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THE THEORY O F MEMBRANE EQUILIBRIA'F. G. D O N N A N

Physical Chem istry Laboratory , Univers i ty College, LondonIn 1890 Wilhelm Ostwald (1) drew attention to the pecuIiarelectrical and other effects which must occur in a system inwhich two solutions containing electrolytes are separated bya membrane which is freely permeable to most of the ions, but

impermeable to at least one of them. It is a curious fact that,the interesting ionic equilibria and potential differences whichmust occur in such a system, and which provide a very strikingproof of the existence of ionization in solution, were not morefully investigated at an early stage in the development of thetheory of Arrhenius.In 1911 the author in collaboration with Harris (2) noticedthe occurrence of these peculiar ionic equilibria during thecourse of an investigation of the osmotic pressure of aqueoussolutions of Congo red. As the equilibrium equations result-ing from the second law of thermodynamics are extremelysimple if we assume that the electrolytes are completely ionizedand that the ions act like ideal solutes, a statement of theseapproximate equations in a number of typical cases was givenby the author (3) . The exact equations can, however, bestated only in terms of the chemical potentials of Willard Gibbs,or of the ion activities or ionic activity-coefficients of G. N.Lewis. Indeed an accurate experimental study of the equilibriaproduced by ionically semi-permeable membranes may proveto be of value in the investigation of ionic activity coefficients.It must therefore be understood that, if in the following pagesionic concentrations and not ionic activities are used, this isdone in order to present a simple, chough only approximate,

An essay prepared in connection with the dedication of the SterlingChemistry Laboratory.73

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74 I?. G. DONNAN

Initial c1 Na+A-(1)

Equilibrium C I - x Na+x K+~1 A-

statement of the fundamental relationships. A good idea ofthe nature of the phenomena in question may be obtained byconsidering a very simple case. Let us suppose that a mem-brane M separates tw o solutions (1) and (2 ) of equal and con-stan t volume containing initially only the salts NaL4 and KArespectively, the membrane being permeable to the sodiumand potassium ions but not t o the anion A. Evidently the ionsXa and K can exchange places by means of diffusion across themembrane, but this exchange is subject to the restraint condi-tion of equimolar quantities. The whole system is thereforesubject to two conditions of restraint. At first sight one mightbe inclined to think that the interchange of sodium andpotassium ions would proceed until these ions were equallydistributed throughout the whole volume of solution. Thiswould occur if the membrane were permeable also to the ion A ;but that it will not generally occur in the present case can bereadily seen. We may (assuming complete ionization) repre-sent the initial and final states as follows:

K+ CIA- Ca

(2 )Na+ xK+ ~ 3 - xA- ~2

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THEORY O F MEMBRANE EQUILIBRIA

1- Na+11!! K+11

75

10Na+ -1110011K+ -

Electrical work terms vanish, and we have simply

so that

ThusClCZ1-x 2- - - x =-x cz-x' Q + C

The equilibrium state of the system is therefore:c 9 ClCZG~a+ 1 Na+ -1+ ct

Thus only when c1 = c2 is it true that for equilibrium[Na+11- K+I, and [Na+l~= [K+]z

Suppose cl = 1, c2 = 10. Then the equilibrium state is:

(1)A- 1 A- (2:In other words, 9.1 per cent of the potassium ions originaIIypresent in (2) diffuse to ( l ) ,while 90.9 per cent of the sodium

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76 F. G. DONNANions originally present in (1) diffuse to (2). Thus the fall ofa relatively small percentage of the potassium ions down aconcentration gradient is sufficient in this case to pull a veryhigh percentage of the sodium ions up a concentration gradient.The equilibrium state represents the simplest possible case oftwo electrically interlocked and balanced diff usion-gradients.These ionic exchange equilibria have been experimentally in-vestigated (4) n the case of the ferrocyanides of sodium andpotassium separated by a membrane of copper ferrocyanidegel supported in the pores of a sheet of parchment paper. Theconcentration ratios at equilibrium were found to be in fairagreement with the simple theory sketched above. The sameresult was obtained in the equilibrium between sodium caseinateand potassium caseinate across a membrane of parchmentpapere2A very interesting case occurs when the valencies of the twoexchanging ions are unequal. Thus suppose we substitute thedivalent calcium ion for the potassium ion: the second restraintcondition (equivalent inter-diff usion) requires that for everycalcium ion passing from (2) to (1) two sodium ions must simul-taneously diffuse from (1) to (2). Hence

+6n log-Na+lSn 2 log-, [Na+l~8F =RToror

This relationship has been tested (4) n the case of solutionsof the ferrocyanides of sodium and calcium separated by a cop-per-ferrocyanide gel membrane, and found to be in fair agree-ment with the facts.2 Unpublished resulte.

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THEORY O F MEMBRANE EQUILIBRIA 77

nK+ t--(1 - )A -

Returning to the case of NaA and KA, let us consider thesingle variation

bn mols K+ (2 ) --+ (1)If there existed no electrical potential gradient across the mem-brane, this would lead to the equation of equilibrium [K+I1 =[K+Iz,which is contrary to the experimentally ascertained facts.A potential difference r must therefore exist between solutions(1) and (2). If e denote the excess of the positive potential ofsolution (1) above that of (2) , the variational criterion of equilib-brium now becomes

t- t--- nK+(1- n)A-

W l z[K+I6n RT lo g- bn E =0orRTF=- log x

where

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78 F. G. D O N N A NAs a result (2) loses (1-n)KA and (1) gains an equal amount.The stationary state will be reached when the rates of changeof concentration produced by the current are balanced bydiffusion. This simple state of affairs may, however, be com-plicated by other actions, such as streaming effects due to in-equalities of density, precipitations, electrostenolysis, etc. (com-pare Ostwald, loc. cit.).Another interesting case of ionic equilibria across a semi-permeable membrane arises when the system contains twoelectrolytes with a common ion, the non-dialyzing ion being,however, one of t he other ions (whether cation or anion). Sup-pose, for example, we have the chlorides BC1 and HCl, themembrane being freely permeable to H+ and C1- but not toB+. Considering equal volumes and complete ionization, theinitial and equilibrium states may be represented thus:

M

H+ c gc1- e2

M

Initial EquilibriumThe variation bn mols H+ (2)+(1), bn mols C1- (2)+(1) givesfor the equilibrium state [H+]l.[C1-]l = [H+]2.[C1-l2 so thatz (c l +z) = (cz - z)*, whence z =--- '' , The equilibriumstate is therefore: c1+ 2c2

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THEORY OF MEMBRANE EQUILIBRIA 79Thus

If c 1 be large compared with cz the expression 1 +!! ill be muchgreater than unity. The peculiarity of this equilibrium istherefore that the presence of the ionized salt BC1 with thenon-diffusible cation B+ acts so as to hinder the diffusion ofhydrochloric acid to the side where BC1 occurs, or converselyto expel hydrochloric acid to the other side. The inequality ofthe ionic concentrations on either side shows that an electricalpotential difference must exist across the membrane. Con-sidering either of the single variations

cz

6n mols H+ (2 ) -+( l ) ,6n mols C1- (2) + 1)and calling e the excess of the positive potential of (1) over thatof (2 ) , it follows that

RT [Hlz RT [Cl-], RT=-1Og-=-log- =-logF [H'li F [Cl-]? FIf we keep c 1 constant and increase cz both the ratio of unequaldistribution of acid and the membrane P.D. will diminish, ap-proaching unity and zero respectively in the limit. Similarphenomena will occur in the case of BC1 and NaCl (or KCl),or in the case of KaA and NaCl (where A- is a non-diffusibleanion). If any of the diffusible ions possess different valencies,the equations given above must be modified in a manner whichwill be evident to the reader.These unequal distributions have been investigated in thecases of Congo red and sodium chloride (2), potassium chlorideand potassium ferrocyanide (6), sodium caseinate and sodiumchloride.3 In the latter case a fair agreement between theoryand experiment is obtained if the sodium chloride be assumedto be completely ionized and if Pauli's values for the ionizationof the caseinate be adopted. It will be noticed that in the caseof such ionic equilibria the effect of the unequal distribution of

8 Unpublished results.

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80 F. G . DONNANthe diffusible electrolyte will be to produce an osmotic pressureacting counter to that of the non-diffusible electrolyte. Thusin the example discussed above, the O.P. of BC1 will be (assum-ing ideal ionic behavior) 2RTcl. The counter O.P. due to theunequal distribution of HC1 =

The apparent O.P. of the BCl will therefore be

If cz be large compared to cl, the apparent O.P. of the BCIwill be approximately equal t o RTcl. A high value of theratio 2 will therefore appear to repress completely the ionizationof the salt BC1, even though both c l and cz are individually small.Thus if B were a colloidal cation (or ionic micelle) of molec-ular weight, say, 5000, then for a 1 per cent solution of BC1c1 = 0.002 molar approximately. If cz = 0.02, the apparentosmotic pressure of BC1 would be only 5 per cent greater thanthe value corresponding t o zero ionization.It is worthy of remark that these ionic equilibria may be treatedfrom the standpoint of the second law of thermodynamics inanother manner. Thus in the case last dealt with, supposereversible hydrogen electrodes be placed in both (1) and (2).If the system be in true equilibrium, the total e.m.f. will bezero, and therefore the membrane P.D. must be equal to

c1

A similar consideration of reversible chlorine electrodes leadsto the value

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THEORY O F MEMBRANE EQUILIBRIA 81

c1 B+c1 c1-

(1)

Ne+ c)Nos- c2

( 2 )

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82 F. G. DONNAN

~1 B+x Na+

C l - c1-Y NO,

Naf ce- xc1- zNO, C Z - Y

Thus the deductions as regards membrane P.D. and apparentosmotic pressure of BC1 are the same as in the previous case.Solving the equilibrium equations2 - y - 2Z(C1- ) =z(cz- )zy= (cz - ) cz - Y)we get

4 cz (Cl'CZ) ClCZc 1 + 2cz' =C' c xx =-

The osmotic pressure of (1) against (2) =

If cz be large compared with c l , this expression reduces as in theprevious case to R T c l . If we suppose c1 large compared withcz, say c 1 = 10 cz, then = 11 and therefore

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THEORY O F MEMBRANE EQUILIBRIA 83

Initial statec1 B+

(1)Equilibrium s ta te el B+

z H+~1 +w C1-

y Na+z NO,

H+ czc1- c 2Na+ e8NO,+ cr(2)

H+ C ~ - XC1- Ca -Na+ c a - a,NO, C a - g

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84 F. G. D O X N A Svery unequal distribution of HC1 and a considerable membranepotential, if c3 be large in comparison with cl, these effects willbe practically annulled (A not very different from unity).Interesting types of equilibrium arise when we consider thephenomena of neutralization or hydrolysis across a semi-per-meable membrane, and such cases have proved of interest inconnection with the physical chemistry of the proteins. Sup-pose there is present on one side of the membrane a feeble mono-acid base Bo, the membrane being impermeable both to thebase and to its cation B+. Keglecting the ionization of thebase, the initial state of the system may be represented thus:

c Q (1)o I water2)where co denotes the molar concentration of Bo. The osmoticpressure will be RTc,. If hydrochloric acid be added to thepure water side, H+ and C1- ions will diffuse into (1) and pro-duce a partial neutralization and ionization of Bo, formingB+Cl-. This will increase the osmotic pressure of solution ( l ) ,but the unequal distribution of the hydrochloric acid will pro-duce a counter osmotic pressure tending to lessen the increasedue to the ionization of Bo. The equilibrium state may be

'

represented thus:C O - Z Bo

2 B+v H+

y+z c1-H+ zc1- z

the equation for the membrane equilibrium being thereforeAlso z*- b+z)

C Y2 - -+Y

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THEORY OF MEMBRANE EQUILIBRIA 85K,where K =-> Kb being the ionization constant of the baseKband K , the ionic product for water.these equations gives Elimination of z between

(sa - 2) ( R+y) =cop. . ....................... (1)The osmotic pressure P of (1) against (2) is given by

P =RT [2y + + o - s )which may be written

( 5 -Y)

Equations (1) and (2) determine P as a function of II: and C(at the given temperature T ) .The first term on the right-hand side in the equation for Pdepends on the unequal distribution of H+ ions. It vanishesfor z = 0, and also tends asymptotically to zero for values of xlarge in comparison with c,. Thus P , starting with the valueRTc,, will increase, reach a maximum, and then diminish asymp-totically to its original value as x: increases. These conclusionsdepend of course on the assumptions of complete ionization ofHC1 and BC1 and the laws of ideal solutions.We may arrive a t the same conclusions in another way.Calling the expression (2 y +z +c o - 22), e ,

P = R T - +RTco.. ....................... (2 )Y

P =RTeFrom the equations

z*=y(y+ ) and 2y + + o- 2 =eit follows thatand hence

e =c o - 22 +1/422+za-=RT f 4 4 x 2 + 2 - 2) +RTcoWhen x: = 0, z = 0; and with increasing II:, z tends towards thelimiting value c,. Thus both for x = 0 and for values of II: largein comparison with c, P =RTc,. The variation of the membrane

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86 F. G. DONNAN

Y H+y c1-

potential e with increasing x can be followed in a similar manner.If a denotes the excess of the positive potential of solution (1)52 zover that of solution (2) , then, since- = 1 +-Y2 Y

H+ p:

2 c1- 1Under these conditions we have

+ R T ~2-!I)*P = R T - Y nwhich with increasing z ends towards

P u R T - - )* f RT c ,YIf Bo be a substance of very high molecular weight (say a pro-tein molecule or aggregate) the second term in the equationsfor the osmotic pressure may be relatively small. However,

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THEORY O F MEMBRANE EQUILIBRIA 87even though co be small, the final value of z( = n e,) may notbe negligible, since n may be large (regarding Bfl(+) s an ionicmicelle),Very many interesting investigations based on this simpletheory have been made by Jacques Loeb (8) and his collaborators.In this work, among other things, the effects of acids, alkaliesand salts on the osmotic pressures and membrane potentials ofthe amphoteric proteins have been studied. Loeb has shownthat the simple theory of membrane equilibria is capable ofaccounting fairly quantitatively for a great many of his ex-perimental results, and regards this as a proof that the phe-nomena exhibited by the protein ampholytes are due to simplechemical reactions and not t o the adsorption of ions by colloidaggregates or micelles. While this view may be correct inmany instances, it is necessary to remember that the theoryof membrane equilibria depends simply on two assumptions :

a. The existence of equilibriumb. The existence of certain constraints which restrict the freediffusion of one or more electrically charged or ionised con-

s t i tuentsand that the equations which result from the theory will holdequally well whether we have to deal with colloid units whichhave acquired an ionic character (electrical charge) by adsorp-tion of ions, or with simple molecules which have become ionizedby the loss or gain of electrons. All that is necessary for thetheory is that the simple ionized molecules or the ionic micellesbe subject to the same constraint, namely inability to diffusefreely through the membrane. This constraint then imposesa restraint on the equal distribution on both sides of the mem-brane of other otherwise freely diffusible ions, thus giving riseto the concentration, osmotic, and electrical effects with whichthe theory deals. Thus, in the type of equilibrium last dis-cussed, it is only necessary to assume that of the C1- ions whichdiffuse from (2 ) into ( l ) , certain proportion (of concentration z)are unable to wander freely back and forth because their averagetotal electric charge is balanced by that of constituents which

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88 F. G. DONKANfor some reason or other cannot do so . The adsorption ofcertain ions on the surface of a solid wall would provide anequally effective constraint, and the binding of these ionswould impose a restraint on the free diffusion and mobility ofan electrically equivalent amount of oppositely charged ions.In fact, J. A. Wilson and W. H. Wilson (9) have consideredionic adsorption equilibria from this point of view.H. R. Procter (10) and his collaborators have used the theoryof membrane equilibria to account for the effect of acids andsalts on the swelling of gelatine. The gelatine jelly is regardedby Procter as an elastic structural network of gelatine moleculesinto which molecules of water, acid, etc., can diffuse. Thehydrogen ions of the acid react chemically with the gelatinmolecules which thereby become ionized. Although there isno membrane, the necessary condition of constraint is providedby the inability of the gelatine ions to leave the structural net-work owing to the forces of cohesion which hold it together.This imposes a restraint on the free diffusibility of an equivalentamount of the anions of the acid, and as a result there is producedan unequal distribution of hydrogen ions and anions of the acidbetween the gelatine jelly phase and the surrounding aqueoussolution, and an increase of the osmotic pressure of the jellysolution against this external solution (although the gelatinemolecules and ions of the structural net-work make no contri-bution thereto). On the basis of this theory Proter and Wilsonhave been able to account quantitatively for the remarkableeffect of acids in low concenrations in first increasing and thendiminishing the swelling of the gelatine jelly. As in the casepreviously discussed, x2 = y(y + z ) , and therefore 2y + z ,the sum of the concentrations of the diffusible ions of the jellyphase must be greater than 22, the sum of the ions of the externalaqueous solution. It is this difference of concentrations whichgives rise to the excess osmotic pressure of the jelly, and hencethe entrance of water into it and consequent swelling. Theosmotic equilibrium must be due to an increased hydrostaticpressure inside the jelly, so that it is assumed that the stretch-ing of the molecular net work or frame work gives rise to forces

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THEORY O F MEMBRANE EQUILIBRIA 89which produce this increase of internal pressure. The impor-tant fact is that a t equilibrium the concentration of acid insidethe jelly is less than that of the external solution. From theequation 22 = y(y +z), it follows at once that y

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90 F. G. D O N N A Nscience, we can understand the complex only by analyzing itinto its simplest constituents and ascertaining, one by one, thelaws which govern the operation-the action and the equilib-rium -of these constituent phenomena. Since many livingcells do exhibit selective permeability as regards ions, it is pos-sible that ionic exchanges and equilibria of the sort discussedin this paper may occur between the cell and the external fluid(12)

REFERENCES(1) OSTWALD,.: . hysik. Chem. 6, 71 (1890).(2) DONNAN,. G. , AN D HARRIS,A. B.: J . Chem. SOC., 9, 1554 (1911).(3) DONNAN,. G.: 2.Elektrochem., 17, 572 (1911).(4) DOKNAN,. G., AN D GARNER, . E.: J . Chem. SOC.,16, 1313 (1919).(5) DONNAN,. G., AN D GREEN,G. M .: Proc. R o y . SOC.,A) 90 , 450 (1914).(6) DONNAN,. G., AN D ALLMAND,. J.: J.Chem. SOC.,06, 1963 (1914).(7) LUTHER, . : 2. h y s i k . Chem.,19,529 (1896); NERNST, .:. h y s i k . Chem.,

9, 140 (1892); RIESENFELD, . H. : Ann. Physik, (4) 8 , 600 (1902);HABER, .: Ann . Physik, (4) 26, 947 (1908); HABER, ., A N D KLEMEN-SIEWICZ, Z.: 2. h y s i k . Chem., 67, 385 (1909); BEUTNER,. : Die En-stehung elektrischer Strome in Lebenden Geweben Enke, 1920.

(8) Many papers in the J . Gen. Physiol., 1920-23; LOEB, . : J . Am . Chem. Soc.,44, 1930 (1922); LOEB, .: Proteins and the Theory of Colloidal Beha-viour, McGraw Hill Book Co. , 1922; SORENSEN,. P. L.: Studies onProteins, Compt. Rend. Carlsberg, 1915-1917.

(9) WILSON, J. A ., A N D WILSON,W. H.: J. Am. Chem. SOC., 0, 886 (1918);WILSON,J. A, : J .Am. Chem. SOC.,8, 1982 (1916).(IO) PROCTER,. R.: J . Chem.Soc., 106,313 (1914); PROCTER,. R., A N D WILSON,J. A,: J . Chem. SOC.,109, 307 (1916); PROCTER,. R., AN D WILSON,J. A.: J . Chem. SO C. ,09, 1327 (1916); PROCTOR,. R. , A N D WILSON,J. A , : J.Am . Leather Chem. ASSOC. ,1, 261, 399 (1916).(11) TOLMAN,. C., AND STEARN, . E.: J.Am . Chem. SOC., 0, 264 (1918).(12) WARBURG,.: Biochem. J., 6, 307 (1922).