262
Gerald
The rare earth oxides were purified by ion-exchange methods by Dr. J. E. Powell's group.
Supplementary Material Available. Tables I and 11, listings of the density coefficients and conductance data, will appear immediately following this article in the microfilm edition of the journal. Photocopies of the supplementary material from this paper only or microfiche (105 X 148 mm, 24X reduction, negatives) containing all of the supplementary material for the papers in this issue 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 $4.00 for photocopy or $2.00 for microfiche, referring to code number JPC-75257.
(26) (27)
References and Notes
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(1) This work was performed at the Ames Laboratory of the U. S. Atomic Energy Commission. This paper is based, in part, on the Ph.D. dissertation of J. A. Rard, Iowa State University, Ames, Iowa, Feb 1973. (2) G. R. Choppin, D. E. Henrie, and K. Buijs, Inorg. Chem., 5, 1743 (1966). (3) K. Bukietynska and G. R. Choppin, J. Chem. Phys., 52, 2875 (1970). (4) G. R. Choppin and W. F. Strazik, horg. Chem., 4, 1250 (1965). (5) R. Garnsey and D. W. Ebdon, J. Amer. Chem. Soc., 91,50 (1969). (6) H. B. Silber, N. Scheinin, G. A. Atkinson, and J. J. Grecsek, J. Chem. SOC., Faraday Trans. 1, 68, 1200 (1972). (7) F. H. Spedding and S.Jaffe, J. Amer. Chem. Soc., 76,884 (1954). (8) D. J. Heiser, Ph.D. Thesis, Iowa State University, Ames, Iowa, 1957. (9) I. Abrahamer and Y. Marcus, lnorg. Chem., 6, 2103 (1967). (IO) J. Reuben and D. Fiat, J. Chem. Phys., 51, 4909 (1969). (11) K. Nakamura and K. Kawamura, Bull. Chem. SOC.Jap., 44, 330 (1971). (12) R. E. Hester and R. A. Plane, lnorg. Chem., 3, 769 (1964).
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(21) (22) (23) (24) (25)
(30) (31) (32) (33) (34) (35) (36) (37) (38)
S.Manning
J. Knoeck, Anal. Chem., 41, 2069 (1969). D. L. Nelson and D. E. Irish, J. Chem. Phys., 54, 4479 (1971). Y. Marcus and I. Abrahamer, J. horg. Nucl. Chem., 22, 141 (1961). I. M. Rumanova, G. F. Volidina, and N. V. Belov, Kristallograflya, 9, 624, 642 (1964); translated in Crystallography, 9, 545 (1965). D. F. Peppard, G. W. Mason, and I. Hucher, J. lnorg. Nucl. Chem., 24, 881 (1962). N. A. Coward and R. W. Kiser, J. Phys. Chem., 70, 213 (1966). A. Anagnostopoulos and P. 0. Sakellaridis, J. lnorg. Nucl. Chem., 32, 1740 (1970). F. H. Spedding and S.Jaffe, J. Amer. Chem. SOC., 76, 882 (1954). C. F. Haie and F. H. Spedding, J. Phys. Chem., 76, 2925 (1972). F. H. Speddingand J. A. Rard, J. Phys. Chem., 78, 1435(1974). F. H. Spedding, M. J. Pikal, and B. 0. Ayers, J. Phys. Chem., 70, 2440 (1966). F. H. Spedding, P. F. Cullen, and A. Habenschuss, J. Phys. Chem., 78, 1106 (1974). F. H. Spedding, L. E. Shiers, M. A. Brown, J. L. Baker, L. Gutierrez, and A. Habenschuss, to be submitted to J. Phys. Chem. See paragraph at end of text regarding supplementary material. F. H. Spedding and G. Atkinson in "The Structure of Electrolytic Solutions," W. J. Hamer, Ed., Wiley, New York, N.Y. 1959, Chapter 22. F. H. Spedding, J. A. Rard, and V. W. Saeger, J. Chem. Eng. Data, 19, 373 11974). F. H.'Spedding, L. E. Shiers, and J. A. Rard, accepted for publication by J. Chem. Eng. Data. F. H. Spedding and M. J. Pikal, J. Phys. Chem., 70, 2430 (1966). F. H. Spedding, D. L. Witte, L. E. Shiers, and J. A. Rard, J. Chem. Eng. Data, 19, 369 (1974). F. H. Spedding, L. E. Shiers, and J. A. Rard, accepted for publication by J. Chem. Eng. Data. L. L. Martin and F. H. Spedding, submitted to J. Chem. Phys. L. L. Martin and F. H. Spedding, unpublished results. D. G. Karraker, J. Chem. Educ., 47,424 (1970). J. A. Sylvanovich Jr., and S. K. Madan, J. lnorg. Nucl. Chem., 34, 1675 (1972). F. H. Spedding, L. E. Shiers, and J. A. Rard, unpublished results. The sulfate solubility curve in ref 20 was inadvertently misplotted and all points should be displaced to the next lowest rare earth.
A Limiting Law for the Conductance of the Rod Model of a Salt-Free Polyelectrolyte Solution Gerald S. Manning School of Chemistry, Rutgers university, New Brunswick, New Jersey 08903 (Received July 9, 1974)
The polyion conductance is calculated for the rod model a t low concentration. The electrophoretic effect is taken from Henry's results, while the relaxation effect is computed from the effective field, previously calculated, which retards the motion of the counterions. Taken together with a previous calculation of counterion motion, the present result leads to a complete theoretical equation for the equivalent conductance A of salt-free polyelectrolyte solutions with counterions of arbitrary valence. Since the expression contains the radius of the cylinder which represents the polyion, an alternate form is obtained, which depends only on the concentration and charge density of the cylinder, and which predicts that a certain function of A is linearly dependent on log c a t low concentration.
Introduction The following equation has been used for many y e a r ~ l - ~ to describe the conductance of solutions of a polyelectrolyte salt with no added simple salt: A = J ( X C o + A,) (1) where A is the equivalent conductance of the salt in solution, ACo is the equivalent conductance of the counterion in pure solvent, A, is the equivalent conductance of the poThe Journal of Physical Chemistry, Vol. 79, No. 3, 1975
lyion species in the solution, and f is a parameter which in general is equal to (A, A,)/(A,o A,), since the numerator, where A, is the equivalent conductance of the counterion in solution, is a generally valid expression for A. From standard electrochemical definitions it is easy to see that eq 1 follows from the a s ~ u m p t i o n lthat , ~ the structure of the solution is such that the fraction (1 - f ) of the counterions is bound to the polyion while the fraction f is free (uninfluenced by interactions with the polyion). How-
+
+
Polyion Conductance of a Salt-Free Polyelectrolyte Solution
263
ever, modern polyelectrolyte theory indicates that this description of the structure is certainly incorrect, since even if a fraction of the counterions is bound, the remaining fraction must interact with the uncompensated charge on the polyion and hence cannot be “free.” Equation 1 may also be derived by a phenomenological approach5 which makes no a priori assumption about counterion binding; the phenomenological theory indicates that
where D, is the self-diffusion coefficient of the counterion in solution and D,O is the corresponding value in pure solvent. Finally, eq 1 has been derived by a molecular theoryG37which also indicates that eq 2 is satisfied and, moreover, provides an expression for f in terms of molecular properties. The key parameter of the molecular theory is
(31 where e is the protonic charge, t the bulk dielectric constant, k Boltzmann’s constant, T the absolute temperature, and b the spacing between charged groups taken along the axis of the polyion chain. The theory indicates that for E < I z J - l , where z c is the valence of the counterion, polyion-counterion interactions are governed entirely by Debye-Huckel atmosphere effects, while if t > I z J - l , counterions condense on the polyion until Enet = Iz cl-l, the remaining counterions again being in a Debye-Huckel atmosphere. Then
5
/lp*
4
>
/zc1-‘
(4)
a more complicated expression,6 not reproduced here, ob. 4 has been tested for univtaining for E < I z J - ~Equation alent counterions against tracer diffusion data, with use of eq 2, and against conductance-transference measurements, by using measured values of both A and A, in eq 1, and found to give close agreement with an a priori choice of E based on structural knowledge of the hai in.^,^ A full theoretical expression for the right-hand side of eq 1, then, reduces to a theory for A,. The purpose of this paper is to give a plausible calculation of A, which takes into account both electrophoretic and relaxation effects.
Calculation The basis of the calculation of A, is the assumption that the polyion chain may be modeled by an infinitely long cylinder, a choice which is discussed in the next section. It is then possible to make use of Henry’s calculation,s as developed by Gorin? of the electrophoresis of long cylinders. This approach has also been used, for excess added salt, by RosslO and by Takahashi, Noda, and Nagasawa.llJza For a long cylinder oriented parallel to an external electric field, Gorin obtains = eL(a)/4a?7
(5 1
where u I * is the electrophoretic mobility of the parallel cylinder, E is the bulk dielectric constant, 7 the bulk viscosity, and $ ( a ) the electrostatic potential a t the surface of the cylinder of radius a , For a cylinder oriented perpendicular to the electric field, Gorin’s result is i9(0)/87117 (6) Actually eq 6, unlike eq 5 , is valid only a t infinite dilution (see eq 62 and Table 8 in chapter 5 of ref 9); however, it is precisely this limit which concerns us here. If one now asI[_* =
= ~&(~)/6~17
(7 1
Asterisks have been used on the mobilities because Gorin’s expressions, like Henry’s, do not take into account the relaxation effect; eq 7 for the polyion mobility includes only the intrinsic frictional drag of the solvent on the cylinder combined with the additional drag (electrophoretic effect) imparted to the solvent, and hence to the cylinder, by the directed velocity of the counterions due to the external field. The relaxation effect may be calculated indirectly from the flux j of uncondensed counterions j , = 3 . 8 6 6 ~ z , / e E n c ( t , ? “ - 0.134u,EnC
= e2/ekTb
f = 0.866/~,/-’[-‘
2111*
sumes that the electrophoretic mobility up* of randomly oriented cylinders is adequately represented by weighting ull* by % and u l * by 2/3 (which would appear to be a more straightforward averaging procedure than that used by Gorin)
(8)
where E is the external field, lCo the friction coefficient of the counterion in pure solvent, n the concentration of uncondensed counterions (equal to I z 4 -2[-1 times the total equivalent polyion concentration), and u the actual electrophoretic mobility of the polyion (written without an asterisk because it includes the relaxation effect). This equation is given for the practically important case [ > and may be obtained from eq 2 of ref 6 if the latter is written for the uncondensed counterions only. The numerical factor 0.866 is merely the value o f f from eq 4 when 4 is set after condensation, it being this equal to its net value Iz value which governs the motion of the uncondensed counterions. The physical meaning of eq 8 is as follows. If the polyions were immobile ( u p = O), application of an external field E would perturb the cylindrical symmetry of the counterion atmosphere, and a restoring force would be set up which impedes the motion of the counterions in the direction of the field; hence the numerical factor in the first term of eq 8 is less than unity. The finite mobility of the polyion increases the asymmetry of the counterion atmosphere (since the directed motion of the polyion is opposite to that of the counterions) and the second term of eq 8 accounts for the resulting increment of restoring force on the counterions. The “polarizable atmosphere” is also the origin of the relaxation effect in classical ionic solution theory. 12b Now eq 8 may also be written as
,
j, =
I Z , ~ e(t,?“n,E*
(9)
where
E # / E = 0.866 - O.13421,(kC0//zc~ e)
(10)
In this form it is recognized that the motion of the counterions is governed, not by the external field E, but by a effective field E* which is diminished relative to E by electrostatic interaction with the polyion. By Newton’s third law it must be the same field E* which drives the directed motion of the polyion; that is //&
= /(,*E*
(11)
where the left-hand side by definition equals the drift velocity of the polyion. Thus, u p equals up* reduced by the relaxation effect measured by E*IE (which in turn, according to eq 10, depends on up). Combination of eq 10 and 11then yields the following result for up: The Journalof Physical Chemistry, Vol. 79, No. 3, 7975
Gerald S.Manning
264
21,
= 0.866~,*/[1
+
0.134(bCo//z,le)u,*]
(12)
where u p * is given by eq 7. I t remains to obtain an expression for $ ( a ) in eq 7. To retain consistency with the dilute solution approximation used for the validity of eq 4 and 6, $ ( a ) must be a limiting form for dilute solutions. Moreover, since the kinetic unit referred to above as the polyion was assumed to include the condensed counterions, $ ( a ) is the dilute solution form of the electrostatic potential at the surface of a cylinder of radius a and surface charge density corresponding to Snet = Iz J-l. That is, it is simply the limiting form of the DebyeHuckel potential13J4 $(a) = 2 lz,l - l ( k T / e ) In ~a
(13)
where K is the Debye screening constant ,y2
= (4ii~~/~kT)
