A. FUNDAMENTAL STUDIES ON THE THEORY OF THE DONNAN MEMBRANE EQUlLIBRIUM BY TERRELL L. HILL Naval Medical Research Institute, Bethesda, Maryland, U.S.A. Received 27th June, 1955 The Donnan membrane equilibrium is discussed using (i) the classical Donnan method (but introducing Debye-Huckel activity coefficients) and (ii) the McMillan-Mayer method. In most of the paper all ions are treated as point charges. If the potential of average force between ions in the outside solution (containing d8usibIe ions only) of the osmotic system is assumed to be of the Debye-Huckel limiting form, the second, third and fourth osmotic pressure virial coefficients are easy to calculate from the McMillan-Mayer equa- tions. The McMillan-Mayer method leads directly to the osmotic pressure, by-passing the membrane potential and other details.However, in a special case, these can be calculated by combining the Donnan and McMillan-Mayer procedures. Both the Donnan and McMillan-Mayer methods have certain limitations, which are pointed out. 1. INTRODUCTION The McMillan-Mayer solution theory 1 is particularly suited to osmotic systems, that is, systems with a semipermeable membrane. The theory has been used to calculate the osmotic pressure of dilute solutions of large molecules by, for example, Zimm 2 and Onsager.3 Zimm considered uncharged polymer mole- cules while Onsager discussed, for the most part, “ hard ”, charged, colloidal particles of various geometrical shapes with electrostatic repulsion between the charged particles entering implicitly through the effective “ size ” of the “ hard ” particles.In the present paper we apply the McMillan-Mayer theory to the Donnan membrane equilibrium. The charge on and electrostatic repulsion between the nondiffusible ions enters explicitly in this treatment. The McMillan-Mayer method is an alternative to the now classical procedure of Donnan. The Donnan method is discussed in 5 2, and the McMillan-Mayer method in Q 3. In 3 4 the two methods are combined in a special case, and their limitations are discussed. A very condensed version of this paper has been published elsewhere,4 as has also an application of the McMillan-Mayer theory 1 9 5 to the binding of ions and molecules on proteins.6 The present work makes no pretence of being an exhaustive analysis of the Donnan membrane equilibrium problem from the McMillan-Mayer point of view.As will be apparent in Q 3, the detailed computa- tions will be different for each choice of the potential of average force. This will depend on molecular size, shape and charge distribution, on electrolyte concentration, etc. We confine ourselves here largely to ions represented by point charges. Also throughout the paper, the solution (in the osmotic equilibrium) containing diffusible ions only is assumed dilute enough to follow the Debye- Hiickel theory. 2. DONNAN METHOD A. GENERAL ELECTROLYTE-DILUTE SOLUTION.-on one side (the “ outside ”) Of a semipermeable membrane we have an aqueous solution of an electrolyte, the j-th species of which has a valence zj and a concentration (molecules/cm3) cj”.3132 THEORY OF DONNAN EQUILIBRIUM On the other side (the “ inside ”) of the membrane, thej-th species has a concentra- tion cp In addition, there is (on the inside) an ion of valence z and concentration p which cannot pass through the membrane. Let $in and #out be the potentials of the two sides, and t,b = +in - t,bOut; a,h is the membrane potential. Assuming the solutions are dilute (activity coefficients unity) and that all ions have negligible size (so that p Y terms may be omitted), cj = CJ exp (- X Z ~ ) = C; (1 - xzj + + ~ 2 ~ j 2 - . . .), (1) where x = c+[kT and E = I electronic charge I. We also have the neutrality conditions I;: C; zj = 0, (2) i and 2 cjzj + pz = 0, i and the osmotic pressure equation (3) where IIin is the osmotic pressure for the system : inside solution - membrane permeable to water only - water.flout has an analogous meaning for the outside solution. From eqn. (I), using eqn. (2), we find (to the term in x2) and hence The first term on the right-hand side of eqn. (6) is the contribution of the non- diffusible species while the second term (z Cj - c/”) is due to the different con- centrations of the diffusible ions on the two sides of the membrane. If eqn. (1) (to the term in x) is substituted for Cj in eqn. (3), we obtain, i so that eqn. (6) can also be written where K is the Debye-Hiickel parameter for the outside solution, i The above derivation of eqn. (8) is of course not new, but is included for reference below. ible electrolyte be uni-univalent with outside concentration c. On the inside, the positive ion of this electrolyte has a concentration c+ and the negative ion c-.We again use the Donnan method here, but introduce Debye-Huckel activity B. 1 : 1 ELECTROLYTE-DEBYE-Hi'r CKEL. ACTIVITY COEFFICIENTS.-Let the diffus-T. L. HILL 33 coefficients for both inside and outside solutions, again considering all ions as point charges. The Debye-Huckel parameters are ~2 (outside) = 8n&lDkT, Ki2 (inside) = (4&/DkT)(c+ + c- + pz2). (1 1) (12) We now equate the inside and outside electrochemical potentials for each of the two diffusible ions : On combining these equations, using the neutrality condition to eliminate c- , we have c- = c+ + z p C+C- = c+(c+ + zp) = c2 exp a { [(: f f g)' - I]} (16) - 11) (17) where a = &/DkT = (d/DkT)% (8nc)*.(1 8) We have written the linear form, eqn. (17), to be self-consistent, since the Debye- Huckel theory is valid only when 7 cc = &/DkT < 1 (outside) and &ci/DkT < 1 (inside). When a = 0, c+ = +[- zp + (z2p2 + 4c2)'], so we write for 0 Q a < 1, where 8 3 0 as cc -+ 0. If eqn. (1 9) is substituted for c I / c in eqn. (17) and we keep only linear terms in a and f , we find a i((!! + [(g)'+ lI))* - l}. (20) = 2[($ + 1 1 Eqn. (20), when substituted in eqn. (15) and (19), gives c+ and c- as functions of p. To obtain the osmotic pressure l7, we use = (.+ + c- + p - .I 24n That is, again to be self-consistent, we use the Debye-Huckel osmotic pressure expression for n i n and flout in eqn. (22). Introducing eqn. (1 l), (12), (15), (19) and (20) in eqn.(22), we obtain, I7 - = p + 2c[@2 + 1 1 4 - 2c - 'It2 + [(z)2 + l]*)i - l} kT 3 2c + ac *{t;+[@--+l]y-l}. B [(E)2 + 1134 THEORY OF DONNAN EQUILIBRIUM In the limit u -+ 0, the last two terms drop out and we have the usual Donnan result (1 : 1 electrolyte, dilute solution). In the limit c -+ 0, This is the osmotic pressure (according to the Debye-Hiickel theory) of a solution containing the non-diffusible ions and their counter-ions, the outside solution in this case being pure water. At the other extreme, when p is small, expansion of eqn. (23) yields 8 The term z2p2/4c agrees, of course, with eqn. (8). To find the membrane potential #, we write eqn. (14) in the form exp (E$/kT) = - exp - 5 DkT(K' - K) (26) " " Substitution of eqn. (15), (19) and (20) in eqn.(26) yields (to linear terms in a, as always) - (27) 1 exP (E#/kT) a((; -+ [@2 + 13")'- l}E [(3'+ 1]+ = (E + [($2 + 1 1 7 ,l - - As c + O , I # I 3 co. Asp +O, we find that The leading term on the right-hand side agrees, as it should, with eqn. (7). eqn. (13) and (14) we have here C. GENERAL ELECTROLYTE-DEBYE-HeCKEL ACTIVITY COEFFICIENTS.-In place O f where and K is given by eqn. (10). Now we write cj = C; + ajp + . . ., (3 1) E$/kT= A p + . . ., (32) where the aj and A are to be determined. From eqn. (lo), (30) and (31), we find where (34) On expanding the exponentials in eqn. (29) and using eqn. (33), there results so thatT. L. HILL Now we note, from eqn. (2), (3) and (31), that 35 or If eqn. (36) is multiplied through, first, by Zj and summed over j [using eqn.(37)] and, second, by zi2 and summed over j , we obtain two linear equations in A and 2: The solution of these equations gives and, from eqn. (36), If the " inside " Debye-Hiickel activity coefficient for then species j is denoted by Fj, The McMillan-Mayer activity coefficients (see Q 4 ~ ) satisfy If we let y/y" refer to the non-diffusible ion then we see from eqn. (43) that the coefficient of p in the expansion of y/y" in powers of p is obtained by putting z for zj in the expression for - [eqn. (41)J. Thus, from the (thermodynamic) argument of eqn. (83) through (87), the second virial coefficient is This agrees of course, with eqn. (8) when a = 0 and with eqn. (25) when the dif- fusible electrolyte is uni-univalent (Z(3) = 0). Since we have limited ourselves in this section to linear terms in p and a, eqn. (44) is believed to be exact to the linear term in cc.3. MCMILLAN-MAYER METHOD McMillan and Mayer showed that if the inside solution in an osmotic system contains a single non-diffusible species, then the osmotic pressure virial coefficients B,, (n/kT expanded in powers of p) are determined by the same formal expressions which Mayer 9 developed for the computation of the virial coefficients of a one- component imperfect gas. In a gas, the virial coefficients depend on the potential of the force between gas molecules ; in an osmotic system, the virial coefficients depend instead on the potential of the average force between non-diffusible mole- cules immersed in the outside solution (which is the same as the inside solution i n the limit p --f 0).36 THEORY OF DONNAN EQUILIBRIUM A.POINT cHARciEs.-We now apply this general result to the Donnan mem- brane equilibrium (i.e. to an osmotic system in which the non-diffusible particles are charged). We consider specifically the system of Q ZA and c with all ions regarded as point charges. The outside solution is assumed sufficiently dilute so that the Debye-Huckel theory is applicable. No such assumption need be made about the inside solution; furthermore, we can calculate the osmotlc pressure virial coefficients directly without considering at all quantities such as Ki, $, n i n , cj, etc. The required potential of average force in the present case (DebyeHiickel theory valid in the outside solution) for a pair of non-diffusible ions immersed in the outside solution is W(r) = z2d exp (- KY)/DY, (45) and for a set of n non-diffusible ions (n > 2) the total potential is a sum of pair potentials.The K in eqn. (45) is defined by eqn. (10). Although we shall calculate below correct virial coefficients (to linear terms in ol) for the potential of eqn. (43, as Mayer 10 has pointed out in another con- nection, these expressions are not necessarily the exact virial coefficients (to linear terms in a) for a system of point charges, since (unknown) higher terms in W(r) [omitted from eqn. (45)] may make additional contributions to the virial co- efficients. In fact, in Q 4 we shall see that the third and higher virial coefficients obtained here are certainly not exact in this sense. To avoid confusion, in the rest of this paper “correct” and “ exact” will be distinguished as in this paragraph.Let us write the osmotic pressure as n/kT=p+B2p2+B3p3+. . ., (46) (47) where Bn is the nth virial coefficient, and let us define hj = exp [- W(rij)/kT] - 1. where drl is the volume element for particle 1, etc., and the integrations are carried out over the volume V. On expanding the exponential in eqn. (47), we have Two terms in this expansion suffice to provide the linear term 11 in a = &c/DkT for B2, and only the first term is necessary for B3 and higher virial coefficients. Substituting eqn. (45) and (50) in eqn. (48), we find after elementary integrations (51) B2 = Bill + .Eli2) The first term, B;’), in eqn. (51) agrees with eqn.(44) while Bi2) also agrees with eqn. (44) when 12 Z(3) = 0. In the Donnan method B(i) is regarded as arising from the unequal diffusible ion distribution ; as an alternative but equivalent point of view, we see here that we may attribute B$’) to the repulsive forcesT. L. HILL 37 [eqn. (45)] between non-diffusible ions in the limit p -+ 0. The agreement of eqn. (51) with eqn. (8) [or (44)] as regards B$') is interesting because the Debye- Hiickel theory is used to obtain Bil) in the McMillan-Mayer method but not in the Donnan method. Although this appears to be a confirmation of the Debye-Hiickel limiting law, it is actually only a verification of the fact that electrical neutrality has been properly taken into account in the Debye-Hiickel theory (as it has in the Donnan treatment of the membrane equilibrium problem).This can be seen as follows.* Let W,&) be the potential of average force between ions of species i and j in the outside solution. We do not assume that Wij(r) has the form of the Debye-Hiickel potential, then is the local concentration of an ion of species j in the neighbourhood of an ion of species i. The total charge in the ion atmosphere of an ion of species i must be - ZiE (neutrality condition) : If we substitute eqn. (a) in eqn. (b), keep only the first two terms in the expansion of the exponential, and use eqn. (2) (neutrality condition), then where we have written Turning now to From eqn. (48), Wij(r) = zizj+b(r) + . . . (4 W(r) = zW$(r). (4 the calculation of &(I), the corresponding W(Y) here is (50) and (e), we find Putting the " neutrality relation ", eqn.(c) in eqn. (0, we obtain B$') E z2/22(2), ( g ) just as in eqn. (8) and (52). That is, we can derive eqn. (9) using the neutrality condition only (as in the Donnan method), and without committing ourselves as to the form of $(r). We may anticipate that when the correct potential of average force is used for any pair of molecules with a fixed charge z, regardless of how dilute or con- centrated the outside solution, B2 will consist of B$') (which is in dependent of T, D, molecular size and shape, etc.) plus other terms (not independent of these factors) ; see, for example, eqn. (66). The third virial coefficient follows from eqn. (49) and (50). The method of Bird, Spotz and Hirschfelder 13 may be used, or one may adopt the following alternative procedure.Write R = r12, s = ~ 1 3 , Y = r23 and introduce bipolar co-ordinates. Then, in eqn. (49), put dr3 = (2nsr/R) dsdr, r roo and J ( )drldr2 = V Jo ( )47~R2dR. v *The analysis in this paragraph was presented in the General Discussion, and was stimulated by the remarks of Prof. G. Scatchard (see Discussion remarks),38 THEORY OF DONNAN EQUILIBRIUM Thus ~3 = - Jrf(R)RdR Jr f(r)rdr S::Ir,f(s)sds. Inserting f = - W/kT, we obtain (54) This disagrees with the Donnan third virial coefficient in eqn. (25) (where there is an additional term in 24). One can be certain that the third virial coefficient in eqn. (25) is not exact since it is negative for small enough I z 1, which is impossible [from the nature of eqn.(49)] with strictly repulsive forces between non-diffusible ions. Also, as already mentioned, we shall see in $ 4 that eqn. (55) as well cannot be exact. The fourth and higher virial coefficients involve 9 integrals with several terms in the integrand instead of just one term as in eqn. (48) and (49). However, it is easy to show that, of the several terms, only one term for each B, (n > 3) gives a contribution to Bn which is linear in cc (using f = - W/kT for each pair interaction). The use of higher powers of l/kT in eqn. (50) or of other terms in the integrand, or both, leadst o contributions to B, involving am, where rn = 2, 3, . . . . The contribution to B,, linear in a, referred to above, arises from the “ cyclic ” term in the integrand ; for example [see also eqn.(49)], (56) and in general (n > 3) B, =(- A,/V) J (product of n f ’s) (product of n dr’s) = - A, 1 (product of n factors - W/kT)(product of (n - 1)dr’s) 2 2 9 = - An(- 1)” ( - DkT) S3J (product of n factors -- KR where A, and A,’ are positive constants [for example, A3’ = 1/24 in eqn. (55) and A4’ = 3/128 in eqn. (58)]. It is easy to calculate B4 but higher virial coefficients present a more serious problem, and are not discussed further [beyond eqn. (57)]. We rewrite eqn. (56) as ~4 = (- 3/8 v J fi4f24j23fi 3drld~zdr3dr4 9 and put dr3 = (27~xr/R)dsdr, dr4 = (2my/R)dxdy, Iv ( ) drldrz = V Then With f =- W/kT, 234 = - 3z80(/128(ZQ))3.T. L. HILL 39 In the special case of a 1 : 1 (diffusible) electrolyte with c: = c: = c (outside solution), as in 0 2 ~ , we have n a26 3 a28 kT -- - P + E(1- 7)]p2 + 9 j ~ 3 - s 3 p 4 + .. . . (5% B. IONS OF DIAMETER a.-Suppose all the ions, diffusible and non-diffusible, instead of being point charges, have a hard core of diameter a. Then, according to the Debye-Hiickel theory, the limiting potential of average force for a pair of non-diffusible ions immersed in the outside solution is W(r) = + GO, r < a a)1, r > a . - 2242 exp [- K ( r - - Dr(1 + Ka) We shall calculate B2 and B3 below for this potential, but we digress here to point out that eqn. (60) may also be regarded as the (approximate) potential of average force between a pair of non-diffusible particles of uniform surface charge z and diameter a, with the diffusible ions considered again as point charges.This is a useful model for charged, spherical colloidal particles or other large molecules (non-diffusible) where the electrolyte (diffusible) consists of ions of ordinary size. Venvey and Overbeek 14 discuss this model at length. Eqn. (60) is only a first approximation 14 to the potential in this case. B2 and B3, below, therefore have the additional significance of approximate virial coefficients for the model described in this papagraph. Essentially following Mayer,lo it is convenient to define w* and @ as follows : w w* kT - kT @? _ _ _ - w*lkT = 4- co , r < a = 0, r 2 a a) = 0, r < a, Also, we define - 1, r <a, 0, r > a ' k* = exp (- w*/kT) - 1 = = k * + @+%+. . . ) ( k * + l ) ( @2 Eqn. (48) and (65) (to the term in a)2) give where the first term is the second virial coefficient of a gas of hard spheres of radius a and the second term is the same as found for point charges [eqn.(52)]. The hard sphere term is negligible for small ions (a 3A) at ordinary electrolyte concentrations but may be important for large (colloidal) particles. Thus, the first term is of the same order of magnitude as the second term if a3 l/Z(2). For example, taking 2 3 2 ) = 0.1 M, a g 25 A.40 THEORY OF DONNAN EQUILIBRIUM We now consider B3. With R = yl2, s = ~ 1 3 and r = r23, the integral in eqn. (49) becomes (k*(s) + [@(s) + + W s ) + . . .])drldr2dr3 = k*(s)k*(r) k* (R) drldr2dr3 + 3 k*(s)k*(r)Q(R)dvldr2dr3 + 3 1 k*(s)@(r)~(~)drldY2dv3 + 1 @(s)~(r)Q(R)drldr2dr3 + . . . (67) Terms not listed in the last line of eqn.(67) can be shown to depend on a higher power of l/kT than 3J2 (linear in 01). The integrals in eqn. (67) are straight- forward but the limits require some care because of the intervaIs in which the functions k* and @ vanish. As an example, the third integral is [see eqn. ($411 The final result for B3, including terms in (l/kT)+, is B3 = 52 + &5[ 22E2 ](: - g K a -1. . .) 18 DkT(1 + .a) 2 exp(2Ka) [ Dk<y+ ~a)] K (1 + . .I where the order of terms corresponds to that in eqn. (67). The first term is the " hard sphere " term and the last should be compared with eqn. (55). For small ions (a E 3&, only the last term is important at ordinary electrolyte concentra- tions. For the first three terms to be of the same order of magnitude as the last, the criterion is again a3 g 1/Z(2).In the example already given, a g 2 5 A. 4. COMBINED DONNAN AND MCMILLAN-MAYER METHODS APPLIED TO The McMillan-Mayer theory, applied to the Donnan membrane equilibrium, gives the osmotic pressure directly. We cannot expect that purely thermodynamic manipulations on the expression for the osmotic pressure so obtained will permit a calculation of the membrane potential (and other details by-passed in the McMillan-Mayer method). This follows because the separation of the electro- chemical potential (a thermodynamic quantity which can be related to the osmotic pressure) into an " electrical " part and a " chemical *' part is an extra-thermo- dynamic procedure. 1 : 1 : 1-ELECTROLYTET. L. HILL 41 All ions are treated as point charges and the outside solution follows the Debye-Huckel theory.The inside solution contains singly charged diffusible ions with concentrations c+ and c- and the singly charged (z = + 1) non-diffusible ion with concentration p, while both diffusible ions have a concentration c in the outside solution. If we use (i) the McMillan-Mayer osmotic pressure, (ii) the fundamental Donnan equa- tions for the diffusible ions, and (iii) the extra-thermodynamic assumption (which is obviously correct on symmetry grounds) that the " usual " activity coefficients (denoted by Y k below) of the three ions in the inside solution are identical, it becomes possible to obtain the membrane potential and other details referred to above. The procedure below for the 1 : 1 : 1-case applies also (by redefining E) when 1 z I $: 1 provided all ions have the same absolute charge (2 : 2 : 2-, 3 : 3 : 3-, .. . electrolyte); Z(3) = 0. If the three ions do not have the same absolute charge, a further assumption must be introduced specifying the dependence of activity coefficient on absolute charge. We confine ourselves here to the sym- metrical case in order to avoid introduction of any approximations or guesses. A. ANALYSIS.-We now introduce the various activity coefficients required below. In any solution, McMillan and Mayer define an activity c'k of the k-th species as being proportional to exp (pk/kT) ( p k = electrochemical potential for a charged species) with a proportionality constant chosen so that Zk + p k , the concentration (number per unit volume) of the k-th species, as the system becomes infinitely dilute with respect to all species (perfect gas). The activity coefficient yk in the solution is then defined by ~k = Zk/pk.For the positive diffusible ion we have We consider here the system of § 2 ~ , taking z = + 1. F+ = c+y+ (inside, $in) = cy: (outside, $ouJ. (69) Let f+ be the limiting value of 7: when c -+ 0, keeping #out constant (all potentials are relative to the potential of the perfect gas used in the definition of zk). Then if we rewrite eqn. (69) as the quantity 7: = y"+yt for the outside solution is the " usual " concentration activity coefficient which approaches unity as c + 0. Since the outside solution is assumed to follow the Debye-Huckel theory, (71) - yy = exp (- E ~ K / ~ D ~ T ) = exp (- a/2) where K and cc are defined by eqn.(1 1) and (18). On the other hand, y+/yS, contains exp [($in - $,,t)/kT)] as well as ?+, so that y-/y: = 7- exp (- qh/kT). (74) We denote the McMillan-Mayer activity coefficient for the non-diffusible ion (no subscript) in the inside solution by y and in the outside solution by yo (that is, y -+ yo as p --f 0). Then, for the outside solution, yo = y:, ys = y: , (75) - yo = 7: = 7: = exp (- 4 2 ) , and for the inside solution, y = y + ,42 THEORY OF DONNAN EQUILIBRIUM Eqn. (76b) is the extra-thermodynamic assumption (3) referred to above. Thus the relation c+y+ = cy: becomes, after cancelling ys, (77) c+y exp (E$/kT) = c?', and c-7- = cy0 becomes, after cancelling yL9 Eqn. (77) and (78) correspond to eqn. (13) and (14) of 3 2B.c-7 exp (- qb/kT) = c r o . Eqn. (77) and (78) can be written as r+=r=-= c r exP ( E W n Y: Y O c+ 7" ¶ (79) and therefore c+ -+). c- To c c The general procedure may now be outlined. We can obtain y/yo as a power series in p from the osmotic pressure virial coefficients (as shown below). This gives [eqn. (79)] c/c+ and hence c+/c as power series in p. Since (neutrality of inside solution) (82) we also have c-/c and therefore [eqn. (Sl)] cyO/y)2 and r/ro as power series in p. Using this last result (with c/c+) in eqn. (79), we find Et,bJkT as a power series in p. The osmotic pressure may be expanded 5 in powers of z/yo(z/yo + p as p -+ 0) instead of p. Denote the coefficients by bj : c- = c+ + p , n/kT = 1 bj(S/y0)jy (bl = 1). i r l The thermodynamic relation 317/kT = Z ( - 3 ; - ) C T .then gives, P = 2 ibj (z/yo>J. 1 2 1 I f eqn. (84) is substituted for p in the virial expansion and the resulting series com- pared with eqn. (83), the bj can be related to the Bn. Eqn. (84) can then be inverted, (85) and the coefficients aj' also expressed 5v9 in terms of the Bn : al' = 1, a2' = 2B2, a3' = gB3 + 2B9, a4' = QB4 + 3B2B3 + %B23, etc. (86) Finally, since z = py, s/yo = al'p + a2'p2 + a3'p3 + . . .¶ yly" = al' + a2'p + a3'p2 + a4'p3 + . . . . (87) From eqn. (59), (86), and (87) we find (to linear terms in a) _ - y - c = 1 +(:-:):+ (!-??)(!?)2+ 8 64 c ( A - 2 ) p ) 3 + . 48 256 c ., (88) yo c+ and [see eqn. (82)]T L. HILL Then, using eqn. (81), (89) and (90), Finally, from eqn. (79), (88) and (91), 43 (91) (93) Eqn.(89), (90), (92) and (93) summarize the properties of the inside solution as a function of p. solution, N,, N-, N, n, A'" and r by : c+ = N+/V, c- = N-/V, p = N/V, Jlr = N+ + N- 4- N, n = X / V = 2c-, CORRECTION TO THE DEBYE-H6CKEL THEORY.-Let US define, for the inside 4 m 2 3 (N+ + N- + N ) r= (*) (12T)2 v ' (94) Let Fez be the contribution of interionic forces to the Gibbs free energy of the inside solution. Then, from dimensional considerations,~~ we may state that Fel/NkT should be a function of r only.16 In the Debye-Hiickel limit, this function is - r*. We therefore write - Fez/JVXT = r* + f(r), (95) where f(r) is a correction to the Debye-Huckel limiting expression. Writing Y = N,v, in eqn. (94), where N, and vs are the number and molecular volume of solvent (water) molecules in the inside solution, and defining Dez, the electrostatic contribution to IIin, by and Eqn.(97) and (98) can be rewritten in the form, In eqn. (99) and (loo), c-/c is given by eqn. (90) and In 7 by eqn. (92). Also we find IIel/kT from Since l7/kT and c-/c are known,44 THEORY OF DONNAN EQUILIBRIUM Eqn. (99) and (100) then become -r-=- df '"(p>, - + (a, _ _ - 3(3'. . . .) d r 32 c (103) f+rg=-z(c)2+ 1 5a p 3 + . . . . d r 32 c The term in (p/c)4 can also be found, as follows. In view of eqn. (57), let us write B5 = ccB/1280c4, where 0 is an unknown positive number. We now extend all the series starting with eqn. (88) to the term in (p/c)4, leaving 0 unspecified. We find eventually that eqn. (102) and (103) have added terms on the right-hand side of (- -& + !k 1024 + E ) ( 3 " a n d 12288 c ($ - respectively.Now, on adding eqn. (102) f = - " ( 3 3 + (- 64 c The thermodynamic self-consistency solving eqn. (102) for df/dp, integrating eqn. (105) : we substitute the relations and (103), 8 cancels and we have of the calculation can be checked by with respect to p, and again obtaining (; - g)($2 + . . .I, into eqn. (102) and find which, on integration with respect to p (f = 0 when p = 0), gives eqn. (105). If, instead of eqn. (95), we write - F,I/JC/lkT = G(F), (108) then a procedure analogous to that in eqn. (96) through (107) (or the use of G = f + F*) leads to B. DIscussIoN.-The McMillan-Mayer calculation would be exact (to linear terms in a) for a system of point charges if the exact potential were used in eqn.(45) (as already pointed out, higher terms which can affect the results are presumably omitted in this equation). The Donnan method (equating electrochemical poten- tials of diffusible ions) would, of course, also be exact if the exact osmotic pressure equation and activity coefficients were used (including higher than Debye-Huckel limiting terms where necessary). We have already seen that the Donnan method with Debye-Hiickel limiting expressions is not exact for third and higher virial coefficients, since B3 can be negative in eqn. (25).T. L. HILL 45 In 3 4A above, we have carried out a thermodynamically self-consistent cal- culation of the function f using the McMillan-Mayer method and eqn. (45). A necessary condition on f in eqn.(105), in order for the calculation to be exact, is that f should be expressible as a function of r only. It is clear from eqn. (106) that this is possible only through the term in p 2 ( f = 0 to p2). To arrive at the term in p3 in eqn. (105), one has to use the third virial coefficient. We can there- fore conclude that starting at least with the third virial coefficient [and with the terms in p 2 in eqn. (88) through (93) and (lol)] inexact results are obtained from eqn. (45). The discrepancy between eqn. (44) and (51) with respect to Bi2) when Z(3) $. 0 also appears to be due to eqn. (43, since eqn. (44) is believed to be exact (to the linear term in a). The self-consistency of the Donnan calculation of 5 2 can easily be checked; for simplicity we consider here a special case.Using eqn. (22), with expansions to p3, eqn. (25) gives, taking z = 1, 5a 9 6 c 2 B 3 = - - On the other hand, on expanding eqn. (19) to p2, with z = 1, we find c+ But, as in eqn. (88), c/c+ = y/yo so that the coefficients in eqn. (111) are a2' and a3'. The virial coefficients follow then from eqn. (86) and we find again eqn. (1 10). The method (in the Donnan case) based on the use of eqn. (86) is in general to be preferred since the virial coefficients can be obtained using series carried out to one less power of p than in the method based on eqn. (22). 1 McMillan and Mayer, J. Chem. Physics, 1945,13,276. 2 Zimm, J. Chem. Physics, 1946, 14, 164. 3 Onsager, Ann. N. Y. Acad. Sci., 1949, 51, 627. 4 Hill, J. Chern. Physics, 1954,22, 1251. 5 For a review of the McMillan-Mayer theory, see Hill, Statistical Mechanics (McGraw- 6 Hill, J. Chem. Physics, 1955, 23, 623, 2270. 7 Let Y(r) be the potential at a distance t from an ion in the outside solution. The Debye-Huckel assumption, cY(r)/kT< 1, when applied at t = 1 / ~ (the value of t at which the net charge in a shell of thickness dr is a maximum), is essentially dK/DkT 4 1. 8 The term in p2 is also contained in Hill, Faraday SOC. Discussions, 1953, 13, 132, eqn. (276). 9 Mayer and Mayer, Statistical Mechanics (John Wiley and Sons, New York, 1940). In neither case is it necessary to assume that the total intermolecular potential is the sum of pair potentials. Hill, New York, 1956). 10 Mayer, J. Chem. Physics, 1950, 18, 1426. 11 The linear term in CL corresponds to the term in (llk7')Q if the expansion is regarded 12 This relationship is discussed further in 9 4 ~ . 13 Bird, Spotz and Hirschfelder, J. Chern. Physics, 1950, 18, 1395. 14 Venvey and Overbeeck, Theory of the Stability of Lyophobic Colloids (Elsevier 15 Fowler and Guggenheim, Statistical Thermodynamics (Cambridge University Press, 16 For our special case, r is the same quantity as 7 3 of Berlin and Montroll, J. Chem. as one in powers of llkT. Publishing Co., Amsterdam, 1948). 1939), p. 384. Physics, 1952, 20, 75.