (17) for the component vjr. of electrophoretic velocity in the field direction, giving =
vj2
where
r =
I=
-- ejxX/6a7
+ (ejX?/16~7P)F(b) (61)
2.3100 2.1413 1.9912 1.8564 1.7346 1.6239 1.5225 1.4294 1.3434 1.2638 1.1896 1.1203 1.0554 0.9944 0.9370
3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8
0.8827 .7825 ,6920 .6095 .5340 ,4645 .4001 .3404 .2847 ,2325 .1837 .1377 .0944 .0536 ,0149
=
56O/&Dz5 -F ( b ) 6.0 0.0217 6.5 .lo54 7.0 .1793 7.5 .2451 8.0 .3043 8.5 ,3579 9.0 .4069 9.5 ,4524 10 .4954 11 .5785 12 .6454 13 .6965 14 .7430 15 ,7857
If one back-tracks through the equations, he finds tha,t R(r), for example, has one term depending on e-r. Now R(r) has a multiplier e l in the electrophoresjs integral-the product el. e-[ is of course unity. The other term in eQ(r), which does not contain the factor e-r, is multiplied by the constant A and the latter contains a factor e-h; again the exponential behavior is suppressed.1° Appendix We now re-examine the integralll J ~ ( K= )
sam
(e-r - 1
+ {)ecx'(l +
XT)
dr
which appeared in .the previous discussion. First
P x / 2 and F(b) is given by
F ( b ) = 2/b
+ (b/4)[(1 - Ti)-'
- (7'2
- I)-'] Fl@) (62)
I n turn Fl(b) is given by inspection of (57) and (58); the functions j,(b) and j p ( b ) are defined as
j,(b)
=
E,(b)
+ In b -tI?
j&) = E,@) - In b -
-
+
because the screening function e-%' (1 xr) isalways less than unity for N # 0 and equals unity for N = 0. To find the order of the error made in approximating J l ( x ) by the constant H(O), consider
(63)
r
(64) The "constantd' M and A (actually functionc; of b) are defined by (22) and (30); the corresplonding functions Tz(b) and Tl(b) are given explicitly by (24) and (32). As will be shown in the Appendix, for very large values of €1, F ( b ) asymptotically decreases as the logarithm of b F(b)
h
F(b)
3/4 - r / 2 - 1/2b - (1/2) In b (65)
The first term of (61) will be recognized immediately as Onsager's 1926 value for the electrophoretic term of the limiting tangent. The second term of (61) is proportional to the concentration through r 2 and will contribute a linear term in h ( c ) in addition to the Jlc term of the 1957 equation.8 It will be noted that the function F ( b ) , for which numerical values are given in Table I,9 is not a sensitive function and involves no exponential dependence on b = a2/aDkT, despite the fact that the explicit Boltzmann functions e-+': were used in the derivation. (We recall that l ( a ) == b P K + . .. .) (8) R. M. Fuoss and L. Onsager, J . Phys. Chem., 61, 668 (1957).
(9) We are grateful to Mr. James F. Skinner of this Laboratory, who programmed the function For computation on the IBM 709.
-dH/dx
=
Jam
(1 - e-5)(-bp/bx) dr
Then we see that 0
5
(- b H / d x )
