J. Phys. Chem. 1987, 91, 3920-3923
3920
Rigorous Theory of the Diaphragm Cell When the bmusion Coefficient Depends upon Concentration James K. Baird* Department of Chemistry, University of Alabama in Huntsville, Huntsville, Alabama 35899
and Richard W. Frieden Department of Anatomy, University of Utah Medical Center, Salt Lake City, Utah 84105 (Received: December 1 , 1986)
A diaphragm cell consists of two well-stirred solution compartments separated by a sintered glass diaphragm. When the compartments are filled with different concentrations of a solution, diffusion sets in across the diaphragm. If the relaxation of the concentration in one of the compartments is followed, the diffusion coefficient may be determined. Since the early work of Gordon and Stokes, which firmly established this technique, there has been no rigorous theory for the cell in the important case where the diffusion coefficient depends upon concentration. For a cell of the NorthropMcBain design with both compartment volumes identical, we have derived three rigorous formulae, which may be used to obtain directly from the data the differential diffusion coefficient. For a Lewis design cell, we have found a single, simple formula for determining this coefficient.
1. Introduction
In the more than half century since its the diaphragm cell3 has become widely used for the measurement of the mutual diffusion coefficient in two-component solutions. The variety of systems investigated has proven to be broad indeed, including, for example, recent measurements of diffusion coefficients for aqueous alkali fluoride solutions,“*5aqueous ascorbic acid,6 aqueous triglycine sulfate,’ and ethanol in benzene.8 The heart of any diaphragm cell consists of two well-stirred solution compartments on opposite sides of a porous membrane, which in practice is usually a glass f r k 3 When solutions of differing concentration are placed in the compartments, diffusion occurs across the membrane. If the porosity of the membrane is known, measurement of the concentration difference across the membrane as a function of time is sufficient to determine the diffusion coefficient. Barnes9 and later Mills et a1.I0 showed theoretically that, if the interstitial volume of the frit is small compared with the volumes of the compartments, the initial concentration distribution within the frit decays rapidly to a steady state. This permits the concentration gradient, dcldx, within the frit to be treated in steady-state approximation and relegates the time, t , to the status of a parameter. Most solutions are thermodynamically nonideal. As a result, within the frit, Fick’s first law assumes the form, J = -D(c)(dc/dx), where J is the molecular flux and D(c) is a diffusion coefficient depending upon the concentration, c.I1 Because of solute mass conservation within the frit, dJ/dx = 0, so that J is a constant independent of the spatial coordinate, x. If cl(t) and c2(r) are the concentrations of solute in the compartments of volume VI and V2,respectively, and if cl(t) > c2(t), diffusion occurs across the frit from Vl into V2. If we let 1 be the ( I ) Northrop, J. H.; Anson, M. L. J . Gen. Physiol. 1929, 12, 543. (2) McBain, J. W.; Liu, T. H. J. Am. Chem. SOC.1931, 53, 59. (3) (a) Mills, R.;Woolf, L. A. The Diaphragm Cell; Australian National University Press: Canberra, 1968. (b) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworths: London, 1959; pp 253-261. (4) Sood,M. L.; Kaur, G. 2. Phys. Chem. (Leipzig) 1978, 259, 585. (5) Sood, M. L.; Bala, M. Electrochim. Acta 1982, 27, 1239. ( 6 ) Shamin, M.; Baki, S . M. A. Ausr. J . Chem. 1980, 33, 1857. (7) Kroes, R. L.; Reiss, D.; Silberman, E.; Morgan, S . J . Phys. Chem. 1985, 89, 1651. (8) Albright, J. G.; Edge, A. V.J.; Mills, R.J. Chem. Soc., Faraday Trans. 1 1983, 79, 1327. (9) Barnes, C. Physics 1934, 5, 4. (10) Mills, R.;Woolf, L. A,; Watts, R. 0. AICHE J . 1968, 14, 671. (1 1) Onsager, L. Ann. N.Y. Acad. Sci. 1945, 46, 241
effective thickness of the frit, and A be its cross sectional area, then the equation of the motion for the cell is3
where
is a constant characteristic of the particular cell. Equation 1.1 is an integro-differential equation, which has been solved exactly in only two instances: (1) D(c) is constant, or (2) D(c) = D(Q)(l ac), where a is a constant, and D(0) is the value assumed by D(c) in the limit of infinite dilution.12 To achieve an approximate integration of this equation for the case of arbitrary D(c), Gordon introduced the “integral diffusion coefficient”, b,defined in terms of the ‘differential diffusion coefficient”, D(c), by
+
where s(t)
= Y2[ci(t)
+ c,(O)],
i = I, 2
(1.4)
and ci(0) is the initial value of ci(t).12 Gordon12and Stokes13have noted that in many cases observed experimentally, D is nearly independent of time. Gordon suggested that eq 1.1 could be integrated in the form3,’* (1.5)
The random internal structure of a glass frit dictates that the thickness, 1 (see eq 1.2), must be determined by some indirect means. Standard procedure is to let diffuse through the cell some aqueous 0.5 M KCI a t 25 OC, a system for which D has been accurately measured by absolute methods3 Analysis of the KCI data according to eq 1.5 is sufficient to determine p for the cell. Values lying in the range 0.1 to 0.5 cm-2 are ordinarily encountered. Once calibrated in this fashion, the cell may be used to determine D for another solution. Stokes studied values of D (12) Gordon, A. R. Ann. N.Y. Acad. Sci. 1945, 46, 285
0022-3654/87/2091-3920$01.50/00 1987 American Chemical Society
The Journal of Physical Chemistry, Vol. 91, No. 14, 1987 3921
Rigorous Theory of the Diaphragm Cell obtained by using various initial concentrations, usually cl(0) > 0, cz(0) = 0, and developed a technique based upon successive approximations which he used to estimate D(c).13 Stokes has emphasized, however, that for general D(c), the integration summarized by eq 1.3-1.5 has no theoretical basis.13 Some improvement on this state of affairs is highly desirable, especially in view of the important role played by D(c) in transport theory,l4.l5the theory of solution^,'^*'^ and in the theory of crystal gro~th.~J~ 2. Theory of the Northrop-McBain Cell 2.1. Solution for Finite Times and Concentration Differences. Equation 1.1 is an integro-differential equation, which for arbitrary D(c) appears to have no exact solution. In the special case that VI = V2,however, substantial progress is possible. When VI and Vz are identical, transfer of solute mass from VI to V2 diminishes cI by the same amount that it increases e,; hence, the cell mean concentration, E E = Y2[Cl(O)
+ c2(0)] = Yz[c,(t) + C,(t)]
(2.1.1)
is unchanged with time and constitutes a constant of the motion.I2 In the integral on the right-hand side of eq 1.1, we exploit the constant of motion by changing the variable of integration from c to Through eq 2.1.1, the value of E is a constant established by the initial conditions, which implies that dy = dc. We note that, when c = cl, y = (l/z)(cl - c2), and when c = cz, y = -(1/2)(c1 - cz); hence, if eq 2.1.2 is substituted into eq 1.1, there results
Now define YZ(Cl
-1 [l pD(~)x
--
+ z(x)]-l
(2.1.8)
where
To the right-hand side of eq 2.1.8, we apply the geometric series (1 + z)-l = 1 - z + zz - z3 + ... Equation 2.1.9 is used to replace z , and each power of z is expanded. There results
:jx3
-
... (2.1.10)
Both sides of eq 2.1.10 may be integrated with respect to x to obtain t(x) = constant -
1 lnx+ -
[
D(2)(E) ]x2+
12P[D(r)12
(2.1.2)
y=C-E
x =
dt(x) -dx
- c2)
(2.1.4)
and note that the definition of the derivative permits us to write dx(t)/dt = l/[dt(x)/dx]. Use of these definitions transforms eq 2.1.3 to the form
Equation 2.1.1 1 constitutes a series solution of eq 2.1.5 and is the principal mathematical result of this section. As such, its analytic properties warrant discussion: First, because the integral on the right-hand side of eq 2.1.5 extends for -x to +x, it depends only upon the even part of D(E + y) after expansion about E. This accounts for the fact that the odd order derivatives, D(2m+1)(p) (where m = an integer), which are implicit in eq 2.1.6, fail to appear in eq 2.1.7. Consequently, dt(x)/dx is an odd function of x (see eq 2.1.10). Second, if D(c) = Dowhere DOis a constant independent of c, all of the derivatives, D(")(c),vanish. Equation 2.1.1 1 reduces to 1 t ( x ) = constant - - In x (2.1 .12) PDO
To obtain a symbolic integration of eq 2.1.5, we expand D(E
+ y ) in a Taylor series about D(E
-
+ y ) = D(E) + ,,=IE-y"D(")(E) n!
(2.1.6)
where D(")(?) [d"D(c)/dc"],=, If eq 2.1.6 is substituted into the integral in eq 2.1.5 and the integration performed term by term, we obtain JJXD('
+ y ) dy = x4 +
...
1
(2.1.7)
where the monomial, 2D(?)x, has been factored out for later convenience. After substituting eq 2.1.7 into the right-hand side of eq 2.1.5 and forming the reciprocal of the result, we obtain (13) Stokes, R. H. J . Am. Chem. SOC.1950, 72, 2243. (14) Clarke, D. J. J . Chem. Phys. 1957, 27, 29. ( 1 5 ) Lin, P. 0.; Pelton, A. D.Eer. Bunsen-Ges. Phys. Chem. 1977, 82, 1243. (16) Cullinan, Jr., H. T. Ind. Eng. Chem. Fundum. 1966, 5 , 281. (17) Leffler, J.; Cullinan, Jr., H. T. Ind. Eng. Chem. Fundum. 1971, 9, 84. (18) Baird, J. K.; Meehan, Jr., E. J.; Howard, S . B.; Xidis, A. L. J . Crysrul Growlh 1906, 76, 694.
a simple form, which can be obtained directly by integrating eq 2.1.5. When D(c) is a function of c, however, we see by comparison with eq 2.1.1 1 that Doin eq 2.1.12 gets replaced by D ( P ) , and there appears a series of terms involving the powers, xZm (where m = an integer), multiplied by coefficients involving D(E) and various D(2m)(E).These differences between eq 2.1.1 1 and 2.1.12 may be thought of as corrections to eq 2.1.12, which account for the c dependence of D(c). Equation 2.1.1 1 may be compared with experimental data and the coefficients determined by least squares. Employing eq 2.1.4 to introduce the usual Ac concentration notation AC E CI - ~2 = 2x (2.1.13) we may write eq 2.1.1 1 in the form t = A.
+ A , In (Ac) + A2(Ac)' + A4(Ac), + ...
(2.1 .14)
The coefficient, Ao,contains the constant from eq 2.1.1 1 and a term involving In (1/z)/PD(E) which comes from replacing x by Ac in the logarithmic term. The coefficient, A I , is the most important, since by comparison with eq 2.1.1 1, it satisfies 1 D(E) = -(2.1.1 5 ) PA I
Of course, because D(c) is positive, A , must be less than zero. The remaining coefficients, A Z ,A,, ...,are complicated functions of D(c) and its derivatives and are less important from a theoretical point of view. According to eq 2.1.1 1 through 2.1.14, a plot of t vs. In (Ac), which is not a straight line, indicates that the diffusion coefficient depends upon c.
3922 The Jourval of Physical Chemistry, Vol. 91, No. 14, 1987 The experimental procedure, therefore, is to measure t as a function of Ac and to fit the results to eq 2.1.14, treating the coefficients, A,, as least-squares parameters. The series beyond the logarithmic term on the right-hand side of eq 2.1.14 can contain only even powers of Ac. The number of A,, n 2 2, employed is at the discretion of the experimenter but cannot exceed the number of data points. If the cell constant, p, is known, then through eq 2.1.15 the least-squares evaluation of A , is sufficient to determine D(?).T o map out the function D(S), F may be changed by altering the initial concentrations, and the experiment repeated. 2.2. Solution in the Long Time Limit. Although valid for finite times and concentration differences, eq 2.1.11 is not a closed form. A closed form expression for D(c) is possible, if we consider the limit of infinite tiwe. Differentiate both sides of eq 2.1.5 with respect to x, using Leibnitz's ruleI9 to calculate the derivative of the integral on the right-hand side. The integrand, D ( E y), does not become differentiated, because it is independent of the time, t(x), and is, hence, also independent of x. The result is
+
[ 4[ -4-' d2t(x)
dt(x)
/3
=p ( S
+ x) + D(F - x)]
(2.2.1)
Equation 2.2.1 holds for general values of x but is unwieldy for determining the functional form of D(c), because the righthand side depends upon D(c) evaluated at two different concentrations, c x and c - x. In the case, x = 0, however, which m, this problem is removed. Setting x = 0 on the implies t right-hand side of eq 2.2.1 and using eq 2.1.13, we obtain
-+
1
D(?)= - lim
p
AFO
[d2t/d(WI [dt/d(Ac)J2
(2.2.2)
On the right-hand side of eq 2.2.2, Ac has been allowed to approach zero only in the limit, because both dt/d(Ac) and d2t/d(Ac)2 become infinitely large, although their ratio remains finite. For illustration, evaluate dt/d(Ac) and d2t/d(Ac)2 using eq 2.1.14. One finds, in the limit of vanishing Ac, dt/d(Ac) a A,(Ac)-I and d2t/d(Ac)2 a -A,(Ac)-~. After substitution of these results into eq 2.2.2 and use of eq 2.1.15, an identity is obtained. We conclude that eq 2.1.14 and 2.2.2 contain the same information. If eq 2.2.2 is to be used to analyze measurements o f t vs. Ac, it would be helpful to have a functional form for t(Ac) from which to compute the indicated derivatives. Because of eq 2.1.14, dt/d(Ac) is an odd function of Ac, while d2t/d(Ac)2is an even function. A general form for t(Ac) consistent with these requirements is t(Ac) = a. al In (Ac) +f(Ac) (2.2.3)
+
wheref(Ac) = f(-Ac). The coefficients, a, and a,, may be regarded as least-squares parameters. The function, f(Ac), may be expanded in even powers of Ac or a closed form may be guessed. For eq 2.2.3 to achieve the proper limiting form when substituted into eq 2.2.2, the logarithmic term must dominate f(Ac) for small Ac. This implies thatf(Ac) must satisfy Jdf/d(Ac)l < (Ac)-l and 0. Idtf/d(Ac)21 < (Ac)-* as Ac 2.3. Solution i n the Short Time Limit. Rewrite eq 2.1.5 in the form
-
(2.3.1) Differentiate both sides of eq 2.3.1 with respect to t using Leibnitz's rule. Note that derivatives of D(e y ) with respect to t are zero because T is a constant of the motion. One obtains
+
d2x/dt2 p = --[D(T dx/dt 2
+ X) + D(S - x)]
(2.3.2)
(19) Arfken, George Mathematical Methods for Physicists, 3rd ed.; Academic: New York, 1985; p 478.
Baird and Frieden As in eq 2.2.1, the right-hand side depends upon D ( c ) evaluated at two different concentrations. For the special case c2 = 0, which can occur only at t = 0, we note by reference to eq 2.1.1 and 2.1.4 that x = S. Using eq 2.1.13 to reexpress x on the left-hand side of eq 2.3.2 and setting x = on the right, we obtain 2 d2(Ac)/dt2 D(2S) = -D(O) - - lim P 1-0 d(Ac)/dt
(2.3.3)
Equation 2.3.3 might prove useful in cases where the mobility, K, is known and D(0) can be computed from the Nerst-Einstein
relation, D(0) = pkT, where k is Boltzmann's constant and T i s the absolute temperature. Unlike eq 2.2.2, both derivatives in eq 2.3.3 are finite. The special case, D(c) = D(O)(l + ac), helps illustrate the meaning of eq 2.3.3. This form for D(c) permits eq 2.3.1 to be integrated exactly. One finds, using eq 2.1.13, that Ac(t) = Ac(0) exp[-pD(E)t], where Ac(0) is the initial value. When this Ac(t) is substituted into the right-hand side of eq 2.3.3,one obtains D(24 = D(O)(l + 2a?), as is required.
3. Theory of the Lewis Cell Lewis suggested a design for a diaphragm cell which, in principle at least, fixed c2 at zero for all t.20 His method for establishing this condition was to replace continuously the contents of V2with pure solvent. Since its introduction, the Lewis cell does not appear to have gained wide acceptance. Putting aside for the moment whatever experimental difficulties this may imply, we derive here the simple analysis for D(c) which the Lewis cell permits. To obtain the equation of motion for the Lewis cell, we set c2 = 0 in eq 1 . 1 . The result is
The cell constant in this case is given by eq 1.2 with V2 = m. Differentiation of both sides of eq 3.1 with respect to t leads to 1 d2c,/dt2 D(Cl) = - - -
P dci/dt
In eq 3.1, the time serves as a parameter. Experimentally, one should monitor cl(t) as a function of t . From the data, the instantaneous first and second derivatives are calculated and substituted into eq 3.2 to determine D(c) evaluated at the instantaneous value of cl. The entire functional form of D(c) may thus be mapped out in a single experiment. 4. Discussion
Except for the Lewis cell, where V2 plays no role, the simple results we have obtained depend upon the condition, V, = V2. We have not made any attempt here to determine what effect small deviations from this condition might have on our formulae. Careful calibration and filling of the cell volumes will, of course, be required to avoid errors. However, even if V, and V2are not exactly equal, 2 will still be approximately constant. If the values of Ac(t) encountered are large compared with the drift in F , one would expect that our formulae should still hold. Although we have not investigated it in detail, we can expect the series in eq 2.1.14 to be rapidly convergent. Our basis for this conjecture is the structure of eq 2.1.11 from which eq 2.1.14 was derived. Through expansion of D(? y ) about S in eq 2.1.6 and the term-by-term integration in eq 2.1.7, the coefficients of the powers of x in eq 2.1.1 1 are entirely independent of D(l)(c). Since we can expect, in general, this first derivative of D(c) to be larger than the rest, the terms involving D(2m)(S) ( m = an integer) constitute small corrections to the first two terms on the right-hand side of eq 2.1.1 1. This implies that eq 2.1.14 should be usable with rather large values of Ac, perhaps of the order of
+
(20) Lewis, J. B. J . Appl. Chem. 1955, 5 , 228.
J. Phys. Chem. 1987, 91, 3923-3925 magnitude of C. Although mass conservation equations3 exist for calculating Ac from c , ( t ) ,a differential method of measurement might prove best for evaluating Ac for use in eq 2.1.14. The dependence of D(c) upon c at high concentration can most easily be determined by using eq 2.3.3, since it gives D(2E). This equation depends upon D(O), which, in the case of electrolytes at least, can be determined from the Nernst-Einstein relation and
3923
accurate measurements of the mobility. Acknowledgment. Support from NASA Contract N A S 835986 with Marshall Space Flight Center and from the Consortium for Materials Development in Space at the University of Alabama in Huntsville under NASA Grant NAGW-812 is gratefully acknowledged.
Solvent Effect on the Dimerization of Some Lithium Compounds A. M. Sapse* City University of New York, Graduate Center and John Jay College, New York, New York 10019, and Rockefeller University, New York, New York 10021
and Duli C. Jain York College of the City University of New York, Jamaica, New York 1 1 451 (Received: March 25, 1986)
A modified Born equation is used to investigate the solvent effect on the dimerization energy of some lithium compounds. Since the solvent stabilizes the monomers preferentially to the dimers, the effect of the solvation is to make the energy of dimerization less negative.
Introduction Association is a well-known property of lithium compounds. In the gas phase, solution, and crystals, aggregated species are found in preference to monomers. Such compounds as alkyllithiums, lithium amides, and ionic compounds of lithium are present as dimers, trimers, tetramers, or even higher oligomers. Gas-phase association energies which are usually quite large can be obtained from J A N A F Tables’ for (LiF)2, (LiF)3, and (LiOH)2, while numerous theoretical calculations are providing association energies for compounds such as LiCH3, LiNH2, LiNF2 and Li2NH, in addition to LiF and LiOH.2 Experiments in solution show the degree of association to be dependent on the nature of solvent; for instance, a dimer-tetramer equilibrium is favored by hydrocarbon ~ o l v e n t while ,~ in tetrahydrofuran, a monomerdimer equilibrium is found for [(CH&Si] 2NLi, results which cannot be accounted for by quantum chemical calculation in the absence of the solvent. This work is the first in a series which examines the solvent effect on oligomerization of lithium compounds. The model used consists of the formation of a cavity in a solvent in which the lithium compound is to be placed. The solvent is (1) Stull, D. R.; Prophet, H. JANAF Thermochemical Tables, 2nd ed; National Bureau of Standards: Washington, DC,1971; Report NSRDS-NBS 37. Wagman, D. D.; Evans, W. H.; Parker, V. B.; Schumm, R. H.; Halow, I.; Bailey, S. M.; Churney, K. L.; Nuttal, R. L. J . Phys. Chem. Ref. Data 1982, 11, Suppl. 2. (2) (a) Raghavachari, K.; Kaufmann, E.; Clark, T.; Schleyer, P. v. R., to be submitted for publication. Cowley, A. H. White, W. D. J. Am. Chem. Soc. 1969, 91, 34. Baird, N. C.; Barr, R. F.; Datta, R. K. J . Organomet. Chem. 1973, 59, 65. Guest, M. F.; Hillier, I. H.; Saunders, V. R. J. Organomef. Chem. 1972,44,59. McLean, W.; Pedersen, L.G.; Jarnagin, R. C. J . Chem. Phys. 1976,65, 2491. Clark, T.; Schleyer, P. v. R. Pople, J. A. J. Chem. Sac., Chem. Commun. 1978, 137. McLean, W.; Schultz, J. A. Pedersen, L. G.; Jarnagin, R. C. J. Organomet. Chem. 1979,175.1. Clark, T.; Chandrasekhar, J.; Schleyer, P. v. R. J . Chem. Sac., Chem. Commun. 1980,672. Graham, G.; Richtsmeier, S.;Dixon, D. A. J. Am. Chem. Soc. 1980,102, 5759. Herzig, L.; Howell, J. M.; Sapse, A.-M.; Singman, E.; Snyder, G. J . Chem. Phys. 1982, 77,429. (b) Sapse, A.-M.; Kaufmann, E.; Schleyer, P. v. R.; Gleiter, R. Inorg. Chem. 1984.23, 1569. (c) Sapse, A.-M.; Jain, D. C. Chem. Phys. Left. 1984, 110, 251. (3) Kimura, B. Y.; Brown, T. L. J. Organomet. Chem. 1971, 26, 57. Mootz, D.; Zinnius, A.; Bottcher, B. Angew. Chem. 1969, 81, 398.
0022-3654/87/2091-3923$01 SO10
treated thus as a continuum characterized by its dielectric constant. The stabilization or destabilization energy provided by the solvent for the oligomers is defined as the difference between solventoligomer electrostatic interaction energy and the sum of solvent-monomer interaction energies. The compounds examined are the monomers and dimers of LiNH2, LiCH3, LiNF2, Li2NH, LiF, and LiN(SiH,),. Gas-phase calculations have predicted geometries, net atomic charges, and dimerization energies for these compounds, within the HartreeFock approximation, using Gaussian basis sets, and these results will be used for the description of the solute. To examine the effects of molecular-level “lithium bonding” where a water molecule (used as a model for ether) is bound to the lithium atoms in monomers and dimer, we optimized the geometry and calculated the energies of the following species:
Method and Results The interaction energy between a Li compound and a solvent is calculated through the use of a modified Born equation. Since the compounds under consideration do not possess net electric charge, the Born equation4 would not predict any electrostatic interaction. However, the corrected equations derived in some studies5 take into account dipoles and higher poles, writing the energy of interaction as:
where Qjdenotes the net charge on the j t h atom of the solute, N is the total number of atoms, Pn(r,,rk)is the Legendre poly(4) Born, M. Phys. 2,1920, 1, 45. ( 5 ) Gersten, J. I.; Sapse, A.-M. J. Am. Chem. Sac. 1985, 107, 3786. Harrison, S.W.; Nolte, H.J.; Beveridge, D. L. J. Phys. Chem. 1976,80, 2580. Felder, C. E. J. Chem. Phys. 1981, 75, 4679. Rinaldi, D.; Ruiz-Lopez, M. F.; Rivail, J. L. J. Chem. Phys. 1983, 78, 834.
0 1987 American Chemical Society
