Irreversible Processes in Electrolytes. Diffusion ... - ACS Publications

IRREVERSIBLE PROCESSES I N ELECTROLYTES. DIFFUSION, CONDUCTANCE, AND VISCOUS FLOW I N ARBITRARY MIXTURES OF STRONG ELECTROLYTES* BY LARS ONSAGER AND RAYMOND M. FUOSS

1. Introduction Scope and Aims. I n an electrolyte, the Coulomb forces between the ions will affect the thermodynamic and other physical properties of the system; these effects can be calculated, because the force law is known. The mathematical difficulties involved were first overcome by Debye and Huckel,' who derived limiting laws for the variation of molecular freezing-point depression, etc., and of molecular conductance with the concentration, c, all of the form I .I.

F(c) = F ( 0 ) - ad/C

(1.1.1)

where F(c) stands for any one of these properties, while F(o) and a! are constants. The term -a!.\/: represents the influence of the ionic forces. It is characteristic of the theory that the factor a! can be computed. For functions F(c) describing irreversible processes, the value of a! will in general depend on the (limiting) ionic mobilities; in addition, only the charges of ions and the ratios of their concentrations are involved, besides properties of the solvent (and universal constants). The present work deals with conductance, diffusion and viscosity; explicit formulas of the type (1.1.1) are derived. Earlier computations, which were limited-to electrolytes containing only two species of ions, will be extended to electrolytes of arbitrary composition, although the scope of the investigation is still limited by the condition that the total concentration of ions must be low. As a consequence of mathematical approximations, we shall obtain in the end asymptotic formulas of the type (I.I.~), while more exact computations, which we have not attempted on account of the mathematical complications involved, would, judging from tentative calculations, lead to expressions of the type (1.1.2) F ( c ) = F(0) a!&pc log c yc .* *

+

+

+ +

.

The limiting l/c laws for the conductance and transference numbers of simple electrolytes (Le. containing only two species of ions), have been computed by one of us in a previous treatise; which amounted to a refinement of the original theory given by Debye and HuckeLa

* Contribution from the Chemical Laboratory of Brown University. 1

8

Debye and Huckel: I, Physik. Z., 24, 185 (1923); 11, 24, 305 Oneager: I, Physik. Z., 27, 388 (1926); 11, 28, 277 (1927). Debye and Hackel: 11, loc. cit.

(1923).

2690

LARS ONSAGER AND RAYMOND M. FUOSS

The only published computations for electrolytes of a more general composition have been given by Bennewitz, Wagner and Kiichler,’ who considered the case of a third ionic species present in very small quantity. Within the limited scope outlined above, the modern electrostatic theory of electrolytes has performed remarkably well in accounting for a complex pattern of experimental data on the basis of a simple picture.2 The present calculations extend the range of phenomena thus covered. The most interesting effect connected with electrolytic conduction in mixtures, as contrasted t o simple electrolytes, was first observed experimentally by Bray and Hunt,3 who measured the conductance of NaC1-HCl mixtures, and showed that the isoionic principle does not hold; that is, the conductances for a given total concentration of ions (and the same total “ionic strength”) are not additive in this case. No corresponding effect is observed in the thermodynamic behavior of electrolytes, and curiously enough, the theory predicts for simple binary electrolytes, in accordance with an empirical rule due to K o h l r a ~ s c hthat , ~ the mobility of any ion a t a given concentration is independent of its partner ion in the solution, e.g., A,, is the same in 0.01n. HC1 as in 0.01n. KCl. I n a HCl-KC1 mixture, as we shall see in the following, the theory predicts that the fast H+ ions will be slowed down more than usual, while the slow K+ ions, “stealing their thunder” as the ions migrate past each other, will be accelerated in proportion, and the C1- ions will move with (practically) their usual velocity. The nature of the effect was recognized by Bennewitz, Wagner and Kuch1er;j it has been demonstrated experimentally by these authors and by Longsworth,6 The present computations permit a quantitative comparison between theory and experiment. When due allowance is given for mathematical approximations, which become rather serious for, say, 0.1n. concentration, the experimental verification may be called satisfactory. The investigation of diffusion is possibly the most important part of the present contribution to the theory of electrolytes. Here the main effect of electrostatic forces between the ions is a thermodynamic one: The gradient of the thermodynamic potential, which represents the “driving force” upon the ions, is less than it would be in an ideal solution for the same concentration gradient,’ However, we shall see that the velocity of diffusion for a given “driving force” may also be affected by the electrostatic (and hydrodynamic) Bennewitz, Wagner and Kiichler: Physik. Z., 30,623 (1929). In this discussion we shall concentrate on the problem in hand, and make no retense of doin justice to maas-action effects, or in eneral to specific interaction of ions. %e procedure for taking such effects into account is fairly obvious; the questions involved have been taken up by several authors and have, by and lar e been given adequate consideration. See N. Bjerrum: Kgl. Danske Videnskab. Selskab. k t h . fysis. Medd., VII, No. 9 (1926); Gronwall, LaMer and Sandved: Physik. Z., 29, 358 (1928). 3 Bray and Hunt: J. Am. Chem. SOC.,33, 781 (1911). 4 Kohlrausch and Holborn: “Leitvermogen der Elektrolyte”; MacInnes and Cowperthwaite: Trans. Faraday SOC.,23, 400 (1927). 6 Bennewitz, Wagner and Kuchler: Iac. cit. e Longsworth: J. Am. Chem. Soc., 52, I897 (1930). 7 At higher concentrations, where short range repulsive forces become noticeable, an incream may result. For thermodynamic properties of concentrated electrolytes, see Harned: Trans. Faraday SOC.,23, 462 (1927) and Huckel: Physik. Z., 26, 93 (1925) (theory). 1

2

IRREVERSIBLE PROCESSES I N ELECTROLYTES

2691

interaction of ions. Where more than two species of ions are present, the latter effects gain in importance (because the condition of zero electric current no longer forces all ions to move with the same velocity). Fundamentally, the problems of diffusion and conduction belong together; we thereby consider the most general migration of ions. From this point of view, the special case of pure conduction is not entitled to any preference. The general problem offers no additional obstacles to the theory; on the contrary, the essential characteristics of the problem are more readily recognized by avoiding an unnatural specialization. The most general description of the laws of diffusion (and conduction) in a multi-component system will be formulated. The theory of electrolytes furnishes a good example in that all the terms in the phenomenological equations are important. We shall only be concerned with the formulation of the differential equations, which even for a simple electrolyte lead to a variable constant of diffusion, and make no attempt to integrate them for any practical case. For an actual test of the theory it would seem necessary to carry out experiments in which the difference between the concentrations of the interdiffusing solutions were relatively small, so as to measure the differential coefficient of diffusion, which is considered by the theory. The differential equations for diffusion take the simplest form when expressed in terms of “driving forces,” namely the gradients of thermodynamic potentials.’ A system of linear equations

- Ji =

Z Q* grad pk, (i = I,

k=- I

2,

. . . , s),

(1.1.3)

...

relate the velocities VI, v8 (relative to the solvent) of the different ionic species to the gradients of the total potentials pi = p’i

+ e&,

(1.1.4)

where pi is the ordinary thermodynamic potential of Gibbs, 11. the electrostatic potential, and ei is the charge per unit amount of a species of ions. As pointed out by Guggenheim and others,2 only the sum (1.1.3) enters into thermodynamic relations, and into the laws of diffusion. The matrix Q is symmetrical: = Qkj.

(1.1.5)

This symmetry relation, which is responsible for a well-known relation between transference number and electromotive force of a concentration-cell , ~ it can be derived with transference,S is not a thermodynamic t h e ~ r e m but 1 Cf. W. Nernst, who ex ressed the driying forces in terms of osmotic pressure, which is directly related to thermo&namic potentials. Nernst: 2. physik. Chem., 2, 613 (1888). The thermodynamic relations in question have been discussed by N. Bjerrum: 2. physik. Chem., 104, 406 (1923). 1 Gug enheim: J. Ph 8. Chem., 33,842 (1929);34, I540 1758 (1930); J. Am. Chem. SOC., 52, 1315 $1930);P. B. dylor: J. Phys. Chem., 31, 1478 (1927). 8 H. v. Helmholtz: Wied. Ann., 3, 201 (1876); W k . Abh. 1,840;MacInnes and Beattie: J. Am. Chem. Soc., 42, 1117(1920). 4 Cf. J. W. Gibbs: “Collected Works” I, 425 (1928);W. D.Bancroft: J. Phys. Chem., 7, 416(1903).

2692


from the very reasonable assumption that molecular dynamical systems, like those known to our macroscopic experience, possess symmetry in past and future.' The following computations will be based on certain fairly general assumptions regarding the mechanism of resistance to migration of ions. The assumed mechanism is compatible with the principle of microscopic reversibility, and, as one should expect, the results conform to (1.1.5). An alternative form of ( I . I . ~ )incorporating , the symmetry relation ( I , I . s ) , is obtained by the construction of a iidissipation-function." The problem of viscosity has been included in this treatise mainly because it yields to the same mathematical methods as the conductance-diffusion problem; we thereby avoid a duplication of the calculations, which are to a great extent identical for both problems. The electrostatic contribution to the viscosity 7 of an electrolyte obeys the formula 7(c) - 7(0) =

dY,

(1.1.6)

as has been shown by earlier calculations and experiments2 by other authors for simple electrolytes. I n extending the theory to cases where 3 or more species of ions are present, we find that no new spectacular effects are to be expected. In some respects, the present treatment of the viscosity problem brings simplification as compared to previous computations. In regard to the physical picture, we disagree with FalkenhagenS on one detail which, fortunately, does not affect the result for viscosity. He considers cases where the velocity gradients for the ions differ from the gradient for the solvent and, in our opinion, a material difference cannot occur. Falkenhagen arrived at his conclusion in searching for a mechanism of force transfer between ions and solvent. We have therefore investigated the mechanism of force transfer, an interesting problem in itself (although a barren one for experimenters), and we can show that there must indeed be a motion of ions relative to the solveut, but of another type than that suggested by Falkenhagen: The transfer of force that takes place in any given volume element, and consequently the local flow density of ions relative to the solvent, will be proportzonal to the dzvergence o j the velocity gradaent. I n addition, there may be a transfer of force in the boundary layer; in the simplest case of a constant velocity gradient, the entire transfer of force takes place along the boundary. I .2. Present and Possible Future Advances in Mathematical Methods. In the present work, certain mathematical methods which are well known to other branches of theoretical physics will be applied for the first time to the theory of electrolytes. On this basis alone a brief survey of mathematical methods seems advisable and we have another, more important reason: while the scope of the present computations will be limited so as to obtain only the Onager: Phys. Rev., 37,405 (1931); 38, 2265 (1931). Grinnell Jones and M. Dole: J. Am. Chem. Soc., 51, 2950 1929); Falkenhagen and Dole: Z. physik. Chem., 6, 159 (1929); Physik. X.,30, 611 (1929); alkenhagen: Physik. Z., 3% 365, 745 (1931). 3 Falkenhagen: Physik. Z., 32, 365 (1931). 2

k


2693

first (&) term of the expansion (1.1.1) correct, we believe that a refinement of these calculations would be practicable, though laborious, and we have definite ideas about the procedure. The computation consists of two parts: first (in section z), the derivation of fundamental equations for the “ionic atmospheres,” simplified by certain approximations, and, second (in sections 3 and 4), the specialization and the solution of the fundamental equations for the different cases to be considered, namely the problem of viscosity and that of conduction and diffusion. The derivation of the fundamental equations will be complete in itself, and accessible to a reader familiar with the theory given by Debye and Huckel for the thermodynamic properties of electrolytes, which comprises the “statics” of the ionic atmosphere. We believe that a certain measure of simplification as compared to a previous treatise’ will be welcome. Full mathematical rigor is a distant ideal for the theory of electrolytes, but in our opinion, some advances towards this ultimate goal are possible.2 It is already very difficult to justify rigorously inevitable approximations, and to establish an upper limit for the error thus introduced. The “static” theory of an electrolyte in thermodynamic equilibrium has been carried further: but even there, the so-called “fluctuation-terms” in the differential equations are neglected: and the effect of these terms is very difficult to estimate. Tentative calculations of our own tend to show that the “fluctuation-terms” are of minor importance, but it would be necessary to go more carefully into the question. For the time being, it would be a great advance, in a certain sense a completion of the electrostatic theory (dealing only with Coulomb forces), if a method could be devised to compute the fields of the “ionic atmospheres” by a series of successive approximations, making it reasonably certain that the method would converge to the desired result, even though a mathematical proof of the convergence might be too much to ask. I n this sense, the original calculations of Debye and Huckel, and for the theory of irreversible processes those of the present work, may be considered as a first step in a series of approximations, which proceeds by developing the potential in the neighborhood of one particular ion, or a pair, triple, etc. of ions5 in a power series of its charge (their charges). The distribution-functions can likewise be developed in power series. The two sets of functions are combined (in the “static” case) by the Boltzmann principle and the Poisson equations, and the terms in the power series can be calculated successively, whereby each Oneager: Loc. cit. We are only concerned with the mathematical aspects of the derivation, and not aa yet with the quastion of desi ning a physical picture that will account properly for aU experimental results. Until we%ave mathematically reliable theory, leading to definite conclusions on the basis of a reaaonable icture, we do not know whether the experiments tast the ph sical picture or merely our magematical dexterity, or poasibly a deceptive compensation of toth. Gronwall, LaMer and Sandved: Physik. Z., 29, 358 (1928). R. H. Fowler: Trans. Faraday SOC.,23, 434 (1927); L. Oneager: 11, loc. cit. By considering the neighborhoods of pairs, triples,-etc. of ions, it is possible to evaluate the fluctuation terms.

2694


new term may add corrections to the preceding. This method looks promising from the point of view of convergence, because the functions to be calculated, qua functions of the charges, are probably entire functions, whose series converge like e'. It also seems probable that the method can be generalized to deal with irreversible processes, whereby the equation of continuity, which is the condition for a stationary state (in a statistical sense), replaces the Boltzmann principle. I n applying this method of computation to electrical conduction, a complication arises because the external force on an ion is proportional to its charge. The complication is only apparent, and may be disposed of by replacing the electrostatic force temporarily by an arbitrary field of forces, which is not supposed to be a function of the charge, until the necessary approximations have been made. This procedure, which will be employed in the following, is necessary in order that the simplicity and the self-consistency of the principle of approximation become evident. As a further means of emphasizing the uniform character of the calculations, all approximations will be introduced before specialization to particular problems: viscosity and conduction diffusion. I n fact, a starting-point for calculating conductance in strong electric fields (Wien-effect), as given by Joos and Blumentritt,' may be found in our ( 2 . 5 . 7 ) , which contains the approximations referred to, but which is not as yet specialized for weak fields. Likewise, the dispersion of conductance, as calculated by Falkenhagen, may be obtained from our calculations by retaining in the equation of continuity the left-hand member afldt (instead of O ) , throughout the derivation. In order to obtain a wieldy solution of the differential equations for the ionic atmospheres, we shall introduce matrix notation. Once this powerful mathematical tool is applied, we can deal with a system of linear differential equations as if it were one, and the intended generalization from an electrolyte containing only 2 species of ions to the case of s species becomes practically trivial. Moreover, this is the only way to obtain solutions that are formally compact, and even the methods for a numemeal cornputahon are most easily derived from the final results as they appear in matrix notation. For a possible future refinement of the calculations along the lines indicated above, it will be necessary to know the solution of the equations that appear in the first approximation for the general case of s ions, because the principle of the method is to develop the potentials, etc. pertaining to each ionic species in power series of their charges, which implies variable charge and therefore variable species. (In order to accomplish anything with the fictitious ions involved in this method of computation, we must be able to deal with real ones.)

+

1.3. Outline of the Physical Picture.

As shown by Debye and Huckel, the effects of the Coulomb forces between the ions on the properties of electrolytes depend on the electric forces preventing a random arrangement of the ions in the solution. Instead, each positive 1 Joos and Blumentritt: Physik. Z., 28, 836 (1927);Compare also Falkenhagen: Physik. Z., 32, 353 (1931).

2695


ion will surround itself with an “atmosphere” containing on the average more negative ions and less positive ions than the bulk of the solution, thus inducing a negative charge density in its neighborhood, while a negative ion will surround itself with a positively charged “atmosphere.” The total (mean) electric charge of an ionic atmosphere equals that of the central ion, with opposite sign, which suffices exactly to consume the force lines from the ion, so that its electric field beyond the bounds of the atmosphere is completely sbreened off. The most importint quantity characteristic of the ionic atmosphere is its mean radius, which can be calculated from the equilibrium between electric attraction and thermal agitation to be approximately I / K where

.

Here the electrolyte is supposed to contain inns of the species I , , ., s with the charges el, .,e, in the concentrations nl, . ., n,, while D denotes the dielectric constant, T the temperature, and k Boltzmann’s constant, the gas constant per molecule. For the potentiai in the neighborhood of an ion of charge e, Debye and Huckel have derived the formula

..

.

(1.3.2) which may be compared with fi = e/Dr for an isolated ion. According to (1.3.1), the inverse mean radius K of the atmosphere is proportional to the square root of the concentration n, or exactly to

.

where 21, , ., z, are the valences of the ions; hence the frequent occurrence of the power n’/*,as in (I.I.I), in the theory of electrolytes. For instance, on the basis of (1.3.2), the potential a t an ion due to its atmosphere will be

#.3 = - Kej, and this quantity determines the electrostatic contribution to the thermodynamic potential of an ion, which is =

f D

The numerical value of 291’ abs. amounts to

s e = ej

$(e)

S=O

K

K

de =

--*

.et

ZD

for a univalent binary electrolyte in water at =

0.332

x

1 0 ’ 6

where c is the concentration in mols/liter. Thus for c = 0.1, I / K = 10-7 cm. and for c = I , I / K = 3 x IO-* cm. Two other quantities of the dimension length are of interest in the theory of electrolytes, namely the diameter a of an ion, for small univalent ions in water usually in the range

2696

LARS ONSAGER AND RATMOKD M. FUOSS 2

X IO-^ < a < 4 X IO-^ cm,

and the distance do = e 2 / D k T ,

(1.3.3)

a t which two ions of charge e have the mutual potential energy IzT. For univalent ions in water, approximately, do = 7 X

IO-* cm.

The approximations employed in the following computations, as well as (1.3.z), will be admissible’ as long as the radius I / K of the ionic atmosphere is large compared t o a and do ; according to ( I . I , I ) , this is indeed the case for suffiicently low concentrations. I t is seen from the numerical data given above that for a univalent binary electrolyte in water, I / K = do for a molar concentration c = 0.2 and I / K = a for a concentration c near 1.0. We shall discuss one more quantity of general interest pertaining to the ionic atmospheres, namely the time o j relaxation. The importance of this quantity for irreversible processes in electrolytes has been emphasized already by Debye and Huckel. When the ions in the electrolyte migrate under the influence of an electric field (conduction) or a concentration gradient (diffusion), the symmetry of the ionic atmospheres will be disturbed in a manner that one may visualize by considering one ion under the influence of an external force: The ion will try to run away from its atmosphere, which becomes asymmet,ric with respect to the central ion. A velocity gradient present in a moving electrolyte will cause a somewhat different type of disturbance : In laminar motion, spheres are deformed into ellipsoids, and the ionic atmospheres will assume a corresponding asymmetry. Now the “normal” structure of an ionic atmosphere is an equilibrium between electric forces and thermal agitation of the molecules. These agencies are continually operating, and will tend to restore equilibrium after every disturbance. The time 7 that is necessary for this process of relamtion is given by the linear dimensions of the region in question, here I / K , and the coefficient of diffusion, which equals k T / p where p is the frictional force on an ion moving with unit velocity. On the basis of the general equation for diffusion

ac 2c --w _a _ at

ax?

the time of relaxation becomes

1 Our description of the part of the ionic atmosphere that is inside do (and a ) will be inaccurate; hut &s long as I / K > > do, the interior region (T ionary condition that (2.2.4) divl (fijvij) divz (fjivii)= 0.

+

The Equations of Motion. I n our problems, three effects contribute to the motion of the ions: ( I ) forces on the ions, (2) thermal motion, (3) flow of the solution itself. If pi = l / w i is the friction coefficient of an ion of the i-th species, then a force Ki gives the ion a velocity Kiwi. We shall use the generalized equation' for Brownian movement to calculate the velocit,y due to thermal motion. The diffusion constant is assumed to be k T w so that a concentration gradient will cause a current of strength-kTw gradf. Then the average diffusion velocity is - k T w grad In f. If the bulk velocity of the solution a t the point located by r2 is2 V(r2), then vji = V(r2) wt(Kji - k T grad2 Inhi) 2.3.

+

gives the total velocity vji which appears in (2.2.4). We have written Kji for the (mean) force which act,s on an i-ion in the vicinity of a j-ion. The equation of continuity specialized for a steady state now takes the form divl {fijV(rd wj(filKij - kT grad1 fd} (2.3.1) div2 (fjiV(rz) wi(fjiKji - kT grad2fji)) = 0, where f,i = r d , fij = f i j ( r 2 , r12) Kji = Kji(rl, r d , K, = Kij(r2, r d .

+

+

+

1 Debye and Huckel: 11, loc. cit. 2 We here neglect the hydrodynamic motion of the fluid at dVz caused by the presence of a moving j-ion a t d VI, This approximation amounts to dropping terms of the order c log e in the final results.

IRREVERSIBLE PROCESSES IN ELECTROLYTES

2701

2.4. The Results of Debye and Huckel f o r an Undzsturbed Electrolyte.

For the case of an electrolyte a t rest under the action of no external forces, the ionic atmospheres are spherically symmetric with respect to the central ion, and the functions introduced above become pure functions of the distance r = r21 = r12 We therefore write

I

I

I 1.

f oj2(r)= n3n03% (r) =

woe,

(T)

=

fot3(~)

(2.4.1)

to indicate this situation, An ion of charge e, in a liquid of dielectric constant D produces a field derivable from the potential e,/Dr. I n an electrolytic solution, the fields due to the various ions superpose and any particular ion will be screened by the ions in its atmosphere. We designate by $,(rl,rZl) the potential due to an ion and its atmosphere in the general case and by PJ(r)the same for the undisturbed case. The potentials are connected with the charge density

I:n,& I

by the Poisson equations =

-

njiei D i

(2.4.2)

If we assume that the concentration noji of i-ions near aj-ion is determined from the energy1 Eji of an i-ion in the vicinity of a j-ion by the MaxwellBoltzmann distribution, we have noii = ni exp(-Eji/kT). For this equilibrium, the average velocit,ies must vanish, so that

(2.4.4)

Kji = Kij = kT grad Infoji. From (2.4.4), we find kT grad lnfoji = -grad Eji = -grad E,.

(2.4.5) (2.4.6)

So far, no approximation has entered. If now we assume linear superposition of the potentials and consequently of the charge densit.ies induced by two ions near one another (superposition of the atmospheres), we assume that Eji = eif'j and Eij = ejVi. In other words, the average force acting on one ion is given by the potential of the atmosphere of the other. Then from (2.4.6), we see that our assumption leads to the result that

eiPi = ejVi * Eji is most sim

ly defined by Kji= -grad& to be the potential of the force Kji; according to (2.4.5), t h s force haa indeed a potential. It is also possible t o define E$ >do the inaccurate description of the innermost region will cause a negligible error in our final results. Integration of (2.4.8) yields, considering the boundary conditions,

ej e-rr J.5 = _.

D r

(2.4.I 0)

This procedure was developed by Debye and Huckel. With the above a p proximation (2.4.7), the distribution function becomes

j".. = njnoji= njni(I - e i V j / k T ) = 1% = njni( I - doe+* = njni ( I

T)

(2.4.11)

Perturbation of the Ionic Atmospheres. If a force acts on the ions in the solution under investigation, then there exists a preferred direction, and both distribution functions and potentials become asymmetric. We shall assume that the applied forces are small, so that we may consider the various functions to be changed by a small perturbation factor from their values in the undisturbed electrolyte. Accordingly, we set 2.5.


2703

For small forces, the (primed) perturbation terms will be proportional to the force. The asymmetric addition to the potential is connected withyji by the Poisson eauation:

The perturbing force sets up an irreversible process in the solution, and the distribution naturally is no longer given by the Boltamann equation. Hence the general condition (2.2.4) must apply, since vji no longer averages to aero. I n accordance with the approximation principle of $2.4, we shall assume that the asymmetric addition to the field of a given ion is proportional to the charge of the ion, and that we may compute the field due to two ions by linear superposition of the separate fields. For the total force $ i acting on an i-ion in the vicinity of a j-ion, we then have I(1% -=

&

- ei grad2 +i

- ei grad2 1c;. (rl, nl),

(2.5.4)

(0)

where ki is the applied external force, -ei grad2 #i(o) is the force due to the ion's own atmosphere and -ei grad2 fij is the force due to the j-ion and the latter's atmosphere. If we substitute (2.5.4) in (2.3.1)~we obtain for the first term div2 (Ami) = div2(AiV(rz) w&kd - eifji grad&(o)

+

-e&

grad2 &(r1, rZl) - kT grad2 Ai)).

(2.5.5)

I n order to expand this, we note that V(r2) and ki are divergence-free, and make the following simplifications. We neglect the term grad&(o) since (of. 2.4.10) it is of the order e? while the other terms are of the order ei. We replace fji in the grad2 II;. term by njmd since - njni ei and the term already has a coefficient e;. We thus obtain div2 (fiivji) = ( V b ) , gradzhi) wi(ki, grad2 Ai).

-

+

-eiwininiA&

- wikTA2 Ai

(2.5.6)

We now substitute (2.5.1) and (2.5.2) and note that for k, = 0, vji = 0 ; hence the terms i n f o and $' must, cancel one another except in the two scalar product terms which vanish by hypothesis. We approximate f in these terms We thus obtain the equation of continuity (2.2.3) specialized to suit our by f particular problems in the form

".

+

+

( V W , grad1 f"ij) (V(rd, grad2 f'ji) w&, grad2f'ji) w&, grad1 f'ii) - eiwininjAz#'j -e.w.n.n.A 1 1 1 3 1+'i - wikT A2f'ji - wjkT Alf'jj

+

=

0.

(2.5.7)

The first four terms of this equation contain the perturbing factors V and k ; the last four represent the asymmetric contributions to potential and dis-

2 704

LARS ONSAGER A N D RAYMOND M. FUOSS

tribution thereby produced. This gives us a system of s2 differential equations connecting the s2 distribution functions f:>with the s potentials and the known function f " ( r ) . Our general procedure in subsequent derivations will be to specialize (2.5.7) to a particular problem, eliminate thef:,'~ by means of the Poisson equation ( 2 . 5 . 3 ) and solve the resulting system of differential equations for the s potentials +',. 3. Viscosity Descrtption of Problem and Method. I n this section, we shall consider a solution containing s species of ions and shall derive a relation giving the viscosity in terms of the various concentrations. I n order to define clearly the physical conditions of the problem, we shall review briefly the theory of motion in a viscous fluid, bearing in mind the fact that the fluid in question is an electrolytic solution, and shall first show qualitatively how the electrostatic forces between the ions will add to the forces transferred in such motion and estimate their order of magnitude. We picture the solution (see Fig. 2 ) confined between two parallel plates, h centimeters apart, of which the lower is fixed and the upper moves with constant velocity v' in the z direction, and shall assume for the moment a constant velocity gradient 3.1.

in the y direction, so that the velocity a t the level y will be

v,

=;

(v'/h)y; vu =

0,

vz

= 0.

Recalling the definition of coefficient of viscosity 7 as the stress transferred per unit velocity gradient, a stress equal to

per unit area will be transferred from each layer of liquid to the layer beneath it. In the case of an electrolyte, we shall show that a part of this stress is due to electrostatic forces between the ions. I n an undisturbed electrolyte, an ion of charge +e will be surrounded by an atmosphere of total charge - e , situated a t an average distance (I/K) in which the average distribution is spherically symmetrical since the ions have no preferred orientation. A velocity gradient in the liquid will deform a sphere into an ellipsoid, and the ionic atmospheres will be deformed accordingly. The electrostatic forces and the thermal motion, whose equilibrium determine the structure of the ionic atmospheres, will tend to restore the original structure, but this process of relaxation will not be instantaneous. If the relative rate of deformation is


2705

a stationary deformation r

9 will persist. dY

and the time of relaxation is

7,

I n (1.3.4)~ it was shown that T = p / ( ~ ' k T ) . The deformation of the ionic atmosphere will then be of the order

Two ions of charge e, situated at a distance ( I / K ) exert a Coulomb force on each other equakto e2K2/Dwhere D is the dielectric constant, so that the total

dVL -

FIG.2 Sketch for Viscosity Calculation

transport of force (stress) between an ion and its atmosphere without regard to direction is ( I / K ) . (e2K'/D) = e?K/D. As long as the spherical symmetry of the ionic atmosphere is not disturbed, the stresses in different directions will compensate each other. A velocity gradient in the solution will cause a deformation of the order (3.1.1) as shown above, and we may expect that the resultant directed fraction of the total stress will be of the same order of magnitude. On this basis, the stress transferred between one ion and its atmosphere should be of the order

and that transferred by the interaction of the order

212

ions and their atmospheres of

ne'p-dv, K D ~ Tay

or, substituting (2.4.9) for

K?

and allowing a numerical factor 8 ~ ,

2706


This result is expected to be correct except for a proportionality factor which must be evaluated by considering the exact type of deformation of the ionic atmospheres. Indeed, our final calculations yield for the case of s species of ions, where w1 = w2 = , . . = w, = w = ~ / p , the stress I av, S",,= PK - *

ay

480a

(3.1.3)

This stress will be added to the stress transferred by friction between the solvent molecules, namely

Consequently, the electrostatic forces cause an addition

to the viscosity 7l0 of the pure solvent. The result of the above consideration is identical with that obtained by Falkenhagen.' The essential feature in (3.1.3) is the factor K ; on account of this factor the electrostatic contribution to viscosity varies as the square root of the concentration of ions. For example, for KCl at zs0C. in water, ? * = 44.7

x

= 5.0

I O " ;

x

10-3qo&.

For 0.I n KCl, the electrostatic increase of viscosity amounts to 0.14% of vo. The electrostatic forces between ions will be transferred to the solvent by friction, whereby theremust be a relative motion between ions and solvent. This relative motion may take place (in part) in the interior of the solution; however, as we shall show immediately, in certain cases, a relative motion of ions and solvent will take place only along the boundary, namely when the velocity gradient is constant throughout the interior. If we now consider the general motion in a liquid where the velocity v at a point located by the vector r has components v,, ,o, v,, the y component of stress per unit area across a plane perpendicular to the z direction is

For a solution of electrolytes, two factors will contribute to the force transfer: first, the friction of solvent on solvent, and second, the interaction of the electrostatic forces between the ions. I t is the latter which we wish t o investigate. The equations of motion of an incompressible fluid are qAv = grad p

div v 1

Falkenhagen: LOC.cit.

-F

= 0,

(3.1.sa)


2707

where p is the pressure and F the force per unit volume. If Somnis the general element of the stress matrix for motion in the pure solvent, then

s,,

-

somn= s*,,

represents the average superposed stress due to ionic forces, and correspondingly q - q o = q* (3.1.6) gives the increase in viscosity of the solution over that of the solvent. On introducing this into the equation of motion, there results qo Av = grad p - (F

+ q* Av),

so that the effect of interionic forces may be interpreted as an addition to the volume force. Furthermore it is evident that wherever Av vanishes in an interior region of the liquid, no force will be transferred from the ions to the solvent in that region. I n the case of a constant velocity gradient, as in our example above, Av = o everywhere in the liquid. Still, the forces acting on the plates A and B (per unit area) equal ( & ) q C3v,/t3y and not qOau,/dy. The difference q* av,/ay can be accounted for if we assume that the ions move relatively to the solvent along the boundaries (up to distances I/K) and nowhere else, with a velocity that is sufficient to produce the required frictional force. The flow of ions along a boundary is not easily calculated by direct methods although our consideration above shows that such an effect must be present. However, we shall calculate directly the relative motion of ions and solvent in the interior of the solution, that is, when Av # 0, and we shall show that the forces derivable from the potentials of the asymmetric “odd”l ionic fields agree with the term q*Av calculated from the stress S*,. The contribution S*,, to the stress due to the presence of ions will be computed by the following method: By means of the Poisson equation (2.5.3) and the equation of continuity (2.5.7), we shall derive a system of differential equations which determine the potentials of the ionic fields, and give their solution for the general case of a mixture of s species of ions. Then E*,, will be computed as the average sum of the corresponding components of the electrical forces between all pairs of ions present, which will require the distribution function f3%(rZ1) = ftJ(r1Z),giving the average concentration of i-ions in the vicinity of a j-ion and vice versa. By means of the Poisson equation, however, f32(r21) may be replaced by the potentials which have been obtained by solving the differential equations, and we shall have in explicit form the result desired. 3.2. Evaluation of Velocity Terms i n the Equation of Continuity. I n accordance with the considerations of the previous paragraph regarding the nature of the velocity field, we shall expand the velocity as a Taylor series in the coordinates and neglect terms of higher order than the second, SO I n the case of a constant velocity gradient, the potential in the ionic atmosphere is an of the distance in the sense that $(s,y,z) = $(-z,-y,-z). A variable gradient w i l l cause “odd” contributions to the field of the atmosphere. 1

eoea function

2708


that the velocity gradient will, t o this approximation, be a linear function of the variables q1 = 2, q2 = y, 43 = z . We then have for the general component of the velocity at a point located by the vector r

For convenience] we shall use the convention that Greek indices always imply summation, and introduce the following abbreviations:

so that our expansion may be written compactly as

d(r)

+

= ~'(0) a i qx

+ 3 b:,

, we have the following data: IOn

3

m3 = ,'I

R

A "a

H+

I

0.02

0. I

K+

2

0.08

0.4

C1'

3

0.10

0.5

315 64.6 65.5

0.8298

w12

=

W1s

= 0.8279 = 0.4965

028

The matrix H is

I

0.0681 0.7348 0.1986

0.2541 0.0830 0.0828

and the invariants are

I1 = 1.6712

1

+ O.ggIOp1 = 1.7703

0.7567 4-1.3772 Then, using (4.9.4), (4.9.7) and (4.9.8) I 1

0.0860 0.2518 0.7814

=

-

+ 0.2186 p12 = 0.8966. -

(I HI") = 0.7080 0.8217 H. Since we are dealing with symmetrical electrolytes, the second term of the vectorrin (4.7.11)vanishes~ andrhasthecomponentsrl = l , r a = 1 , r 3 = - I . The product (4.9.4) is now readily evaluated:

- 0.8217

(I = 0.7080

HI/')$,

r, = 0.7080 rl

(0.2541

(I - H r / 2 ) z u r ,= 0.2429 and (I By (4.7.13) and (4.6.7), with r = 0.2,

- H1/2),uru

- - Ax' = 0.5416 I"/'z',(I

=i

X

- 0.8217 HI, r,

+ 0.0681 - 0.0860) = 0.5139

- H1/')3,,rU = -0.zg71.

0.24232', (I

- H1/')3,rr.

(4.10.1)

Now AXJX measures the relaxation correction to the conductance. Since the electrophoretic correction is independent of pl, we shall find it most interesting to confine our attention to the first correction. For either pure HC1 or KC1 a t a concentration I' = 0.2,

-AX,lX for each ion.' Oneager: 11, equation (65).

0.159

rl/' = 0.0711


2754

We have tabulated in t'he first column below AXJX calculated by means of (4.10.1)for the mixture of HC1 and KC1 under consideration; the second column AAj/Aoj gives the difference between these figures and 0.07 I I , reckoned as percent change of conductance, and the third column the corresponding changes Ah; in conductance. TABLE I j AXj/ X 100AAj/hoj AAj I

0.1245

- 5.34

0.0589

+I.22

-0.09

0.0720

3

- 16.8

+- 0.79 0.07

Assuming that our limiting laws (4.8.1)were valid at a concentration = 0.2, the conductance of hydrogen ion in the 4 to I mixture would be 16.8 units less than in 0.1normal HC1 (r = o.~),t'hat of the potassium ion would be increased by 0.8 units over its value in 0 . I normal KCl and that of the chloride ion decreased by about 0.1 unit. Actually, this concentration is in general too high for applying the limiting formulas of the electrostatic theory, but we may assume that the difereizce in conduct'ance in a mixture and in a solution containing a single solute at' the same ionic strength would be given by (4.8.1) to the right order of magnit'ude at least, and that the calculated difference for a given ion would agree more closely with experiment, the lower the total concentration of ions. We shall see in the following section that these predictions are verified. For s = 4, the equation jhji - Sji ( 1 = o which determines the characteristic numbers is of the fourth degree, and alt,hough one of these numbers = I) is known, it is not convenient in general to compute the ot'hers. Expansion of the function (I-HI") in a series will prove to be the most satisfactory method. We shall illust'rate this method by computing the conductance changes a t 18' in a solution containing 0.04995 mols HC1 and 0 . 0 5 0 2 mols KC1 per liter; the method for s = 4 is entirely analogous. Our data are then as follows:

(zl

rj

Ion

j

mj =

rj

h0j

H+

I

0.04995

0.2495

Kf 2 Cl' 3 The matrices are

0.05016

0.250j

0.1001

0.5000

315 64.6 65.5

(h;J

(zhji

- Sji)

=

0.0860

0,3782

0.0426

0.2070

0.7094

0.2518

0.2066

0.1244

0.8310

-0.~436

=

(

0.0853

0.1721

0.4140

0.4188

0.4132

0.2488

0.5036 0.6620

By means of the recursion formula (4.7.12) we now compute the vectors dn), for example T ( I ) L = (2H

-

I)L~).,,=

(-0.2346

+ 0.0853 - 0.1721) = -0.3304,


2755

The following table gives the first four:

TABLE I1 n

r(d,

0

+I

I

-0.3304 0.1086 -0.0358

2

3 4 The sums

0.0118

r(4,

r(n),

SI 0.3292 0.0011

0.0180 -0.00~7

-1

0.0000

-0.0546 0.0091 -0.0043

2 cn r(n)jmay now be evaluated, using the values of the coefficients ~

n =o

c, given in (4.6.8); the first five terms of the series already give four place accuracy. The results of the calculations are given below, where the symbols are the same as those used in Table I. Ion

Hf K+ c1'

j

TABLE I11 AXj/X 0.4212 0.1020 0.1758 0.0426 0.2982 0.0723

z'&ni"(n)j I

z

3

IoOAAj/A"j -3.09 +2.85 -0.12

AA,

-9.72 f1.84 -0.08

The change of hydrogen ion conductance PA1 for an infinitesimal amount of HCl in 0.1normal KC1 is readily obtained by setting pl zero: the following figures are self-explanatory: (O.I:IZ 0.0851 0.0861 (hJJ = 0.7518 0.2518 0.2483 0,7483 11 = 1.6712; 12 = 0.7567 (I-HI") = 0.7391 0.8922 H = a bH

-

j

Ti

I

I

2

3

I

-1

H,,r,

0.1702 0.5000

0.5000

TABLE IV z'j(ai-bH)j,r, Axj/X 0.5872 0.1423 0.2930 0.0710 0.2930 0.0710

+

IOOAAj/A"j 7.13

AAj

22.4

0

0

0

0

Our result is that the limiting equivalent conductance of hydrogen ion in 0.1 KCI would be 22.4 units less than the conductance of hydrogen ion in 0.1HC1. This is the result calculated by Bennewitz, Wagner and Ktichler who have given a solution of the general equation (4.4.2) for the case s = 3, valid when one of the ions is present in infinitesimal amount. 4. I I . Comparison with Experiment. The change of conductance due to the "mixture effect" has been computed for several sets of published data. Bennewitz, Wagner and Ktichler' have de1

Bennewitz, Wagner and Ktichler: LOC.cit.


2756

termined the transference numbers of Ba++,H+, K+ and C1’ in various mixtures of BaClz and HC1 and of KC1 and HCl a t 18’. The concentrations used in our two examples in the description of the method of computation were taken from their data. Table V compares their observed results with our computed values: TABLE V 0.05 KC1 .- 0.05 HC1 0.08 KC1 - 0.02 HC1 obs.

cazc.

AAi(H+) A A 2 (K+)

-9.72

-10

obs.

-12.8

+ 0.8

+o.g

+1.84

calc.

-16.8

-

Longsworth’ has measured the migration velocities of individual ions in mixtures of HC1 and KC1 at 25’ in which the chloride ion concentration was kept constant at 0.1 normal (r = 0 . 2 ) . Table VI compares his resultswith those calculated2 by means of (4.8.2).

TABLE VI H1

calc. 0.jooa

0.375 0.250

0.125 0.000~

pure HC1

AAi(H+)

0.00

-5.14 -10.81 -17.24 -24.79 b

obs.

-

-2.56 -5.06 -8.06 (-10.96) pure KC1.

AAz(K+)

calc. +3.86

3.00 2.09

obs. (2.36)

AAa(c1‘) calc. 0.000

1.10

2.06 0.96 0.56

0.063 0.097 0.086

0.00

-

0.000

A3

obs. 66.32

66.19 66.07 65.83 65.75

The “observed” values in parentheses were obtained by Longswort,h by linear extrapolation of his data. Fig. 4 summarizes the calculated and observed results for H+ and Fig. 5 for K+. Bennewite’ data are indicated by crosses, Longsworth’s by circles and the theoretical curve is drawn solid. The theorebical curve depends on the mixing ratio in a rather complicated way; both graphs exhibit a slight curvature. Longsworth’s results lie on a smooth curve which is approximately of the same form as the theoretical, although due to the high concentration, the numerical values differ by about a factor of two. Bennewita, Wagner and Kuchler obtained their results by the Hittorf method, while Longsworth employed the method of moving boundaries. The latter method is probably capable of greater accuracy, and in fact,, the effects measured by Longsworth fall beneath the calculated values by amounts that might reasonably correspond to the mathematical approximations allowed in the theory. Longsworth: J. Am. Chem. Sac., 52, 1897-1910 (1930). We used the following values in order to compute the results for the 25’ experiments: A,(HCl) = 426.0,A,(NaCl) 126.42,A,(KCl) = 149.84;t K (KC1) = 0.4905. tk 5 limiting transference number of K+ in KC1. We are indebted to Dr. MacInnes, Df. Longsworth and Dr. Shedlovsky for permission to use these figures in advance of their publication. 1 2

-


00

2757

05 FIQ.4

Variation of Hydrogen Ion Conductance in HC1-KCl Mixtures

Fig. 6 shows the calculated mixture effect for the common ion (Cl' in HCI and KCl); the effect is so small that it was necessary to magnify the scale ( 5 0 times compared to figures 4 and 5). The maximum effect is apparently within the experimental error, and as seen from Table VI, is masked by a specificdifferencein Acl between HCI and KC1; we have therefore tabulated A3 instead of PAS. As a final example, we have chosen the data of Bray and Hunt' on the specific conductance of mixtures of NaCl and HCI a t 25'. If there were no mixture effect,one could compute the specific conductance K of a solution containing ml mols NaCl and m2 mols HC1 from the conductance A N ~ of c ~NaCl in ml normal solution and that of HC1, AHC,,in m2 HC1 by the formula

Fm.5 Variation of Potassium Ion Conductance in HCl-KCl Mixtures 1

Bray and Hunt: J. Am. Chem. Soc., 33, 781 (1911).


2758

These values have been computed from Bray and Hunt’s data on pure NaCl and pure HC1 solutions. Then the observed specific conductances were subtracted from the additive conductances, to give “ A Kobserved”] the deviations from additivity due to the mixture effect. By means of (4.8.2)] we calculated the change of conductance due to the presence of 3 kinds of ions as before, and then computed the sum mlAhl m 3 A h msAAa as ‘(AKcalculated.” The results are given in Table V I I : (concentrations in millimols).

+

+

% 0.10

G!00 0.5

0.0

FIG.6 Variation of Chloride Ion Conductance in HCI-KC1 Mixtures

TABLE VI1 Conc. HCI

Conc. NaCl

A K X 103 obs. 0.94

AK X 1 0 3 calc.

70

70

obs.

calc.

1.56

1.99 I.72

3.31 3.29

0.26

0.95 0,45

1.44

2,50

0.15

0.24

I .os

0.I2

0.64

I .68 0.98

I00

0.08 0.04

0.05

0.35

0.42

50

20

0.17

0.248

0.78

20

20

0.139 0.085

I .02

1.14 I .38

0.040

0.98 0.88 0.68 0.40

1.35 I ,03 0.71 0.41

IO0

IO0

50*

IO0

0.50

20*

IO0

IO

I00

5

IO0

2

IO

20

0.103 0.061

4

20

0,034

2

20

0.021

0 021

I

20

0.011

0.011

5 5 5 5

0.028

0.50

0.55

0.010

0.0311 0.0174 0.0106

0.76 0.62

0.66 0.65

0.005

0.0050

0.49

0.49

12.5

5 2.5 I

0.020


2759

The two experiments marked with an asterisk are in about the same range of concentration and mixing ratio as the KCl-HC1 mixtures discussed above, and again the calculated result is about twice the observed. It will be noted, however, that the calculated figures agree more closely with experiment, the smaller 1.1~and I? become. Also, the calculated values for AK duplicate the observed flat maximum of AK at the equal mixture of NaCl and HC1. The experiments of Bray and Hunt indicate that if the individual mobilities in a mixture were measured a t low concentrations, not only the type, but also the absolute magnitude of the mixture effect would be found to agree with the theory.

The General Equations for Diffusion. Fick’s law for diffusion in a two-component system is ordinarily written

4.12.

c v = J = -Dgradc, (4.12. I) where D is the coefficient of diffusion, v the velocity, J the flow and c the concentration (amount per unit volume) of solute. The flow is usually defined relative to a fixed plane in the solution, which is assumed “at rest.” This convention has the esthetic advantage that no distinction is made between solute and solvent, both obeying (4.12.1) with the same coefficient of diffusion D. Moreover, the ideal case of “self-diffusion” (interdiffusion of isotopes) is most simply described in this form; D becomes independent of the mixing ratio. I n a general case, however, it is somewhat awkward to define a solution “at rest” during diffusion, particularly when the mixing of the substances involved causes an appreciable change of volume. We must bear in mind that the concentration gradient determines only the local relative flow of solute and solvent, and that a bulk motion of the solution, dependent on other conditions, may be superimposed. In the following, we shall amume that one (solvent) constituent of the solution is present in dominating quantity. The flow of any solute constituent will be defined relative to a local frame of reference moving with the solvent. This simpler convention seems definitely preferable in the case of electrolytes, which are in any case hardly amenable to a Symmetrical consideration of solute and solvent. If a symmetrical formulation is desired, one may define a “bulk velocity” v = I: JiV, = I: c , v ~ V ~ , (4.12.2)

-

- -

.

-

.

where ( V,),V1, . , V , are the partial vulumes of the constituents (o), I, . . , s; apparently there is no reasonable alternative to this choice. We may assume that either the velocity of the solvent or the “bulk velocity” (4.12.2) obeys the hydrodynamic laws; the difference is of no practical importance. Fick’s law (4.12.1) may be written in the alternative form

J = -Qgrad 1.1,

(4.12 - 3 )

where p is the chemical potential, and the coefficient of diffusion equals


2760

In the special case of an ideal solution

ap/ac = R T / ~ , so that

RT il In general,

(4.12.4)

=

cD.

may be rewritten in terms of activity coefficientsf: c~

=

+ caiogf/ac).

RTQ ( I

(4.12.5)

I n the case of electrolytes, the correction term involving f is negative, except in some cases at relatively high concentrations (rf. 0 4.13). It has been suggested that the variation of D with concentration might be accounted for entirely by this deviation from ideal thermodynamic behavior, and that Q/c = const. As we shall see in the following, the theory predicts that O/c will also vary with the concentration, although much less than D. According to ( 4 . 1 2 . 3 ) , the ratio Q/c equals the “mobility,” that is the velocity caused by a unit force k = -grad p, (cf. 4.1). The natural generalization of (4.r2.3) to the case of several solute substances is Jz = - 2 Oik grad pk, (i = I , . . . , s), (4. I2.6) 9

k-I

assuming a linear relation between velocit,ies and potential gradients for the solute substances I , 2 , . . ., s. The matrix of coefficients (&) is symmetrical Qik

=

a&.

(4.12.7)

This reciprocal relation states bhat the flow of constituent no. 7 caused by a unit force per unit amount of no. 3 will equal t’heflow of no. 3 caused by a unit force on substance no. 7. Relations of this type are included in the “principle of the least dissipation of energy,” which has been derived’ from the assumption of microscopic reversibility. We shall show a t the end of this section, assuming the principle of the least dissipation, that the symmetry relation (4.12.7) applies independently of the frame of reference for the flow, which is not quite obvious. By solving t,he system of equations (4.12.6) for grad p1, . ., grad pa, we obtain an alternative formulation of some interest, namely

.

s

-grad pi = Z Rik Jk, (i =

I,

k=i

.

~

s),

(4. I 2.8)

where the coefficients (Rik)form the reciprocal matrix of (ai&), and on account of the symmetry of the latter, R,k = Rkj. (4.12.9) Thanks to this symmetry, we only need to know the dissipation-function s

2

F(J,J) = 2

i.k=I

Rik

(Ji,Jd

=

i.k

and (4.12.8)may be written grad pi dF/aJi

+

1

L. Onsager: Phys. Rev., 37, 405 (1931).

n i n i , R i k ( ( ~ i - ~ o ) , ( ~ k - V u ) ) , (4.12.10)

= 0,

(i =

I,

. . ., s),

(4.12. I I)


2761

or in a variation-principle for convenient transformation

S(TS where

k is the rate of

- F)

=

(4.12.12)

0,

production of entropy; according to thermodynamics

T dS/dt

=

Ti = - 2 (Ji.grad pLi).

(4.12.13)

The formulation (4.I 2.I 2) satisfies all esthetic requirements. I n the case of an ideal solution containing solute molecules of the species I, . s, the solute molecules might be expected to diffuse independently according to the law Ji = -Qii grad pi, (i = I, . ., s); (4.12.14)

..

.

it also seems probable that the mobilities Qii/ci would not vary more with the composition than the fluidity of the solution. I n the case of a solution that exhibits deviations from the ideal laws due to compound formation between the solute substances, we should expect that one migrating substance could carry another along, and that the coefficients Oih, ( i # k ) , would not all vanish. On this picture, if the simple law (4.12.14) is valid for the different species of molecules and molecular compounds actually present, the reciprocal relation (4.12.14) evidently holds. In many inst’ances, howeyer, attempts to describe all deviations from ideal laws in terms of molecular compounds lead to awkward difficulties, and the interpretation is usually ambiguous. Therefore, where a noncommittal phenomenological description will not suffice, it is preferable, when possible, to construct a more detailed physical picture, and to allow explicitly for the forces between the molecules. This applies particularly to electrolytes. In dealing with electrolytes, it is convenient to describe conduction and diffusion by one combined scheme, so that (4.12.6) now relates the rates of flow J1, . ., J8 of ionic species I, ., s to the gradients of the total potentials

.

..

pi = pi’

+ eip,

(4.12.15)

of the ions, where p is the electrostatic potential, (cf. $ 4.1). The generalization to the case where the electric field does not possess a potential is obvious. In the limit of low concentrations, the ions migrate independently (ideal case)

Oil, =

0, (i #

k);

3.. = n,w. 22 z e,

where wi is the “mobility” of a single ion (in this notation, the electrolytic electrolytic mobility is eiwi where eLiis the charge). At higher concentrations as shown by our calculations in @ 4.1-7,the ions will interfere wit.h each others’ motion; the order of magnitude of this effect is given by

O(nink 4awdo2/K) * O(ninkd,/Kq), with do = leiekl/DkT and otherwise the notations of $5 (1.3) and (4.1); the first term is due t o direct transfer of forces, the second to electrophoresis. There are similar corrections to Oii: Qii = niwi - O(niwiKd,) - o(n?do/Kq). oik

2762

LARS ONSAGER AND RAYMOND M , FUOSS

Since K-n'", the relative effects of interaction are proportional to the square root of the concentration, in the sense that doubling all concentrations increases the effects by a factor d;. The symmetry condition (4.12.7)for (aik)holds for the migration of ions as well, if we allow the assumption of microscopic reversibility. While this assumption is nowhere explicitly introduced into our calculations, we shall see that the results conform to the reciprocal relation ( 4 . 1 2 . 7 ) . The reason is, of course, that the assumed mechanism of int,eraction between the ions is in accord with the general principle. Indeed: The forces between the ions possess a potential, and so does the Brownian motion, because the relation of neighhorhood is a reciprocal one; the hydrodynamic interaction of ions also obeys the same reciprocal relations because the principle of the least dissipation of energy' is valid for (slow) motion of viscous fluids. The equations for migration of ions may be given the form (4.12.8) or (4.12.11-12);in the case of ions, PI, . . .,pa in (4.12.13) &and for the total potentials (4.12.15). The results of our calculations will be summarized in a very compact form by the construction of a dissipation-function. We have promised to derive the reciprocal relation (4.12.7) from the principle of the least dissipation. We shall prefer to obtain its equivalent (4.12.9). Let us consider a case of parallel flow in unit cross-section, so that the volume element dV = dx. The flow Ji is defined relative to a frame of reference (g,y,z) which may move with the velocity of the solvent, or with the "bulk velocity" ( 4 . 1 2 . 2 ) ; in the latter case we could assume $ = 2. In any case there exists a dissipation-junction 2 F 2 T9 E f Z R,kJiJb dx. In order to apply the general principle (4. I 2.I 2 )

S(S we must express

A in the variables

@) =

Ji.

0,

For this purpose, write

nkdx/d$ = Ark,

then N k d < is the amount of k ions contained in the deformable volume element d $ . According to thermodynamics

where S*, corresponding to the term in brackets, denotes the external entropy change (omitted above for simplicity). Now the equation of continuity

is valid in deformable coordinates as well, hence 1 This principle may be stated: When a force of magnitude X, applied at a point A in the direction ( I ) , will cause a velocity v in the direction (2) a t the point B , then conversely, the force X applied at B in the direction (2) will cause a velocity v in the direction ( I ) at A.


= 2 [~ckJk]$ -

and

&:

Jk

2 763

dPk

+

T ( $ $*) = - J$ Z Jk dph = - f $’ Z Jk dpk dx. Now the principle of the least dissipation yields 0 = T S($ $* - $) =

+

whence

where R z k = Rkit

because the linear form is obtained by differentiation of a quadratic. 4.12. Diflusion of a Simple Electrolyte.

In an electrolyte containing only two kinds of ions, the condition for no electric current is that anions and cations migrate with the same velocity VI =

~2

J1 = niv;

= V;

J2 = n 2 ~ .

(4.13.1)

I n the limit of low concentrations, the interaction between ions (except the space charge that prevents an electric current) may be neglected, and we have

v = klwl = kzw2 = -wl grad pl = - w 2 grad p z . (4.13.2) I n this “zero” approximation, the “forces” kl and kz on the individual ions are given by (4.13.2) and k = vlkl v2kz = -grad where p is the chemical potential of a molecule dissociating into v1 anions and vz cations. We obtain

+

kl =

+

grad p =

-pl VIP1

e,

VZPZ

+

grad p ;

-wz VlW2

Y2Wl

kz = (pz/pi)ki = (Wi/Wz)kz,

(4.13.3)

and

v =

- w1wz VlWl

+ vzwz grad p.

(4. 13.4)

Here, if the solution is assumed ideal in regard to its thermodynamic properties, so that grad p = (VI VZ) ( k T / n ) grad n, Nernst’s formula’ J = nv = - (” ”) wlwz kT grad n (4. ‘3 * 5 ) VlW vzwz

+

1

Nernst: LOC.cit.

+

2764


results. When the forces between the ions are taken into account, the main effect consists in a contribution to the thermodynamic potential; we have now grad p = (vl

+

YZ)

kT grad log (fn),

(4. 13.6)

where f is the mean activity coefficient of the ions, for low concentrations‘ given by the limiting law log f = elez ~ l 2 D k T . (4.13.7) We thus allow for the interionic attraction reducing the “thermodynamic” tendency of the ions to separate by diffusion; it remains to compute the effect of interaction on the mobality of the migrating ions. We first note that the diffusion of a simple electrolyte will not be obstructed by the effect which Debye and Huckel named “ionic forces,” the retardation of a migrating ion by an asymmetric field in its atmosphere. The reason is that all ions migrate with the same velocity, so that an atmosphere needs no coercion to follow its central ion; in consequence, the atmosphere is displaced as a whole, undisturbed, and develops no asymmetry. This can be verified already in our fundamental equations (4.3 . z ) by substituting

kiwi = kzwz; the perturbation terms in (4.3.2) then cancel each other. (In calculating the first approximation for the effect, we may compute the perturbation terms from the “zero” approximation (4.13.z).) Of course, our final formulas for the “ionic forces” also yield a vanishing effect for this case; we may refer to formula (4. I 4. I 2) below. Otherwise with the electrophoresis, this effect depends on a volume force attacking in the ionic atmosphere. According to (4.2.2) and (4.2.6), if the potential at a distance r from a j ion is +,(r), then the force density at this distance is F = - (nlelkl nzezkz) (rC;lkT) (nle?kl nze2kd (+JlcT)z- . . (4.13.8)

+

+

+

.

Substituting (4.13.2), the first term in (4,133) vanishes when and only when w1 = wz; the second term is always positive. I n $ 4 . 2 , we calculated the effect of the volume force (4.13.8) on the motion of the ions (assuming k, = vp,) and found for the resulting additional velocity (4.2.12 )

(4. 13.9)

The ultimate result of this effect is of course to alter the forces kl and kzthat are needed to make both ions migrate with the same velocity v : because in 1

Debye and Hiickel: Physik. Z., 24, 185 (1923).


2765

pure diffusion, any velocity difference vl-vz will be eliminated by a readjustment of the electric field (space charge). Since both ions migrate relative to a moving immediate environment, we have now

kj = pj(v - Av,), (j =

I,

2).

(4. ‘3.1 1)

Strictly speaking, this corrected force should be substituted in (4.13.9) in order to calculate Avj, but our calculations are not accurate enough to warrant this refinement. When we use (4.13.g), as stands, for calculating the correction to k, we obtain (of. (4.2.13-14))

- grad p

=

k

= vlpl(v

- Avl) + vZp2(v- AVZ)=

for the ‘(force” per molecule of solute (present in concentration n = nl/vl = nz/vz molecules/cma). The third term in the brackets is the “first order” electrophoretic effect, proportional to 6(for Ka < < I ) . The last term is of the order n log n, (second order), and always negative. The physical interpretation of the effect is interesting. If the ions were arranged at random, then the assembly of ions, moving as a whole with the velocity v against the solvent, would meet the same resistance as if the ions were moving in separate batches of solvent; the last two terms in (4.13.12) are due to deviations from a random arrangement, which affect the combined resistance. The reason for the first order effect, proportional to (pl - p z ) 2 ,is that the anions in the neighborhood of a given anion are partly replaced by cations, and vice versa near cations. If anions and cations are subject to the same frictional force, that is when pl = p z , the replacement does not matter. However, when one species of ions is more mobile than the other, then the sluggish ions will move in a countercurrent (consider the first term in (4.13.9))~while the mobile ions will be aided by a motion of their environment, whereby

nlAvl

+ nzAv2 =

0.

The compensation of the corresponding forces is incomplete; hence the first order term in (4.13.12). The second-order (last) term depends simply on an overall reduction of the mean distances between the ions in the sense that the distances between ions of opposite sign are reduced more than the distances between ions of the same sign are increased. This effect always decreases the resistance to diffusion; we here meet for the first time a case where the Coulomb forces between the ions aid a transport process. We obtain the equation for diffusion in the form (4.12.3) by solving (4.13. 1 2 ) for v and multiplying by n. Within the limits of accuracy of our calculations we may neglect second and higher powers of the corrections Avl and Avz due to Coulomb forces between the ions, which gives

2766

LARS ONSAUER AND RAYMOND M. FUOSS

In converting this formula to macroscopic units, we must remember that n= Nn;

N p = Z ,

n

where N is Avogadro's number, the concentration in g.mols/cm3, and chemical potentia1 in ergs/g.mol.; hence

J = nv =

- (Q/ N 2 ) grad

p =

the

a grad G.

Substituting (3.8.40), we find

where F = 96500 coulomb and the first and second order correction terms are, respectively, Kn A n ' = - ZZAl - ZlliZ (4.13.154 vz)N 6nt](1 KU)'

+

We are still retaining the abbreviation

When we introduce the values given by Birgel for the universal constants and A2 = A, 1000 ?i = m (g.mols/l.), we obtain write Al (4.13.16a) = 1.074 x Io-20 *A ViziA m AP AP~,

+

+

K =

1

3.556 X

109

. m

ZIZZ(Y1

+4

Birge: Phys. Rev. Supp., 1, I (1929).

m

=

+

3.556 X

104

1/DT

1/T.

(4.13.16d)

2767


Here the function c p ( ~ a is ) given by (4.13.1o), and a is the mean diameter (least distance of approach) of anion and cation. For water as a solvent a t 18Oc we have T = 291.2; D = 81.1; q = 0.01055; hence

A

=

0.232

X

10'

U,

r

= ZIZZ(YI

+

YZ) m,

and for a univalent binary electrolyte

- 9.60

Io20fi/m = 1.074 A&/A

(::if)' +$;k +15.6mdAdG). -

I

(4.13.181 The coefficient of diffusion (in c. g. s. units) is

D

=

a dp/dii = io3a d p / d m .

For a binary electrolyte

m dp/dm

=

2

RT

(I

+ m d log j / d m ) ,

wheref is the appropriate activity coefficient; hence at 18OC

D

=

4.842 X

IO~O

(a/m)(I

+ m d logj / d m ) .

(4.13.19)

A useful semi-empirical formula for f has been given by Huckell namely

where A solution a t

= 25'.

NU, applicable

to strong uni-univalent electrolytes in aqueous Differentiating this formula we obtain

+

On the basis of our computations, the factor I md log j / d m should be responsible for the variation of the diffusion coefficient of D with concentration, and we may expect that D will assume a minimum value a t some conoentration in the range o.I-In., say, corresponding to the minimum of (4.13.21). For an experimental test of our predictions, direct measurements of the differential coefficient of diffusions are preferable. Such measurements have been carried out in relatively recent times by B. W. Clack2 who enployed an interferometric method. Most experimenters have been content to measure the rate a t which a given solution diffuses against pure solvent; this procedure 1 Huckel: Physik. Z., 26, 93 (1925). Formulas (roo) together with (27a) and (30), the latter for conversion of concentrations. 2 B. W. Clack: Proc. Phys. SOC.London, 36, 313 (1929).

2768


yields a sort of an average coefficient of diffusion for all intermediate concentrations, and the type of averaging depends on the time when the experiment is interrupted for analysis. We have compared some of Oholm’s data’ with appropriate averages of our calculated D values. His results carry some resemblance to the theoretical predictions, but in several cases (KCl, NaCl), an extrapolation of his data leads to a limiting D value which is definitely higher than that computed from electrolytic mobilities, and we do not know how to interpret this discrepancy. We might mention in this connection that Oholm’s measurements were performed a t temperatures ranging all the way from 4OC. up to 18’, and were reduced to 18” by temperature coefficients. However, it is our impression that the discrepancy in extrapolation cannot be ascribed to an error in the temperature coefficients, because at higher concentrations Oholm’s values agree substantially with those given by Clack. I n the particular case of HCl, Oholm’s D values extrapolate “correctly” and conform to the theory over a range of concentrations from 0.003 n. to 0.06 n. but it is not fair to select the most favorable case. We shall be content to state that reasonable inference from Oholm’s data, to the extent that we have studied them, does not alter the picture which we shall presently derive from Clack’s observations. Clack measured the coefficients of diffusion for NaC1, KC1 and KNOs over a range of concentration from 0.05 n. to saturation. While a lowest concentration of 0.05 n. does not permit a test of our “limiting laws,” our calculations show that the Coulomb forces ought not (greatly) affect Q / mvery much a t any concentration, which justifies a consideration of experimental data even for relatively concentrated solutions. For NaCl and KC1, but not for KN08, accurate measurements of the thermodynamic potentials by e.m.f. are available. Harnedz found thar. his d::ia could be represented very well by Kuckel’s formula (4.13.20); the following values for the constants A and B were derived from his data: NaCl: A = 0.84; B = 0.129; a = 3.62 X IO-* cm. KC1: A = 0.76; B = 0.071; a = 3.28 X IO-^ cm.

(4.13.22)

These values are valid for z so, while Clack’s diffusion experiments were made a t 18’;the difference, which can be estimated from heats of dilutiona is negligible for our purposes. For the calculation of Q/m, the values of A given by Harned were adopted; hence for KCl iozo Q / m = 34.84 f 15.6 m cp(0.76&)

and for NaCl 1022

Q / m = 28.01

,-

- 0.39

d 2 m

I

-

-I- o . 8 4 4 2 m

+ 15.6 m 9(0.844%)

The results of these calculations are represented in Fig. 7 . The curve for KC1 is simply a graph of the function ~ ( K u ) I. n Fig. 8, the solid curves are the calculated values for the coefficient of diffusion (in units cm2/sec.), while circles Oholm: Z. physik. Chem., 50, 309 (1904); 70, 378 (1910). Harned: J. Am. Chem. Soc., 51, 416(1929). 8 Wust and Lange: Z. physik. Chem., 116, 161 (1924). 1 2

2769


and crosses indicate different runs in Clack’s experiments. The general type of the relation between D and concentration is accounted for by the theory, and for the lowest concentrations the agreement between theory and experiment is satisfactory. As regards the theoretical predictions for the variation of Q / m with concentration, the confirmation is only negative, that is, the experiments show that for low concentrations, Q / m does not vary much more than predicted by the theory. At higher concentrations, we meet with quite appreciable deviations, particularly in the case of NaCl, and our theory for Q / mis definitely not adequate.

-u G a5 N

6 Q5

10

15

2.0

FIG.7 Dependence of Mobility Q/m on Concentration

15

2.5

FIG.8 Dependence of Coefficient of Diffusion on Concentration

The discrepancy is too large to be ascribed to Coulomb forces between the ions; while our calculations are admittedly not accurate for concentrated solutions, at least the order of magnitude of the effects due to these forces should be correctly computed. The differences between calculated and observed values for D are nearly linear functions of m, indicating that they are probably due to specific properties of the electrolytes, which have not been allowed for in the calculations. We suspect two effects, namely viscosity and “hydration.” At I n. concentration, the difference (calc. - obs.) amounts to 3.4% (of calc.) in the case of KCl, and 12.5% in the case of NaCl. The fluidity of I n. KC1 is 1.018, that of I n. NaCl is 0.921,reckoned to unity for pure solvent.1 If the mobility is assumed proportional to the first power of fluidity, we obtain residual defects in D of around 5 . 2 % for KCl and 4.6% for NaCl (at 1 These figures are computed from macroscopic viscosities, which contain an electrostatic contribution that should actually be subtracted in calculating the “microscopic” viscosity that appliea to ions; but the difference is too small to be important here.

LARS ONSAGER A S D RAYMOSD M. FUOSS

2770

I n. concentration); this defect must be attributed to other causes, e.g. hydration. A fractional power of the fluidity may be more appropriate;‘ in that event’, the residual effect for KCl becomes smaller, that’ for NaCl greater. In using t,he term “hydration,” we shall not necessarily imply that a given ion has a definite number of solvent molecules firmly attached, but merely that there exists a region surrounding each ion, in which other ions are relatively scarce. For a simplified picture we might assume a “forbidden” region around each ion. We have tried to estimate the “forbidden” volume from the values of A and B in (4.13.zz), and obtained values of the order IO-*^ om3, which corresponds to a few (3) molecules of solvent. A hydration of one molecule of solvent per ion, firmly bound, (or maybe a couple loosely bound), would decrease D/m by 3.670, (that is 100x2X 18.03/1000) at I n. concentration. On this basis it seems quite plausible that defects of the order 5yo,as found above, can be attributed to hydration. At the same time, it becomes apparent that an exact computation would require an exceedingly detailed knowledge of concentrated electrolytes.

4.14.The Dissipation-Function for a Simple Electrolyte. So far, we have consistently sought formulas which gave the velocities of the ions explicitly by the forces. For certain purposes, particularly for a simple formulation of the laws for combined conduction and diffusion, it is more convenient to express the “forces” by the velocities, as follows:

In the limit of very low concentrations, only the friction of the solvent need be considered, so that k3. = P1, V1. = v./w, 3 = Jj/n3uj. (4.14.2)

At finite concentrations, the electrostatic interaction of t’heions must be taken into account; hence kj = plvj - ejAX, - pjAvj, (4.14.3) where A i i is the electric field on an ion due to its at’mosphere, and Avj is the electrophoretic flow of the medium surrounding an ion, which adds -Avj to the relative velocity of the ion with respect, to the surrounding solution. For the case of a simple electrolyte (s = z ) , it is possible to derive fairly simple explicit formulas, as we have in effect done above. In order to calculate the ionic field, we start conveniently with the general formula - AX,

=

grad

(0)

=

I

._

3DkTl/n1wj

(K

- Kx”)jud n x e u k u , (4.6.4)

(summation-sign omitted). The symmetrical matrix panded in terms of its eigenvectors:

K.I

- KII*may be ex-

U-1

This suggestion has been made by other authors for the interpretation of data on conductance. 1


’

277 I

where al,. .‘., a, are the eigenvalues, and It, . . ., E, the corresponding eigenvectors of K . We have shown ( 5 3.8) that for our matrix K always K2 = ff1 > f f z 2 . 2 CY, > 0, (33.4)

..

L i = e, l/n3p,/n,e,2p,.

For s =

2,

(3.8.5)

we obtain simply (K

- K1”)ji =

(K

- d/(yz) E Iz i i t. .

(4.144)

For this particular case (2 dimensions) an orthogonal matrix (Izk) represents a rotation in the plane and has the form cos y

(-sin

so that 121

= 122

-112

=

E11

y

= pK2

),

= -ezdnzwl/(nle12wz = eidnlw2/(nle12w2

We have also seen that for the case s = 012

sin y cos y

+

+ n2e2201)

+ n2e22w1).

n1el2w1 n2ez2w (n1eI2 nzezz) (w1 02)

+

(4.144

2,

+

K2.

(3.8.29)

Substituting these formulas in (4.6.4), we obtain

We note again that AX is proportional to (with K ) , where n is the concentration. From its derivation, ( 4 . 1 4 . 7 ) ~like our other main results, expresses an asymptotic law, and powers of n higher than n“’ have been neglected. In calculating this “first order” term, we may substitute the “zero order” approximation (4.14.2) for kl and k2, as follows:

+

+

- wlkl) = mez(v2 - VI) = n2e2v2 nlelvl = elJl e2J2. (4.14.8) It is seen that the ionic field AX is proportional to the velocity difference between anions and cations. The electrophoretic correction we take from (4.2.15)~omitting the “second order” term with ~ ( K u )thus ; nzez(wzk2

or, with the approximation (4.14.2) for kland k2:

We substitute this formula together with (4.14.7-8) in (4.14.3), and obtain

2992


By comparison with (4.14.1)it is seen that the reciprocal relation holds. We may then write (4.14.10) in the form kj = aF/aJj,

(4.14.11)

with the dissipation-function

The ionic field and the electrophoresis both cause positive contributions to the dissipation-function. The former vanishes when all ions migrate with the same velocity, which in the case of a simple electrolyte means no electric current. The additional dissipation due to electrophoresis vanishes when eJlp1

+ ezpilz

= nlel(vlpl

- ~ 2 ~ 2=) 0,

that is, when anions and cations meet the same frictional resistance per ion, (both migrating in the same direction). When the theory is elaborated to include in the dissipation-function terms of higher order than 6for instance n log n, the picture becomes more complicated. While we have made no exhaustive search for terms of any order higher than the first, we may point out that the second order term in formula (4.2.15) for the electrophoretic effect will lead to the negative term

in the dissipation-function. 4.15. Difusion of a General Electrolyte. Dissipation-f unction.

Diffusion in a system containing three or more ions is a subject of considerable interest. The case which has received the most attention is the interdiffusion of two different electrolytes; both the rates of diffusion and, particularly, the electromotive forces at the boundary have been studied. Unfortunately, the equations of diffusion for such cases are very difficult to integrate even when the variation of activity coefficients and mobilities with concentration is neglected. The velocities of the ions I , ., s are determined by the gradients of concentration, n1 . . n,, subject to the condition of electric neutrality

..

.

Z nzet = z

0,

(4.15.11

and by the electric field; the latter automatically adjusts itself so that the solution remains electrically neutral everywhere in accord with (4.15.I ) .


2773

When the effects of Coulomb forces between the ions are neglected, (except the macroscopic electric field, which depends on space charges in the electrolyte), the equations of diffusion are for a one-dimensional case (parallel flow)

Jt

=

- kT dni/ax)

w(,n,e,X

(4.15.2)

together with (4.15.1).The electric field is related to the current density i = I: eaJP 1

by the equation

AX

=

i 3. kT Z e,w,an,/ax, a

(4.15.3)

where X denotes the specific conductance of the solution: X = Z nae,2w,. a

The integration on (4.15.2) and (4.15.3) cannot be carried out in closed form except in certain special cases, most of which are trivial. The one case of interest which can be conveniently computed is the interdiffusion of two electrolytes (mixtures) of nearly the same composition,’ determined by the differential coefficients of diffusion for the given (nearly constant) composition. The computation of this special case is feasible because nl n, may be considered (for a first approximation) as constant coefficients in a system of equations involving anl/ax, ., an,/ax, linear in the latter. If necessary, the variation of nl n, may be taken into account by successive approximations. The simple theory demands that the differential coefficients of diffusion should depend only on the ratios of n1 n, and remain constant when the concentrations are varied in proportion. When the interionic forces are taken into account, this simple relation no longer holds; our object is to calculate the corrections. We shall develop differential equations which allow for the Coulomb forces between the ions, and discuss the most important relations to the underlying physical picture. For eventual applications, it would be necessary to integrate our fidal equations, which is again feasible for the case of diflerential diffusion, as outlined above. As long as no actual measurements of this type are available, we have not found it worth labor and space to carry out these integrations. We employ the general form (4.12.8)

...

..

...

. ..

&

=

-

B

grad pi =

RikJk,

(4.15.4)

k-r

with the convention (4.12.15)~that the electric field X is included in grad p, thus: - grad pi = - grad pi‘ eiX.

+

The splitting of grad pi into an electrical part eiX and a %hemical” part grad pi‘ has been discussed in 5 4.12. As shown there, the concentration 1

For exam le, one solution containing KCI (0.050 N ) and HCl (0.030 N ) , and another (0.050 N ) and HC1 (0.028 N ) .

containing

KR


2774

gradients yield n- I relations to determine grad pi,, . ., grad ws; the nth relation necessary to complete the determination is derived from the condition of electric neutrality (4.15.1), which again leads to Z e$ an,/&

=

- div 2 eJz

=

- div i

= 0,

or in the case of parallel flow: - Z e, ZQZka p k / a x = Z eJ1. = i = const. z

k

a

(4.15.5)

where i is the electric current. The case of “pure diffusion” is defined by i = 0,for our purposes a specialization of no important consequence, except that it is common in practice. We employ the general formula (4.14.3) k, =

PIV,

-

4x1

- P.,AV,,

(4.154

where the terms on the right represent, in the order named, the friction of the ion against the solvent, the asymmetric electric field of the atmosphere, and the electrophoretic velocity a t the centre of the atmosphere. We can calculate the latter as in the preceding section (4.14.9)~thus

For the ionic field, we use the formula -AX,

=

grad $’,(o) =

or, substituting the “zero approximation” ki

=

vi/wi

=

Ji/niwi

for the force,

Now if we write

4% = (p,/n,) 4, + Rf12 + R’Ijt

where Rf,Land R f f Jare Z given by (4.15.7-S), we have from (4.15.6) k., = R,,

J,,

= (p,/n,)

J,

+ ( R r j a+ R“,u)

Ju.

(4.159)

Here we shall point out in the first place that the matrix R is symmetrical, as required by (4.12.9). The first term in (4.15.9) is diagonal; R’,%is symmetrical according to (4.1j.7), and R“.,$ likewise according t o (4.15.8), on account of the symmetry of the matrix K .


2775

The dissipation function takes the form 2

F(J,J )

=

ZZ Rji

Jj Jj

=

The fundamental equations for combined conduction and diffusion are

- grad pt

=

a F / a J L = Z RzkJ k .

(4.15.11)

k

In the dissipation-function (4.15. IO), the electrophoretic contribution (second term) consists of a single squared term. The contribution from the ionic fields z F” = ZR“,, J, J z , containss-1 squared terms, which is the number of positive characteristic numbers of the quadratic form

where according to (3.8.4),

K2=O(1>CY2>.. . > C Y > O . This implies that F” will vanish for a certain not trivial set of values of J 1 , . . , Js, corresponding to the eigenvector Zl, of K. From (3.8.5), it is easily verified that F” vanishes when

Jl/nl = Jz/nz =

. . . = J,/nn,= VI = vz = . . . = v,,

that is, when the velocities of all ions are equal. This is what we should expect, because a simultaneous displacement of all ions will not disturb the ionic atmospheres. We may, therefore, if we like, express F” in the alternative form 2F“ = 2 R’f,En, n, (v, - vJ2, (4.1.5.12)

4

39%

which is the same as ZR”J,J, because, as is readily verified with the aid of 3.2

(3.8.5)

Z n,R“,, = 2: n,R“, = a

0.

a

The construction of the electrophoretic term F’ in (4.15.10) is apparent on sight; but some interesting properties of F” are hidden. We shall next show that F“, the contribution to the dissipation function arising from the interaction of the ionic fields, is proportional to the 3 / 2 power of the total ionic strength r and that the proportionality factor is determined by the velocities vl,. . v,, the mobilities w1, . . ., a,,and the ratios of ionic strengths pl, ., pa. We have for zF”,according to (4.5.10),

..

.

2F” = L: 2; R’IJzJ,

Jz =

JuR”,,

JT

=

1 %

--3 DkT -

.\/sU

err VU

(I

-

K-l

K1”)U7

d G

e7 VT.

(4.15.13)

2776


We now recall the transformation (4.7.5):

M-’f (K-’K) M h,, =

=

a,,

~-‘.f(Gl

z

=

.f(fO, = m1t

d\/.,w,

0%

-k W,

e,

+ + 1.1, + w, W

1.11

83,

(4.7.5) (4.7.7)

0,

by means of which we introduced a matrix H where the elements are all pure numbers. On applying (4.7.5) to (4.15.13)~we obtain 2F”

=

--!!-n, eU2 vu (I - II~’~),~ vr pr

3DkT and on substituting (4.7.1-3),

Numerical computation by means of (4. I 5.14) is made by a procedure parallel to that developed in $ 4 . 7 for the conductance problem; there is no need to repeat the manipulation here. In (4.15,14),the concentrations of the various ionic species appear only as the ratios pl = Fj/I’, while the total concentration is represented by the factor = ( v z , z ~ ~ ) ~The / ~ ,mobilities enter in the elements of H, by their ratios, (cf. 4 . 7 . 7 ) , and in the vector with components v, p, representing the force on an ion. It is of interest to compute the dissipation function for a special case, corresponding to equal mobilities W’

=

W2.

. . = W,

= w =

1/PJ

because the nature of the various terms of (4.15.10) is then very readily seen. For this case, the term zF” simplifies to

which may be written in terms of a quadratic form C(r, r),

The form C(z, z) is easily determined because the elements of K-*Kreduced to

The second term contains only the eigenvector 11 (cf. (3.8.5)), thus K - 2 k . 3% .

= /1 $. Sji

+%

hj

ki.

Then we have for the quadratic form K(r, r) in this special case 8

K(ZJ2)

K2

( h u xu)*

% K2

f p=o

( i p u Xu)’,


2777

so that the characteristic numbers are

Then C = (I numbers 51

-

K - ~

K”’) has the same eigenvectors and the characteristic

= 0, (2 =

58

. , . = (*

=

= I -

and

C(5, 5 )

=

a

=

2 - 4 2 (52 - ( l I , 2 , ) 2 ) .

(2

- &)/2 (4.15.16)

Substituting (4.15.16) in (4.15.15), we obtain

I-

where

v = nu eu2 vu/nr er2

denotes the average velocity of the atmosphere. Hence, when the coefficients of diffusion, (kT w) are the same for all ions, the entire dissipation-function takes the form

Kp + ___ Z ni e: 6 DlcT, I-

2

112

(v;-

$2.

The first term on the right is the contribution to entropy production (energy dissipation) due to the frictional forces of the ions against the solvent. The second term represents the contribution due to the electrophoretic effect, and is approximately independent of the viscosity of the solvent, because the ratio q/p- ionic radius enters. (cf. (4.2.4)). The last term is the effect produced by the assymetry of the ionic fields, and as might be suspected, depends on the differences in mobility of the various ionic species present. 5. summary The effects of the Coulomb forces between ions on transport processes, namely: diffusion, electrolytic conduction, and viscous flow are considered from a common point of view. I.

2. It is shown how results previously obtained for simple electrolytes can be extended to mixtures containing an arbitrary number of ionic species, and general limiting laws are derived. As in the case of simple electrolytes, the electrostatic effects are proportional to the square root of concentration.

2778

LARS ONSAGER A S D RAYMOND M. FUOSS

3. The fundamental equations of the ionic atmospheres lead to systems of differential equations which are compactly treated by matriz algebra. The numerical evaluation of the formal results is illustrated by examples. 4. The electrostatic contribution to viscosity is computed explicit,ly for the general case, and some simple special cases are discussed in more detail. The mechanism of force transfer is investigated. 5 . The general limiting laws for conduction in mixtures are derived. Kohlrauscb’s rule of the independent migration of ions, valid as a limiting law for simple binary electrolytes, does not apply to mixtures, nor are the conductances additive. The computed deviations are compared wit,h experiment. 6. The general laws of difusidn are discussed with particular reference t o electrolytes. The effects of the Coulomb forces are computed for simple electrolytes and for mixtures. The theoretical results for the former are compared with the available data. ;. The rate of production of entropy in transport processes is a quadratic function of the velocities. With the aid of this dissipation-f~nction,the laws of combined diffusion and electric conduction are formulated and discussed. Providence, R. I .

Irreversible Processes in Electrolytes. Diffusion ... - ACS Publications

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