Conductance-Concentration Function
525
Laplace transforms of C andfi with respect to t Laplace transform of C, with respect to t and z Ci distribution ratio between stationary and mobile k phases E perturbation parameter t time Z coordinate along column length linear velocity of the mobile phase 11 solute amount in units of the delta function tu standard deviation of the gaussian distribution U function s ,p complex variables 5 dummy variable for integral D diffusion coefficient Awn statistical moments kl cumulants rh(IC) incomplete gamma function
6,ji
Then, the inverse transform of (A3) leads to the solution in (s,z ) domain
where a =
(5)' +7 1 + ks
Introducing (A2)
e"&(Z-c)}
dc
-
im e,2(s
References and Notes
,L)e(U/2D)(Z-c)
{e G ( Z - 9 ) -
e -Jii(zcCi) dl]
(A5)
and changing the interval of the last integral in the bracket of (A5) and rearranging, one can obtain eq 28.
List of Symbols Q (t,z ) solute concentration in the stationary phase C ( t ,z ) solute concentration i n the mobile phase Co( t ,z ) unperturbed function of solute concentration in the mobile phase C, ( t ,z ) first-order perturbation function of solute concentration i n the mobile phase
(1) J. N. Wilson, J. Amer. Chem. SOC.,62, 1583 (1940). (2) D. DeVault, J. Amer. Chem. SOC.,65,532 (1943). (3)E. Glueckauf, J. Chem. Soc., 1302 (1947). (4)J. Weis, J. Chem. SOC., 297 (1943). ( 5 ) H. C.Thomas, Ann. N.Y. Acad. Sci., 49, 161 (1948). (6)J. E. Funk and G. Houghton, J. Chromatogr., 6, 193 (1961). (7) F. T. Dunckhorst and G. Houghton, hd. fng. Cbem., Fundam., 5, 93 119661. \----,-
(8)G. Houghton, J. Phys. Chem., 67, 84 (1963). (9)P. C.Haarhoff and H. J. Van der Linde, Anal. Cbem., 38, 573 (1966). (IO) R. D. Oldenkamp and G. Houghton. J. Phys. Chem., 67,597(1963). (11) T. S.Buvs and K. DeClerk. Seor. Sci.. 7. 543 11972). i12j L. Lapidk and N. R. Arnundsoi, J. Phys: Chem., 56: 984 (1952). (13)K. Yarnaoka and T. Nakagawa, J. Chromatogr., 92, 213 (1974). (14)J. J. van Deemter, F. J. Zuiderweg, and A. Klinkenberg, Chem. Eng. Sci., 5, 271 (1956). (15)A. B. Littlewood, C. S. G. Phillips, and D. J. Price, J. Chem. SOC., 1480 (1955). (16)K . Yamaoka and T. Nakagawa. J. Chromatogr., submitted for publication.
Conductance-Concentration Function for Associated Symmetrical Electrolytes Raymond M. Fuoss Sterling Chemistry Laboratory, Yale University, New Haven, Connecticut 06520 (Received September 20. 1974) Publication costs assisted by the Office of Saline Water
A new model for solutions of symmetrical electrolytes is proposed: Gurney cospheres centered on ions of charge f e , and surrounded by a continuum containing a continuous space charge which integrates to Fe. When two cospheres overlap, the corresponding ions are counted as pairs; the pairs are assumed to contribute nothing to net transport of charge and, as dipoles, to be disregarded in the calculation of screening potentials and activity coefficients. Relaxation field and electrophoretic countercurrent for the model are computed using continuum theory. These are combined to give a three-parameter conductance function A (c; Ao, KA,R ) where A0 is limiting conductance, K A is association constant, and R is the diameter of the cospheres. The terms of the function are given explicitly, and a method of deriving values of the parameter from conductance data is described. The new conductance equation replaces the previous equations, now obsolete, which have been proposed earlier by Fuoss and coworkers.
Most current theories of electrolytic conductance have two features in common: (1) they use the same model (rigid, charged spheres in a continuum); and (2) they start
by calculating the theoretical behavior of the model, assuming full participation of all the ions in the long-range interactions and subsequently postulating a mass action The Journal of Physical Chemistry, Vol. 79, No. 5 , 1975
Raymond M. Fuoss
526
equilibrium between free ions (which are assumed to have the theoretical mobility) and ion pairs (which a r e assumed to contribute nothing to the transport of charge). This sequence of operations involves the hazard of counting some of the short-range interactions twice. The resulting relation A( c ) , usually a three-parameter equation, predicts the dependence of equivalent conductance on concentration for the model; these parameters are A0 = lim A (c 0), the association constant K A and a distance R. The experimenter gives a set of conductance data ( c k , h k ; k = 1, . . .n)and the equation A = A ( c ) to the computer as input and asks as output for the values of the parameters which will give the best mean square fit of the observed conductances to the equation. Then a correlation is sought between these parameters (which describe the model) and molecular parameters (which describe the real physical system which was the source of the data). For electrolytes whose ionic volumes are large compared to the volume of t h e solvent molecules (e.g., tetrabutylammonium tetraphenylboride in acetonitrile), that is, systems which might be expected to conform to the model in the same sense that a real gas a t low pressures conforms to the ideal gas model (point masses in motion at given p , V , T ) , good correlation is found between the distance parameter R and a, the center-to-center contact distance of the ions, the latter computed as the sum of the crystallographic radii. Furthermore, values of a calculated by Stokes hydrodynamics from A0 and values of a calculated by Boltzmann statistics and classical electrostatics from K A agree with the value of R; for this case, the conduction function reduces to the one-parameter equation A = A (c; a). But in cases where ionic and solvent volumes are comparable, usually U R # aK # aA,and all three vary in a seemingly erratic way for a given electrolyte in different solvents, or with composition for solutions in mixed solv e n t ~ . ~ The - 3 reason for the disaster is obvious: real ions in real solvents are not rigid charged spheres in continua. While theory based on the primitive model gives a good account of the long-range ionic interactions, it ignores all detailed short-range interactions (ion-ion and ion-solvent) which can only be described by parameters which must be system specific. Conductance data for n given system can provide a t most three parameters. Two of these, A0 and KA, depend directly on molecular parameters; the third is dimensionally a distance. It comes in through the boundary conditions used to evaluate the constants of integration in the derivation of the long-range effects (relaxation field dnd electrophoretic countercurrent). As mentioned above, this distance for the primitive model is R = a, the contact distance; in order to use a more sophisticated model, the R parameter of the conductance function must be uncoupled from short-range effects. This in turn requires a theory for the long-range effects based on boundary conditions which are independent of details of molecular structure. The purposes of this paper are to describe a model which separates the problems of correlating A. and K A with molecular parameters from the treatment of long-range electrostatic and hydrodynamic interactions, and to present a theory for the latter which is based on general boundary conditions which are independent of short-range effects. The conditions we shall use are electroneutrality, continuity of all functions a t r = R, and nonsingularity of all functions in the limit of zero concentration. The parameter R will be identified as the distance r from a reference ion beyond which continuum theory may be applied; ions for which a i r I R will be defined
-
The Journai of Physicai Chemistry, Voi. 79, No. 5, 1975
as paired, provided that unique partners can be statistically defined. The model represents the solution as a continuum containing Gurney c o ~ p h e r e scentered ~,~ on the ions; when two (oppositely charged) cospheres overlap, the corresponding ions are counted as nonconducting pairs and are deleted from the population of atmospheric ions. Friedman’s t h e ~ r yof~ the , ~ thermodynamic properties of electrolytic solution is based on a similar model in which a Gurney potential is included in the energy as a “catch-all term for the missing parts of the cavity term, together with other poorly understood effects related to the molecular structure of the solvent.” 1. Ion Pairs
Ionic interactions may be divided into two categories: short range and long range. Interactions of a given ion with nearest neighbors (ions and solvent molecules) are completely system specific. If one of the nearby molecules is an ion, their mutual energy U is much greater than k T ; distant ions would feel the pair electrostatically as a dipole. The behavior of the solvent molecules near an ion or ion pair is quite different from that of distant solvent molecules. (“Near” and “far” are, of course, relative terms, the former meaning within several molecular diameters and the latter half a dozen or so away.) We shall use Gurney’s designation of cosphere to mean the sphere around an ion outside of which the properties of the solvent are described by the macroscopic dielectric constant D, the density, and the viscosity ?I. The mutual energy of two ions i and j whose center-to-center distance ril is greater than R, the diameter of the cosphere, becomes small compared to hT. Let a be the center-to-center distance a t contact of cation and anion, either or both of which may be aspherical, nonrigid, and polarizable. Imagine the solution to be frozen and count the number of cations whose cospheres overlap anionic cospheres; such pairwise configurations for which a Ir i R are defined as ion pairs. If every cation (charge = e,) is surrounded by a sphere of radius a, electroneutrality demands that
where p j ( r ) is the charge density. For the ion pairs, p j ( r ) is a discontinuous function; the central cation finds a unique countercharge in the interval a i r i R. For the other cations, by definition we must have
and electroneutrality therefore requires that
4a
lm
pjr2 d r = -ej
(1.3)
Therefore neutralizing counter charge for unpaired ions is in the space charge around them; for distances r 1 R, the mutual distances are so large that the charge density is describable by the Debye-Huckel time average which appears in the Poisson equation A$* = -4apj/D
(1.4)
For these ions, the exponential of the Boltzmann distribution n j , = ni exp(-e,i,/W) = niet (1.5) may be approximated by the truncated series
527
Conductance-Concentration Function
=
+
5
+
t2/2 (13) (Here tC/, is the potential a t a distance r from a reference ion of species j , n, is the average number of unpaired i ions per unit volume, and n,, is the concentration of i ions a t a distance r.) Combining (1.4), (1.5), and (1.6), we obtain the
et
1
linearized Poisson-Boltzmann equation A&J =
(1.7)
K2$,
where x2 = 8nnoye2/DkT
(1.8)
where no is the total number of cations per unit volume. Note that y, the fraction of ions which are not paired, enters quite naturally a t the very beginning of the calculation. We now assume that pairs are formed by the diffusion of ions to distances such that their cospheres overlap, and dissociate when they diffuse at the steady state, the equilibrium between free ions and ion pairs is given by the law of mass action
1- y
(1.9)
= KAcy2f?
where c is stoichiometric concentration (no = Nc/1000) and f is the activity coefficient of the free ions. Integration of (1.7) gives the familiar +j
= AemKr/Y+ BeKr/r
(1.lo)
where B must be zero to make the field vanish for r = m. The constant A is evaluated by substituting (1.10) into (1.4) to express p J in terms of and the result into the integral (1.3), giving
+,
AD
Lrn
K2re-K'
whence
A = e,eKR/D(l
+
d.; = e j
K R ) = e,/Dp
(1.11) (1.12)
Inside the cospheres of free ions, where by definition no other ions are present, the screening factor is e-KR,and the potential is e,/Dr(l KR).The potential
+
io= j e,e-Kr/li.DY
1. R
Y
(1.13)
is used in the Debye charging process in the usual way to get the electrostatic term in the free energy; differentiation with respect to concentration then gives
-1nf
+
= e2~/2DkT(1
KR)
(1.14)
To simplify this (and later) equations, the following symbols are defined $ = e2/DkT,
T
= /3~/2,
t = KR
(1.15)
which convert (1.14) into
+
-In f = ~ / ( l t )
(1.16)
In order to understand the dependence of the association constant on molecular parameters, it is convenient to begin by reviewing the pair correlation functionlo for the primitive model. Define a given cation as paired with the nearest anion provided there is no other unpaired anion nearer t o t h a t cation. For example, cation 1 (Figure 1) is paired with anion 4, and not with 2 or 6; while r12 < r14, anion 2 is already paired with cation 3 and therefore by definition cannot be the partner of cation 1, and rI4 < r16 excludes anion 6. In this way, unique partners can in principle be assigned to every cation; the probability G ( r ) that a given cation is so paired with an anion at a distance r islo
Figure 1. Method of pairing ions (two-dimensional example).
G ( y ) = 4anv2 exp[p//r
- 47in
s,'
x2 eXp(9/x) dx] (1.17)
For b = P/a > 2 , this function has an exponential peak at r = a, a minimum near r = p/2 (the Bjerrum radius), and a maximum at a distance of the order of n-1/3,after which it rapidly decreases to zero. The association constant in the usual units (l./equiv) is given by the integral d
$ exp(p/y) dv
K A = (4~iV/1000)
(1.18)
a
Ions whose pairwise separation distance lies between a and d are defined as nonconducting ion pairs. For solvents of low dielectric constant, K A becomes quite insensitive to the value of d. For d = p/2, eq 1.18 gives the Bjerrum constant. In contrast to G ( r ) ,the correlation function for charged spheres in a continuum, the probability curve P ( r ) for real ions in real solvents must have several maxima and minima for small values of r, regardless of dielectric constant. There is a nonzero probability for contact pairs ( r = a ) , even for uncharged particles; for ion pairs, this probability is enhanced by the Boltzmann factor exp ( - U / k T ) where U is the sum of the electrostatic term e2/aD and all other short-range interaction energies u ( r ) . Then the probability must drop to nearly zero for a < r < a d,, where d , is the diameter of the sphere whose volume equals that available to a solvent molecule (actual volume plus free volume); a second maximum, corresponding to the solvent-separated d,, followed by another drop, pair, will appear at r = a 2d,. The discrete strucand a third maximum at r = a ture of the solvent will be felt in the probability curve up to distances R Ia m d s , where m is probably between 2 and 4; for r > R, P ( r ) will approach the smooth curve G ( r ) .It is in the range a < r < R that the intense electrostatic field of the central ion will polarize and orient the neighboring solvent molecules; we thus identify the sphere of radius R with the Gurney cosphere, and are led to choose R as the pairing distance. The following stochastic form is proposed for P ( r ) :
+
+
+
+
P(r) =
where the weighting factors the normalization condition
Wk
and 6' are chosen to satisfy
~ * P ( Ydv) = 1
(1.21)
and the functions f h [ ( r - a - kd,)/fl] are functions which peak strongly at r =: a kd,, for example, gaussians or
+
The Journal of Physical Chemistry, Vol. 79, No. 5, 1975
Raymond M. Fuoss
528
Dirac 6 functions. The corresponding association constant is given by the integral K , = ( N / l O O O ) l R P ( v )dv
(1.22)
Since by definition u ( r ) sums the potentials of short-range forces, exp(-u/kT) must very rapidly approach unity.for r > a, so u ( r ) may be replaced by u ( a ) in the first term of the sum in (1.19) and by zero in the others. If 6 functions are chosen for the peaking functions, we finally obtain for the association constant
where h is chosen so that a
+
kd, 5 R
(1.24)
where R is the third parameter of the conductance function. This formulation provides a natural place for the insertion of system-specific parameters into KA. For solvents of low dielectric constant, the first term in braces in (,1.23) above dominates the integrand; here the asymptotic expansion
K,
= (4aNa3/1000)(eb/b) exp[ - u ( a ) / k T ]
(1.25)
is probably a good approximation. Alternatively, since exp(/3l r ) drops very sharply from its value eb a t r = a when b is large ( D small), it may be used as the peaking function, giving K , = (4nS/lOOO) exp[u(a)/KT]
JR?2
exp(p/r) dv
a
= 8 1 6 . 1 ~ ' / ~ ~ ~ at / ~25" / 0 ~(1.27) /~
t = KR = 2i-/(P/R)
(1.28)
To limit concentrations to less than about 2 X lO-'O3, T must therefore not exceed 0.4; since R can be as large as /3 in practical examples, t must not exceed 0.8. For solvents of higher dielectric constant, where ion pairs are defined by the distribution function P ( r ) , values of R = 2a have been found. Since the Debye averaging is assumed to be valid for r > R, the distance 1 / must ~ exceed R; therefore t must be less than unity. The range of applicability of the final conThe Journal of Physical Chemistry, Vol. 79, No. 5, 1975
2. The Relaxation Field
The conductance function is
A
= y [ R o ( l - A X / X ) - Ah,]
(2.1)
where AX is the relaxation field generated by the asymmetry in the space charge surrounding an ion moving in an external field X,and AA, is the decrease in equivalent conductance produced by the electrophoretic countercurrent. The latter will be derived in section 3; here, we treat the asymmetry effect. The theory parallels the 1957 development12J3 with four significant differences: (1)ion association is assumed ab initio; (2) new boundary conditions are used in evaluating the constants of integration; (3) the higher terms are kept in explicit form, and (4) a number of previously neglected higher terms are included. In order to conserve space, we begin with a condensed review (which omits all mathematical detail) of the derivation of the basic differential equations, which we then proceed to integrate. The goal is the calculation of the relaxation field, which is the component in the field direction of the negative gradient of +'j, the asymmetric part of the total potential +j
=
$Or
+
(2 2)
$1,
The latter satisfies the Poisson equation A i j = - (47i/D?zj)Cieifji
(2.3)
f .3 .1 = ?zjnji= ninij = f i j
(2.4)
where
(1.26)
The integral above becomes relatively insensitive to the value of the upper limit R when the dielectric constant is less than about 20. In this range, where association constants are large ( K A > 100) a fairly wide range of R values near the Bjerrum radius R = p/2 will give almost equally good match between calculated and observed conductances; in effect, the three-parameter equation A (c; Ao, KA, R ) reduces to a two-parameter equation, A (c; Ao, KA). In these cases /3/2 may replace R in the integral and in the conductance function which becomes A (c; '40, KA,p/2). For data in solvents of high dielectric constant, all three parameters are unambiguously determined by the data. The distinction between free and associated ions based on G(r) for the primitive model fails when the concentration exceeds 3.2 X 10-7D3, because the minimum in G ( r ) disappearsll. This sets an upper limit to the concentration range to which we may apply the conductance function which will be derived in section 4. Rather than concentration, the dimensionless quantities T and t will be used as independent variables T = 3K/2
ductance function will therefore be restricted to the concentration range for which t < 0.8 for all dielectric constants.
+flji
= fOji
(2.5)
is the distribution function which specifies local concentrations around a reference ion. The functions f';; and +;', proportional to the field and to the cosine of the angle between the vector r;; and the field direction, are the perturbations in the functions P;i and q0; which characterize the system in the absence of an external field. The functions (2.5) satisfy the equation of continuity div, (fijvij)
+
div, ( f j j i v j = i) 0
(2 8)
where vji is the velocity of an i ion, given a j ion at the origin. The total velocity is the sum of the velocity vi of the medium at the location of the i ion plus the velocity produced by the forces acting on the i ion
vji = vi
+
wi(Kji - KTV, ln,fji)
(2.7)
where wi is the reciprocal friction coefficient of the i ion and Kj; the vector sum of the electrical forces
K.. 3 2 = Xeii -
eiV,4j(r,i) - e,Vz#'i(a)
(2.8)
Combination of eq 2.2-2.8 gives a second-order differential equation for f'ji which may be written symbolically in the form Af
Iz1
-
~2ezf',,/2 = -{n2xeb e - K T ( l+ r2pkT}
+
KY)
cos
e/
@(KY; + ' j , f ' j i )
(2.9)
where
& = e , i o j / k t = ej+Oi/kT
(2 .lo)
and summarizes a number of inhomogeneous terms involving the as yet unknown functions +'j and f'zl. (Up to
Conductance-Concentration Function
529
this point, the details of the derivation are as before; eq 2.9 is a rearranged form of eq 10 of ref 12 and of eq 1.8 of ref 13. It is suggested that the reader review these references up to the cited equations.) Next, expansion of the Boltzmann factor eE on the left
+
eF w 1
+
5
t2/2
(2.11)
and moving the second and third terms to the right gives Af‘21 - X2f’,,/2 = -{2.9)
+
K 2 ( ( 1 4- 5/2)f‘21/2
+
d? (2.12)
2a. Leading Term i n the Relaxation Field. The first approximation F2l to f‘21 is obtained by solving the equation AF,,
- K2FZi/2
= -{2.9}
(2.13)
= 0;
lim F2,(K~)/n2rt
w
KY-
0
lim (Q,/n2) #
- zh‘j)R = 0
(2.15)
The argument leading to (2.15) is exactly the same as before, with a replaced by R; the argument is valid because, by definition, there are no unpaired ions within the sphere of radius R around the reference ion with charge ej, and therefore the potential must satisfy the Laplace equation for r 5 R. Then the first approximations F21 and POcos 0 for f’21 and are substituted for the latter in the higher order inhomogeneous terms of (2.12). This gives
v
AG21 - K2G2,/2 = C k T k
(2.16)
CQk
(2.17)
and the T k ’ s are now explicit functions. Integration of (2.16) gives the corresponding terms Qk of Gzl; substitution of these in the Poisson equation and integration of the latter then gives the higher terms in the relaxation field. The leading term in the relaxation field will be calculated first. Explicitly, the differential equation (2.13) which determines the leading term of the distribution function reads AF,, - K ’ F , ~ = / ~-n2Xep e-Kr(l + KY) cos 0/y2pkT (2.18) where n = Ncy/lOOO (2.19) is the number of unpaired ions per unit volume. Writing QO(V)
COS 0
(2.20)
eliminates cos 0 and reduces the partial differential equation (2.18) to an ordinary differential equation
&1[S0(41= -n2Xep e-Kr(l + where L1 is the operator
-
0
(2.24)
+
K Y ) / T ~ ~ ~(2.21) T
L, = d2/dV2 + (2/Y) d/dr - ( 2 / ~ ’ + K2/2) (2.22) The complete solution of (2.21) is of course a particular in-
qKY)] (2.25)
with q2 = 112. Series expansion of the exponentials in (2.25) shows that (2.24) requires that A0 = -1. Substituting = 8anp = 8nne2/DkT
K’
(2.26)
and using A0 = -1 gives Q(Y)
= - ( n X e / 4 ~ p k T ~ ~ ) [ e ‘ ~+~KY) (l -
e-aKr(l4- qKY)] (2.27) for the first approximation to the perturbation YIzof the distribution function. Designating the corresponding potential by Po(r) cos 0, where
$‘ = P~(Y) COS
e
+ CP,(Y)
COS
e
(2.28)
v
(Po(r)cos 0 is the leading term of = v2 = and the p k ’ s are the radial parts of the higher terms in the asymmetry potential) the Poisson equation A$’ =
F,, =
n
A, e-gKr(l+
where G2, = f ‘21 - F,, =
00
Explicitly (2.22), with Bo = 0, is
(2.14)
The result is substituted in the Poisson equation (2.3), where the terms in lrpl and f 0 2 1 of course cancel; this is then integrated, to give the first approximation to the potential, subject to the boundary conditions that the field must vanish for r = a, and that field and potential be continuous a t r = R. This replaces one of the previous boundary conditions by
(Ya$’Jav
Qo = QO(part.)+ AoQo,(hom.) f B0QO2(hom.)(2.23) where the coefficient A. and Bo are determined by the boundary conditions. Q 0 2 becomes exponentially infinite for r = m; since Qo must reduce to zero for r = m, Bo must be zero. Recalling that the distribution function flz is proportional to n2,and that QOcos 0 is an additive part of f,L, it is clear that the limit of the ratio flLln2a t zero concentration must remain finite. The second boundary condition therefore is
Qo = -(2n2Xep/pkT~2+)[e-Kr(1+ KY)
subject to the boundary conditions F,,(w)
tegral plus a linear combination of the solutions of the homogeneous equation
-(4~/Dn,)C:,e~f’~~
(2.29)
leads to the differential equation which determines PO L~[P~(= Y ) (px/pLy2)[e*Kr(1 ] +
KY)
-
e-lrKr(1+ qKY)] (2.30) where
L? = d2/dV2
+
(Z/Y) d/dv - 2/?
(2.31)
Solutions of the homogeneous equation are Pol = r, Po2 = l/r2. Setting the coefficient of Pol = 0 and combining the other solution with the particular integral of (2.30) gives
Po(Y) = (px/pK2./2)[e-Kr(l + KY) 2e-lrKr(1 + qKY)
& f ~ ](2.32)
Note that the charges el and e2 appear only as their product in p; therefore = = .’)I The boundary condition (2.15) is equivalent to
vl
[d(P/~)/dv]R = 0
(2.33)
which evaluates Mo M , = 2emPt(l+ qt = 1
+
O(t4)
+
t2/6) e m t ( l+ t
+
t2/3) (2.34) (2.35)
For r = R ( K R = t ) , the quantity in brackets in (2.32) becomes The Journal of Physical Chemistry, Vol. 79, No. 5, 1975
530
Raymond M. Fuoss
[2.32] = e - t ( l
+ t ) - 2e-Qt(l +
qt)
+ M,
(2.36)
Expansion of the exponentials and substitution of Mo from (2.34) leaves terms of order t3; division by the K~ in the coefficient produces a leading term of order K . The latter gives the c1'2 term in the limiting tangent; the higher terms in AX, come from terms of order K 4 and higher from MO and the exponentials. (This comment on order of terms was made in order to point out the origin of the relaxation term in the limiting tangent; in the final conductance equation, the exponentials in [2.32] are kept explicit.) The relaxation field is AX = -grad,i'(a) = -((ad'/ax), (2.37) but since the asymmetry potential is constant and equal to P ( R ) for r 5 R (recall that there are no other unpaired charges inside the sphere of radius R around the central cations) (2.38) The potential is of the form P ( r ) cos 0 = xP(r)/r;use of the boundary condition (2.33) shows that AX =
-(a$'/aX),
AX = - P ( R ) / R because
(2.39)
+
k T ( q + w2)Laf'zl(r)+ f O z l ( e i ~ l A I C l ' z- e z o 2 A $ f l )= X(eiwi - ezwz)(?fozl/ax)- ( e l w i v i ' z e2%v$!"l)*vfo2i
AX,/X = -2r[exp(l - q)t - 1]/3t(l where t = KR, T = pK/2
+ t)
(2.41) (2.42)
In the limit of very low concentrations AXo/X
+ 4)
-~/3(1
= c!ci'2
(2.43)
the classical result. In Appendix A, it will be shown that the relaxation field can be obtained directly from the inhomogeneous term on the right in (2.30) by use of the boundary conditions; that is, it is not necessary to solve (2.30) for POin order to obtain AX,. Integration of (2.30) is easy, but the corresponding differential equations for the higher terms of AX lead to very long and complicated functions which can be bypassed by the theorem developed in the Appendix; its use also eliminates the very tedious chore of evaluating the constants corresponding to MO above. It will be applied here to the evaluation of AX0 as a check on (2.41) and to demonstrate its use. The theorem states that AX = -{~[P(Y)
= (K/3)
COS
Jm
R
e]/a~]},
(P (4 d r
(2.44) (2.45)
if
L,[P(Y)l = K Y ( 4
(2 -46)
subject to the boundary conditions [dP(~)/dr], = 0,
[d(P/Y)/dr]~= 0
(2.47)
For AXo, K = PX/K and p(r) is the function in brackets in (2.30) over r2;therefore (px/3p)
i*
[e-Kt(l
+
- f'2~(eiwlA$02+ e 2 w z A 4 J o l )-
( e l ~ l v + O+ze2w2v$0i)*vf '21 + [eiwIV$',(R)
= eje-"'/FDv = njni exp(-eiioI/kT) = n2 exp(pe-"'/pv) $Oj
fojr
~ K Y )dY/$ ]
LHS = k T ( q +
- ( ~ ~ e * / 2 ) f ' (2.54) ~~]
+ +
The truncated series (et = 1 [ 12/2) is substituted in (2.54) and the part of the left-hand side corresponding to ([ [*/2) is moved to the right. (This approximation is justified by the definition of unpaired ions, for which the distance r to the nearest other unpaired ion cannot be less than R . ) After the above manipulations and use of the PoissonBoltzmann equation in the third term on the right in (2.50), division by kT(w1 w2) gives
+
+
Af'21 - (K2/2)f'21 = ( x e / ~ T ) ( a f 0 2 l / W+
(W/2)(1
+ t/2)frZl - [ k ~ ( q+ ~ 2 ) l - i { ( e l w l ~ +-' 2
avpzl +
(elw,1°2 + e z ~ 2 1 ° 1 ) K 2 f f 2 1 + ( ~ I ~ I V ~+C e' 2~q V 4 0 i )*Vf'2, - [ e I q V $ ' I ( R ) (2.55) e2w2V$'z(R)I Vfo2i + (Vi - ~ 2 )Vf '21) (The convention for reading the above equation and others which are longer than one column width is similar to that for reading Fortran; (- [. (. ). 1. .)will, however, be used instead of nesting parentheses.) Differentiation of f o 2 1 gives ezwzVV1)
-
= n2(1
+
= ( p X / 3 ~ R ) ( e -~ e-qt) The Journal of Physical Chemistry, Vol. 79, No. 5, 1975
-- -
R, the velocity derived from (UA u,) correctly describes the velocity field, by hypothesis. For r < R, system-specific parameters must be used to describe the motion; obviously, no general theory can be developed for this region. But we may try continuum theory for r < R to estimate the eventual contribution of u, to AXp Substitution of (3.16) in (3.5) gives
+
+
+
VV, =
-
( ~ e / 4 ~ ) [ i ( - I / 2 ~E / + ) r1(1/2y
+
3E/y3) cos 01 (3.17)
The second term in (3.17) is the radial component; for it to vanish at r = a/2, E must satisfy the condition E = -a2/24
(3.18)
Since the Walden product is not constant, we know that the Navier-Stokes equation does not describe the flow pattern inside r < R exactly; on the other hand, the product is approximately constant so we may at least use (3.17) with (3.18) to estimate the velocity v, in the sum v = VA ve Substituting (3.9) into (3.5). seDaration into comDonents, and addition of (3.17) gives
+
v,
= ( x e cos 0/27117K2Y3)[~ - (e-"/p)(I K'
+
KY) -
a2/24 ] (3.19)
valid for a I r I m for the primitive model. Using C from (3.14) and p = e-KR(l KR)
+
0,.
= ( x e cos Q / ~ T V K ~ Y ~ )-[ C(e-K'/p)(l Y
+
KY)]
>
R
(3.21)
The differential equation for G, (3.14)
For u,, we have AU, = -Xei/4nr
+
(3.19) = (K2R2/2)(1 - a2/12R2) O(K3R3) (3.20) Values of R from data on a variety of systems were found to be at least 6 A and usually larger; if we assume Rla 1 1.5, the E term of (3.17) contributes less than 4% to (3.20); in the usual range of concentrations to which the final equations apply, AX,IX is only about l%of the total relaxation term. Now recall that AX, is to be calculated as an integral over the range R Ir Ia;therefore values of u, for r < R will never be needed. Test calculations showed that complete omission of the short range r-3 term of (3.17) has a negligible effect on the values of the other parameters derived from experimental data. Since it is therefore futile to try to determine a as a fourth parameter from conductance data, we shall drop this term at this point and use simply
(A
- K2/'2)G,
= (vir
- u2r)(dfozl/d4/kT(wi + q )
(3.22) has the last term of (2.55) as the inhomogeneous function on the right. After substituting (3.21) for the radial velocity components, the coefficient can be considerably simplified. n2 is an adequate approxiFirst, since AX,/X 0 on both AXIX and AAe/Ao therefore is to give a curve which lies above the limiting tangent, whose slope is determined by boundary conditions for point charges. 0
+
+'
4. The Conductance Equation The conductance-concentration function (2.1) for associated symmetrical electrolytes can now be assembled, using the functions of t(c, DT, p/R) = KR derived in sections 2 and 3 for the elecrostatic and hydrodynamic terms: A = y[A,(l + RX) + HY] (4.1) where RX is given by (2.117) and HY by (3.47). (The negative signs before A X l X and AAe in (2.1) are included in the definitions of RX and HY.) The limiting slope on a Kohlis rausch plot (A/& against S/A, =
@
+ &/A0
(4.2)
+ Po/ho
(4.3)
= 816.1ci/2yi/2/D3/2 = pt/2R
(4.4)
= ~ / 3 ( 1+ q ) ~ " '
where, for 25O T
and
Pocl'2/A, + 5.854 X 10'3~D/A017 (4.5) The evolution of the complete A/& - t function is shown in Figure 5 for a system for which the following numerical values were chosen: A07 = 1, D = 100, T = 298.16, R = p/2 = 560.4 X 10-81D25, so that
Figure 5. Composition of A(c)/Ao.
(a + P ~ / A ~ ) 5C ~(0.3 ' ~ + 1.0)t/3 (4.6) Note that the electrophoretic term controls about 75% of the initial decrease of A wjth increasing concentration for this example. For calculating y by (1.9), the numerical value K A = 0.5 was used. The straight line is the limiting tangent; for curve 1
y = 0.3AXo/X - /30c*'2/Ao(l + t)
(4.7)
It shows that most of the upward curvature of A - c1l2 plots for salts in solvents of high dielectric constant can be accounted for by the leading terms of A X / X and A&, (2.41) and (3.32), up to concentrations corresponding to t FS: 0.2. At higher concentrations, the higher terms in c, c3l2, c2, . , . ,whose net sum is positive (see Figures 2 and 4)pull the curve still further upward from the limiting tangent; curve 2 shows ?, =
0.3RX
+ HY/A,
(4.8)
where y now includes all of the relaxation and electrophoresis terms. The final conductance curve 3 is obtained by multiplying the ordinates of curve 2 by y,which is unity a t zero concentration and decreases (at first, linearly with c) as concentration increases. Note that curve 3 runs only a little higher than curve 1. If curve 3 were an observed conductance curve, the association constant (here 0.5) would be determined by the difference between (y1- y z ) and ( y l - y3). If some of the positive higher terms had been omitted (or missed) in the theoretical derivation, curve 2 would lie correspondingly lower, and the value of K A derived from The Journal of Physical Chemistry, Vol. 79. No. 5, 1975
538
Raymond M. Fuoss
the “data” of curve 3 would be less than 0.5. Omission or neglect of some positive higher terms in previous (now obsolete) conductance f u n c t i o n ~ ~ ~isJ ~therefore ,~* the presumptive source of the absurd negative values of K A which the computer occasionally derived from data for slightly associated salts, using the earlier equations. For larger values of K A (greater than 10 or so), y decreases so rapidly with increasing concentration that A(c) drops below the limiting tangent and becomes concave down on a A - c1l2 plot in the working range of concentration (c,,, < 2 X 10-7D3). For still larger association constants, the curve becomes sigmoid; often what is then observed in the usual concentration range is only the inflexion region of the S curve a t the low concentration end, followed by the concave-up region as concentration increases. The latter curves (since sigmoids simulate linearity in the inflection region) appear to be linear in c1l2 at low concentrations, with slopes much larger than the true limiting slope (4.2). Given a set of conductance data (c,, A,; j = 1, . . . , n), spanning a concentration range of at least a decade, the three parameters are obtained by solving the three equations A = y[Ao(l - AX/X) - AA,] (4.9) where y = 1 - K,Cy2f2 (4 .lo) and Inf = - p ~ / 2 ( 1 KR) (4.11) for the unknowns Ao, KA, and R. Since the equations are nonlinear in two of these variables, they must be evaluated by the method of successive approximations. The conductance function is written in the form
+
A j = A[cj; KA
+
A, t (aAj/aAA,) AAO, R (aAj/8AR) AR]
( 8 h j / 8 A K )AK,
+
(4.12)
where Ao, KA, and R are estimated preliminary values of the parameters and the added quantities are the first terms of the Taylor series for the expansions of Ao, KA, and R in terms of correcting increments AAo, AKA, and AR. Then the data set (c], Aj) is analyzed to find the values of the parameters which minimize
..
C2 = C[A,(obsd) - Aj(calcd)12 j = 1,. , n (4.13) by setting the partial derivatives of Z2 with respect to AAo, AK, and AR equal to zero and solving the resulting three equations, which are linear in AAo, AK, and AR.The process is iterated until the mth increments satisfy the specifiA K A< ~ 0.02K~,and AR,,,< 0.02R. cations AAom < The initial value of A0 is obtained by freehand extrapolation of a A - c1/2 plot or estimated by Walden’s rule. The starting value for K A is calculated from the data a t the ~~ highest concentration c1 by the F u o ~ s - K r a u sapproximation K A = (l - YO)/c,YOzf02 (4.14) where yo e A , / ( R o - Sc1”2h,i/2/A0f’2)
(4.15)
and of,
exp(-0.42016
x 1 0 7 ~ 1 i / 2 D - 3 / 2 T - 3 /(4.16) 2)
The preliminary value of R is set at R = /3 = e2/DkT for solvents of higher dielectric constant ( D > 40) and at R = PI2 for the lower range of dielectric constants. y is obtained by solving the quadratic (4.10) which has The Journal of Physical Chemistry, Vol. 79, No. 5, 1975
the root y = [(I
+ 4KAcf2)1’2 -
1]/2KAcf2
(4.17) but, since f in turn depends on y implicitly through the transcendental (4.11), the solution must also be obtained by successive approximations (implicitly, because f = f( K), and K2 = ~ N p ~ y / 1 2 5 ) (4.18) The calculation starts by setting y = 1.0 and using this value to calculate f i n zeroth approximation. This value o f f is used in (4.17) to obtain y1,the first approximation to y, and then y1 is used again in (4.11) to get the next approximation for f. The cycle is repeated until two successive values of y differ by less than 5 X 10-5. Only one of the partial derivatives needed to construct the linear equations for (AAo, AK, AR)can be obtained explicitly; it is
a A j / a A h o = ~ (- lAX/X)
(4.19)
The other two are approximated by chord slopes a K ~ / a A Ke~ [ A j ( l . o a K ~ )- Aj(KA)]/0.02KA (4.20) and similarly for aRlaAR. The above operations are of course carried out by computer. For data on solutions of high dielectric constant, the method just described usually converges within three or four cycles. The process involves finding the minimum standard deviation (7 = c / ( n - 3)1/2 (4.2 1) in the ( u ; A&, AK, AR) space. Sometimes (especially for data for solvents of lower dielectric constant), the minimum in the 4 space is quite shallow. For such data, the process fails to converge within 10 cycles; successive increments in K A and R are nearly equal, and of opposite sign, and this futile see-saw hunt continues indefinitely. To handle such data, the values of A0 and K A which minimize
D =
c / ( n - 2)112
(4.22)
are found by solving the set A, = A(cj; Ao, KA,R ) for a sequence of values of R, chosen first to cover the range 1.0 < PIR < 2.5 a t intervals of 0.25 in the ratio PIR. Then u is plotted against PIR in order to locate the approximate value of R which minimizes u. Next, a finer scan (at intervals of 0.1) is made in the vicinity of the minimum, and finally A0 and K A are found for the value of P/R corresponding to minimum u. When K A is small (less than one), it sometimes happens that a calculated increment AKA is negative and larger numerically than the current value of KA, which of course makes the approximation for K A in the next cycle negative; the process then diverges. For such cases, the data are analyzed to find the values of A0 and R which minimize c for the sequence K A = 0.0, 0.1, , . . . 1.0. Then a plot of u against K A locates the minimizing value of Ka and fixes the corresponding values of the other two parameters. The conductance equation (4.1) reproduces experimental data well within the estimated error; typical examples of its application are given in Table I, for potassium nitrate in dioxane-water mixtures.30 The values of the parameters for D = 60.53, 35.13, and 14.56 are given for each set of data; 6A is the difference (in A units) between observed conductance a t concentration c and the value calculated using the values of the parameters listed, which minimize the standard deviation. The plots at the left in Figure 6 show how u
Conductance-Concentration Function
539
TABLE I: Conductance of Potassium Nitrate i n Dioxane-Water Mixtures at 25" ~~
~~~
D = 14.56,
D = 35.13, KA = 12.0
D = 60.53; KA = 1.9, 1 0 4 ~
A
6A
10%
382.72 333.37 288.55 238.42 198.44 158.62 119.59 79.94 48.50 34.11
84.98, 85.68, 88.38, 87.27, 88.091 89:02, 90.10, 91.47, 92.89, 93.73c 98.40, R = 8.73 X lo-'
-0.001 -0.004 0.004 0.003 0.001
86.912 69.881 64.030 55.461 37.143 31.890 27.042 18.191 10.042
A
10,~
6A
KA =
782
A
6A
30.31, 30.96, 32.01, 33.42, 34.98, 37.99, 45.57, R = 16.73 X IO-*
0.000 0.000 0.000 0.001 -0.002 0.001 0.002
~~
0.00
(7.5
15
-0.001 -0 .ooo
-0.005 0.001 0.001 0.002
0 .ooo
/3/3 2.5
9.838, 8.782, 7.303, 5.660, 4.217, 2.220,
0.011 -0.002
52.72, 53.53, 53.85, 34.34, 55.63, 56.07, 56.53, 57.52, 58.76, 62.19, R = 9.97 X
-0.007
-0.009 -0.005
0 .ooo
0 .ooo
0.006 0.014 -0.009 0.008
the corresponding increase in the distance over which the central ionic charge influences the surrounding solvent. As mentioned in section 1, real systems approach the primitive model in behavior as the dielectric constant decreases, due to the dominating exp(e2/aDkT) in the distribution function. For the lower range of dielectric constants, a pretty good fit to the data can in fact be attained by setting R = P/2, which amounts to using a two-parameter equation A(cA0, KA). Nevertheless, the fit is better (as might be expected) if R is retained as a disposable parameter. It should be pointed out that as R is varied across the range corresponding to [u(min) 0.01%], the accompanying values of A0 and K A which minimize u also change, the former very little, the latter by percentages which decrease as D decreases. Table I1 illustrates these changes. The explanation for the relative insensitivity of association constant to R at low dielectric constants is found in the mass action equation (1.9) where the product f y appears: if R is increased, we are counting more ions as paired, thereby decreasing y at a given concentration. But more pairs mean fewer free ions in the space charge and therefore a larger activity coefficient. One closing comment on the conductance function: it may not be applied to data for solvents of dielectric constant lower than about 10, even if c (max) is less than 2 x because interionic Coulomb forces become strong enough to produce overlap of three3I or more32 cospheres. Effects of association higher than pairwise are not included in (4.1).
+
Figure 6. Plots from data30 on potassium nitrate in dioxane-water mixtures at 25'. Dependence of u on P/Rat left; top to bottom, D = 14.56,35.13 and 60.53.Dependence of log KA on E ' ,0 ' s at right: of Ron E ' ,vertical bars at right.
(expressed as percentage of ho)varies as the parameter R is changed. For water-rich mixtures (ie., at high dielectric constants), the a - PIR plot shows a very sharp minimum. Here, the three parameters could be obtained simultaneously; the calculation for a range of R values was made to test the sensitivity of the fit-to-data to the value of the parameter. The double arrow shows the range of R values corresponding to u I [cr(min) 0.010%]; for D = 60.53, R = 8.73 X cm; o(min) = 0.003%. As the figure shows, the best-fit R could be varied by about f 5 % and the equation would still duplicate the data within f0.013%. As the dielectric constant decreases, however, the minima in the u P/R curves become progressively more shallow, as shown in the figure; that is, a progressively wider range of R values could be tolerated for a fit to u < [a(min) 0.01%]. For this system and for a number of others which have ben analyzed, R is 6-8 8, for a given salt in water; as the dielectric constant decreases, R increases at a decreasing rate, as shown by the curves on the right in Figure 6 bracketing the bars. The length of the bars corresponds to the [a(min) 0.01%] range of R values derived from the u - p/R plots. For D less than about 30, R is not greatly different from the Bjerrum radius PI2 (where P/R = 2), and as the figure shows, the tolerance range of R is quite broad. The increase of R, the diameter of the cosphere, with decreasing dielectric constant is of course the consequence of
+
+
+
Acknowledgment. This work was supported by Grant No. 14-01-0001-1308 from the Office of Saline Water. Appendix A Given:
+
( Z / Y ) dZ/dY - 22/$ = V(Y) (A1 Find the solution which satisfies the boundary conditions d2z/d$
z(m) = 0 ; [d(z/~)/dv], = 0 and evaluate P(R)IR, where P ( r ) = Kz(r). The solutions of the homogeneous equation are 21
= Y,
22
= l/y2
(A2
(A31
with the Wronskian W ( Y ) = -3/$ The complete solution is Z ( Y ) = Z, + Az, + B z ~
(A4) (-45)
The Journal of Physical Chemistry, Vol. 79, No. 5, 1975
Raymond M. Fuosa
540
TABLE 11: Interdependence of Parameters
D = 60.53 98.264 98.356 98.434 98.499
0.90 1.oo 1.10 1.20
0.039 0.014 0.008 0.026
1.40 1.50 1.60 1.70
0.026 0.016 0.015 0.021
= 35.13 62.123 62.161 62.193 62.221
2.10 2.20 2.30 2.40
0.012 0.006 0.003 0.006
D = 14.56 45.548 45.567 45.579 45.586
the relaxation field due to a potential Per) cos 0 is simply one third of the integral of the inhomogeneous function-on the right in (A12) over the range R I r 5 a. There is no need to solve the equation in order to obtain AX.
2.14 2.01 1.88 1.73
10.29 9.26 8.42 7.72
13.32 12.67 11.99 11.27
11.39 10.63 9.97 9.38
805 794 782 768
18.33 17.49 16.73 16.04
D
Supplementary Material Available. The derivation of eq 2.87-2.110, the terms of the relaxation field, Appendix B, the table of particular integrals of L l [ y ( r ) ]= cp(r),and Appendix C, the table of definite integrals used in the evaluation of the higher terms in relaxation field and electrophoresis, will follow these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementary material from this paper only or microfiche (105 X 148 mm, 24X reduction, negatives) may be obtained from the Journals Department, American Chemical Society, 1155 16th St., N.W., Washington, D.C. 20036. Remit check or money order for $3.00 for photocopy or $2.00 for microfiche, referring to code number JPC-75- 52.5. References a n d Notes
where z p is the particular integral
zp = (v/3)lr
dv - ( l / 3 v 2 ) J r ? s ~ dv
(A6)
The first boundary condition requires that A = 0. Then
+ zp]
P ( Y ) = K[B/v2
(-47)
Differentiating Plr with respect to r and setting r = R evaluates B: B = -(1/3) J s v 3 ~ ( dr r)
(-48)
R
Then
+
P ( Y ) = H-(1/3v2) Jm?'o d r R
(.i/3)
Ir
@
dv -
( 1 / 3 ~ J~ ') r r 3dv] ~ (A9) Setting r = R gives P ( R ) / R = - (K/3)
*
u (v)d?'
R
According to (2.39), this leads to the result AX = (K/3)
Jec.(r)dv R
(All)
To summarize: given the radial part of the Poisson equation
The Journal of Physical Chemistry, Vol. 79, No. 5, 1975
(1) A. D'Aprano and R. M. Fuoss, J. Solution Chem., 3, 45 (1974). (2)C. J. James and R. M. Fuoss, J. Solution Chem., 4,91 (1975). (3)A. D'Aprano and R. M. Fuoss, J. Solution Chem., 4, 175 (1975). (4) R. W. Gurney, ionic Processes in Solution," Dover Publications, New York, N.Y., 1953. (5) H. S.Frank, "Chemical Physics of Ionic Solutions," B. E. Conway and R. G. Barrades, Ed., Wiley, New York, N.Y., 1966. (6)P. S.Ramanathan and H. L. Friedman, J. Chem. Phys., 54,1086 (1971). (7)Review of J. C. Rasaiah, J. Solution Chem., 2,301 (1973). (8)M. Eigen, Z.Phys. Chem. (FrankfuftamMain), 1, 176 (1954). (9)P. Debye, Trans. Nectrochem. Soc., 82,265 (1942). (IO)R. M. Fuoss, Trans. Faraday SOC.,30,967 (1934). (11) R. M. Fuoss, J. Amer. Chem. SOC.,57, 2604 (1935). (12)R. M. Fuoss and L. Onsager, Proc. Nat. Acad. Sci. U S . , 41, 274, 1010
(1955). (13)R . M. Fuoss and L. Onsager, J. Phys. Chem., 61, 668 (1957). (14)Reference 9,eq 3.3. (15) R. M. Fuoss and K. L. Hsia, Proc. Nat. Acad. Sci. U S . , 57, 1550; 56, 1818 (1967). (16)R. M. Fuoss, "Computer Programs for Chemistry," Voi. 5, K. B. Wiberg, Ed., Academic Press, New York, N.Y., 1975,in press. (17)S. Glasstone, K. J. Laidler, and H. Eyring, "The Theory of Rate Processes," McGraw-Hill, New York, N.Y., 1941,pp 552-561. (18)L. Onsager and R. M. Fuoss, J. Phys. Chem., 36, 2689 (1932). (19)Reference 13,Section 5. (20)R. M. Fuoss, J. Phys. Chem., 63, 633 (1959). (21)Reference 13,eq 5.7-5.9. (22)Reference 13,eq 8.6. (23)P. C. Carman, J. Phys. Chem., 74, 1653 (1970). (24)Reference 13,eq 7.1. (25)E. Pitts, Proc. RoyalSoc., Ser. A, 217, 43 (1953). (26)R. M. Fuoss and L. Onsager, J. Phys. Chem., 67, 628 (1963);eq 7. (27)Reference 26,eq 8. (28)R. M. Fuoss, L. Onsager, and J. F. Skinner, J. Phys. Chem., 69, 2581 (1965). (29)R. M. Fuoss and C. A. Kraus, J. Amer. Chem. Soc., 55, 476 (1933);eq 7. (30)I. D. McKenzie and R. M. Fuoss, J. Phys. Chem., 73, 1501 (1969). (31)R. M. Fuoss and C. A. Kraus, J. Amer. Chem. Soc., 55,2387(1933). (32)R. M. Fuoss and C. A. Kraus, J. Amer. Chem. SOC.,55,3614(1933).
