J. Chem. Soc., Faraday Trans. I, 1986,82, 3293-3305 Osmosis and Reverse Osmosis Part 2.-The Separation Factor of Reverse Osmosis and its Connection with Isotonic Osmosis Gerhard Dickel” and Abdeslam Chabor Institute of Physical Chemistry, University of Munich, 8000 Miinchen 2, Sophienstr. 1 I , Federal Republic of Germany Two variational principles control isotonic osmosis and reverse osmosis : the principle of least dissipation of energy referring to a definite time integral and the principle of least constraint referring to the stationary (extremum) value of a volume integral. An extended form of the Nernst-Planck equation resulting from the time variation has been taken as the basis of our investigations. A theorem of Gyarmati dealing with the minimum conditions of local variation in the presence of constraints yields a lemma concerning the force equilibrium in reverse osmosis.According to Weierstrass’s excess function we have obtained linear relations between the potential gradients (forces). Whilst in isotonic osmosis there is an electro-osmotic equilibrium, an electrochemical equilib- rium takes place in reverse osmosis. The latter controls the separation effect. An inversion takes place if c,, = cF. Experiments have demonstrated that at this point the demineralization effect turns over into a concentration effect. The mathematical correlations between osmosis and reverse osmosis manifest that reverse osmosis represents the inverse of the transcendental function of osmosis. If one considers the flux of water connected with the flux of ions through a permeable membrane as osmosis then the reverse effect, the flux of the dissolved particles resulting from a flux of the solution, can be conceived as reverse osmosis.The advent of Onsager’s relations gave rise to the supposition that osmosis and reverse osmosis could be connected to each other by reciprocal relations. This conception coming on in the fifties, however, has fallen short of expectations. So, Sourirajan’ says on the first page of his monograph Reverse Osmosis: ‘It must be understood that the mechanism of both “ osmosis ” and “ reverse osmosis ” is still an open question, and the distinction between the two terms is entirely one of arbitrary convention and popular usage.’ The importance of reverse osmosis is based on the fact that it represents a thermo- dynamical separation effect.Therefore we have to consider the thermodynamic separa- tion methods rather than theories of the reciprocal effects. A general theory concerning such an operation (the thermal diffusion separation) as developed by Onsager’s school2 and later generalized to membrane separation effects by Cohen3 will be the basis of the following treatment. Onsager’s theory is remarkable in this context because it deals with an important and well clarified reciprocal effect : the diffusion thermoeffect, generally called the Dufour effect. This fact, however, was not taken into account in Onsager’s theory, as the Dufour effect neither exerts an influence on the separation effect, or is of any relevance in this context.Keeping in mind Sourirajan’s statement, even more attention must be paid to this fact as no reciprocal relations could be proved between osmosis and reverse osmosis. Over 32933294 The Separation Factor of Reverse Osmosis VL “t/‘, = source S,, 0 = sink St Fig. 1. Schematic representation of a separation device. and above that we have to deal with the admissibility of this principle. In this context we must take into account the following theorem of Gyarmati:4 ‘If homogeneous linear relationships exist between the fluxes as well as between the forces, Onsager’s reciprocal relations are not necessarily fulfilled ’. An example should explain this : Diffusion in the presence of a temperature gradient is controlled by two independent forces: the concentration gradient giving rise to the familiar diffusion and the temperature gradient giving rise to the thermal diffusion. Concerning the reciprocal effect, along with the familiar heat conduction an additional effect of heat conduction arises from the concentration gradient. In both cases the gradients of concentration and temperature are the common variables of the fluxes of diffusion and heat.In osmosis and reverse osmosis the corresponding forces are given by the chemical potentials of the ions and the solvent. According to the Gibbs-Duhem equation these forces are linearly dependent on each other, and Gyarmati’s theorem must be considered. In this case the electric potential offers itself as a second variable. However, since in osmosis and reverse osmosis the electric potential is a ~onstraint,~ but not an independent variable, such a procedure will not be correct.It was Gyarmati who showed that in this case the application of the extremum principles leads to the goal. Already de Groot and Mazur5 have demonstrated how to treat problems where diffusion potentials as well as outside electric potentials are applied to a system. Theories of the Separation Effects General Remarks Onsager’s as well as Cohen’s theories are based on the application of the condition of continuity to the flux equations. So the master-quantity of a separation theory is not the flux, but the transport z resulting therefrom. To understand this, let us examine a thermal diffusion tube. In the stationary state where the transport vanishes, the fluxes of diffusion and thermal diffusion continue.In this case the separation effect reaches a maximum value which decreases with increasing values of transport. So the dependence of the separation effect on the transport must be taken into account in order to obtain quantitative results. The outstanding result of Cohen’s theory was the finding of a ‘value function’ representing the value of energy expended in order to enrich a gas from any given concentration to a concentration under consideration. This is a potential function which controls all separation effects if an individual coefficient is taken into account. We have showns that Cohen’s value function is identical with Hamilton’s eiconal resulting fromG. Dickel and A. Chabor 3295 the principle of the variation of the endpoint.We note this because Hamilton's method will play a dominant role in the following treatment. Condition of Continuity d the Transport Equation Let us consider a separation device bordered by the reservoirs vb and V, at both ends (fig. 1). If a solution of a particle i of the concentration C&b) enters at the point zb with a velocity uL, in a stationary state the unchanged solution comes out at point zt if neither sources nor sinks of particles are present in the separation device. As shown in fig. 1, it is suitable to imagine a plane source sb instead of the reservoir vb, and a plane sink S, instead of V,. Both must be situated at the inside boundaries of the separation device at z = zb and z = z,. Thus, the separation device consists of the real separative element, e.g.the membrane, bounded by a source and a sink with surface divergence r&) and - r,(z,). According to the relations and 4 divji(zb) = q divji(z,) = -ri(zt) the divergence depends on the flux densitiesj,(zb) andj,(z,), where q is the cross-section of the separation element. According to the principle of continuity, the relation divj,(x,y, z) = 0, where zb < z < z,, (i = 1,2,. . ., N) (3) is valid at any point inside the separation device in the stationary state. As concerns the points z = zb and z = z,, however, eqn (1) and (2) are valid. By integrating eqn (3) over the total volume from zb to zt we get, restricting to the stationary state, for any component i: Transforming the volume integral with the help of Gauss's theorem into a surface integral, we obtain considering eqn (1) and (2): ~~~' divji dxdydz = j i ( x , y, 2,) dxdy - j i ( x , y, 2, ) dxdy + zi(z,) - &) = 0.( 5 ) It was taken into account that fluxes enter or leave the separation device only at z = zb and z = zt in the direction of the abscissa. After integration over the cross-section q we obtain, considering dxdy = dq: ss" 1s" qjkzt) - qjdzb ) = - 1. (6) In the absence of sources and sinks we havej,(z,) =ji(zb) as stated at the beginning. Because r, represents the number of moles i leaving or entering the reservoirs vb and vt in unit time, it is called the transport. In order to express rc(Zb) in terms of the fluxes, we integrate eqn (4) from z = zb to an arbitrary position z inside the separation device (zb < z < z,).Instead of eqn (6) we get (because the sink S, is excluded from the integration) : j,(z) represents the flux moving at any arbitrary position z under the influence of the separation effect through the membrane and ji(q,) the flux superimposed on it. Replacing (7) ) = qjdZ) - qjdZb 1. in eqn (6) by eqn (7) we obtain: Q(Zt 1 = qjdz) - qjdz, 1. (8)3296 The Separation Factor of Reverse Osmosis Concerning the boundary conditions, the flux j i is subjected to the ‘separation force’ at all points z > zb and z < z,, but not at the points zb and zt. At these points we have a break of the forces, but not of the fluxes. According to the condition of continuity we have where J?(z,) and J?(z,) represent the fluxes at the solution-membrane boundaries. Concerning the velocities of the particles entering or leaving the separation device we have stating that in a solution, in the absence of discrimination, solvent and dissolved particles have the same velocity.The fluxes necessary for the calculation will be discussed in the next section. J%) = J?(zn), (n = b, t) (9) v,(z) = vL(z) in (2, > z > 2,) (10) Flux Equations and the Canonical Representation of Reverse Osmosis The base of our investigations of osmosis7 is Onsager’s principle of least dissipation of energy represented by the variation of the double integral rAt rAz 6,6,E= StdZ J, J dLdt = 0. 0 dL is the Lagrangian in the volume element qdz given according to the model of viscosity by N ri, represents the number of moles of dissolved particles of type i passing the volume element qdz in unit time and iiL the corresponding numbers of moles of the solvent.N is the number of the different types of dissolved particles,f,, are the frictional coefficients as defined in the preceding paper, ci and cL the concentrations of the dissolved particles i and the solvent L, respectively, and q is the cross-sectional area. As the relations furnish the extremum value of the condition (1 1). In this case j , represent the observable fluxes, p is the chemical potential and 4 the membrane potental. F is the Faraday constant and z, the valence including the charge sign. Instead off,, we have writtenf,. Eqn (1 3) can be conceived as an extended form of the Nernst-Planck equation. Already in our first paper’ we found the drag coefficient a = 1 for Donnan-ions, and a = ?j for counter-ions.As discussed in the preceding paper, investigations of Mearesg, lo have confirmed this. In order to understand how a separation effect comes about, let us take generally a = 1. Thus eqn (13) entails a Galilean transformation of the Nernst-Planck equation as represented by the flux of a solution, exhibiting a concentration gradient, through a tube or the pores of a membrane. This transformation is realized only in the absence of constraints resulting from specific interactions between the walls of the pores and the penetrating solution. In this case the solution penetrates the membrane unchanged as postulated by a Galilean transformation. Conversely, these considerations demonstrate that specific interactions (constraints) are necessary in order to bring about a separation effect.Phenomenological theories can point out such effects. As shown by G~armati,~ the application of the variational principles to the thermodynamics of irreversible processes yields a method to treat effects resulting from constraints. Concerning variational principles, relation (1 1) represents the general form of Onsager’s principle of least dissipation of energy because it involves the time as well as the local variation. The time variation yields Prigogine’s principle of least production of entropy, whilst the local variation deals with the constraints.G. Dickel and A. Chabor 3297 The following theorem of Gyarmati4 will be helpful in solving the local problem under consideration : ‘ In any thermodynamic system in the case of given thermodynamical free forces and in the case of given local constraint conditions, the only irreversible processes which take place are such that the “constraint” C is minimum for them.’ Concerning the constraint controlling the fluxes of the cations and anions in reverse osmosis, we have to concede that the drag force of the flux of water must be taken as the driving force of the ions.Whilst in isotonic osmosis, where dp, = 0, the membrane potential is the driving force of the Donnan-ions,ll in reverse osmosis dp, represents a free force. Therefore, according to Gyarmati’s theorem, dp, furnishes the minimum constraint C = 0 if we postulate Whilst grad p, gives rise to an irreversible diffusion process, when the conditionj,.= j , is applied to eqn (1 3), the intrinsic membrane potential dpD+ZDFd(b = 0. (14) Fdd = - (CC ~ P C / ~ C -cD ~ P D / ~ D )/(CC/~C + CD/.D 1 follows. In order to evade the constraint resulting from the membrane potential, the Donnan-ions give rise to a concentration gradient compensating it. The same result can be obtained from Hamilton’s method. According to Hamilton, a potential representation can be obtained, if a transport process is controlled by an extremum principle. In the case of the permeation of a solution through a membrane it is natural to take the principle of least constraint as such a principle. In order to understand the influence of this constraint on the transport process we consider the stationary-state situation, where a pressure gradient is applied to the solution bordering a membrane.Having a fixed electrochemical potential jib = F,(z,) of the Donnan-ion at the solution-membrane boundary, the value P,(Z, + dz) = jib + dji, at the opposed membrane-solution boundary represents a free boundary condition. In this representation dji, is called the variation of the end-point. The idea of free (natural) boundary conditions is the basis of Hamilton’s method of the variation of the end-point. To understand this method, let us consider the electrochemical potential &.at the boundary point zo of the membrane and determine the change of the value of pD(z) in any volume element along the path z in order to furnish the principle of least constraint. The answer can be found easily: if pD evades the constraint along the path z, surely the principle of least constraint is fulfilled.This means the vanishing of the variation along the path z, which is given by 6BD = 0. (144 Indeed, this is in accordance with eqn (14), called the canonical representation of reverse osmosis. Concerning the Donnan-ion, from eqn (13) and (14a) eqn (16) follows immediately j~ = (CD/CL)~L (16) a relation demonstrating a Galilean transformation. The Separation Factor Substituting eqn (1 5) into the canonical representation (14), we obtain regarding dp, = (dpD/dc,)dc, and dc, = dc, the differential equation of the separation effect3298 The Separation Factor of Reverse Osmosis The slope dc,/dz can be interpreted as the simple process factor within the membrane element dz. Extending eqn (14) to an arbitrary number of ions we can obtain instead of eqn (1 7) a relation involving multicomponent systems.In the point cF = c,, dc,/dz vanishes and the effect changes from a demineralization effect to a concentrating effect. This is a consequence of the factor a, = 4 concerning the counter-ions. Arbitrarily setting a, = 1 in eqn (13) we obtain the extended version of the Nernst-Planck equation which was discussed by Schlogl12 in order to explain anomalous osmosis. In this case we get C, instead of 8 (c, - c,) in eqn (14) and (1 7) and therefore no inversion occurs. If dc, @ C, we can replace dc,/dz by the difference quotient AcD/Az. Taking into account that the difference Acs of the solutions adjoining the membrane is the unique measurable quantity, we replace C, by 8 with the help of the Donnan-equilibrium as follows.Considering p = RT In c and the relation of the Donnan concentration as well as its differential, we get the equation of the separation effect where Instead of j , we can represent the separation effect as a function of the pressure. This will become necessary in the following investigations. For this sake we start from the relation:" l3 N N Z (ci/cL X j L - X aihji = - CL grad (& + V L PI. (21) Relation (21) furnishes the extremum value of the condition (1 1) with respect to j,. Considering eqn (16) we obtain This equation is the reciprocal counterpart of the relation (23) resulting from a transposition of eqn (13). It must be emphasized that the summation in eqn (22) involves the counter-ion C only.By neglecting j i and setting p: = 0 in eqn (22), we conclude that the permeation is nearly inversely proportional to the frictional coefficientf, of the counter-ions. An influence of the matrix or the membrane is not included in this concept, explicitly. The latter, however, is contained in the frictional coefficients f, representing the only measurable quantities. Measurements never can discern between the frictional action of the fixed ions and their counter-ions. fa lit - S ( c i / c ~ ) j L 1 = - ci grad (pi + zi F4) Experimental The Compensating Method The separation factor given by eqn (19) is derived under the condition of a stationary state and the vanishing of the transport z. As the familiar osmotic cells are composed of two reservoirs separated by the membrane, the following procedure will be in accordance with these conditions.Having filled at the beginning of the operation the reservoir V, (see fig. 1) with the original solution, the reservoir V, was filled up with a solution exhibiting the presumed end-concentration. If this condition is fulfilled, theG. Dickel and A . Chabor I capillary 3299 t I I Fig. 2. Schematic representation of the osmotic cell. M = ion-exchange membrane, V, = reservoir of the initial solution, V, = reservoir of the separated solution. concentration in the reservoir remains unchanged in the course of the separation process and so the condition z = 0 is fulfilled. Otherwise an increase or decrease of the original solution in the reservoir takes place until the stationary state is reached.With the help of interpolation we can find out quickly the point of compensation where the concentration remains unchanged. This method avoids the concentration fluctuations occurring in the case of the outflow method, and so it can be applied successfully to the small effects in the neighbourhood of the point of inversion. Apparatus The osmotic cell used in our investigations is represented in fig. 2. The membrane M of type Nepton C 61 AZL 183 was a cation-exchange membrane of condensed phenolsulphonic acid reinforced with an inert support. In order to resist high pressure differences a perforated disc of Plexiglas was placed between the two cell halves as a mechanical support. The operating pressure was 6 bar. The original solution enters the reservoir Vb at the taps E and leaves simultaneously at tap 0 in order to avoid a decrease in the concentration due to diffusion through the membrane. The solution, having penetrated the membrane, passes along the capillary, and so the flux j, can be determined.On average the fluid in the capillary moves 10 cm h-l, corresponding to 0.8 cm3 per day. After a run of ca. 12 h, the solution in the reservoir V, was analysed and subsequently a solution, exhibiting the concentration found in the preceding run, was filled into V, using tap A. This operation was repeated until no change of the concentration could be found. Finally V, was filled with a solution of concentration exceeding that necessary for producing the separation effect in order to check whether equilibrium was really attained.A decrease of the concentration was found in this case. In the first phase of our investigations, relaxation effects resulting from a change of the concentration of the solutions bordering the membrane were found. So no point of inversion but a concentration interval, depending on the preliminary treatment of the membrane, could be determined. However, having applied the compensating method and approaching from both sides to the point z = 0, satisfactory results could be obtained.3300 The Separation Factor of Reverse Osmosis Table 1. General specifications of the membrane Az 5.00 x cm fU 27.50 x lo8 J s mol-1 [ref. (2)] fK 13.50 x lo8 J s mol-1 [ref. (2)] j,(Li) 0.28 x mol cm-8 s-l jL(K) 0.50 x mol cm-2 s-l cF(Li) 1.22 mol dm-3 [ref.(2)l Table 2. Values of the flux of solvent (in 108cm s-1) as a function of the normalized concentration of the solution 0.16 0.40 0.82 0.94 1.25 1.40 1.43 1.65 1.68 1.89 2.04 0.25 0.26 0.28 0.28 0.28 0.28 0.26 0.26 0.49 0.50 0.49 ReSults In table 1 the values necessary to evaluate eqn (19) are quoted. Whilst the value of j , in table 1 corresponds to the point of inversion of an LiCl solution, a number of values of j , concerning additional points and those of a KC1 solution are represented in table 2. The curve in fig. 3 represents the separation factor according to eqn (19) with cs/c, taken as the abscissa. Note that the values in table 1 yield fLi A.z/RTcL = 1.00 x log. Along with the theoretical curve we find in fig. 3 the measured values of LiCl marked by + and those of KCl marked by *.Each point represents a mean value of more than ten runs performed using the compensating method. Whilst in the point of inversion there is a good agreement between the values found experimentally and the theoretical ones, the other values deviate from the latter. This is not surprising, because on the one hand relaxation effects falsify the results and on the other hand our model is an idealized one and so neither inhomogeneities nor activity coefficients have been taken into account. Considering that the degree of accuracy of the measurements of the flux of the solution is much higher than that of the concentration, the fact thatj, exhibits a maximum value in the point of inversion (see table 2) can be taken as a consequence of the fact that no diffusion film bordering the membrane is present if the solution passes unchanged through the membrane.A diffusion film reduces the fluxes in all cases.G. Dickel and A . Chabor 3301 \ * * 2 . 0 \ Fig. 3. Dependence of the separation effect on the concentration of the solution. Graphical representation of the normalized separation effect A8/c, as a function of the normalized concentration 8 / c , of the solution in the reservoir V,. +, LEI; *, KC1. Frictional Coefficients According to the preceding paper our frictional model contains only the frictional coefficients between the particle i and the solution. Considering that according to Nernst only these coefficients give rise to the diffusion potential, it is possible to determine these with the help of electric measurements. A representative example is the determination of the frictional coefficientf,, in isotonic osmosis.14 Since the only driving force of j,, is the membrane potential, it is easy to determine fcl with the help of measurements of the flux j,, and the membrane potential.Concerning the cations, the influence of the chemical potential must additionally be taken into account. The values quoted in table 1 are determined in this way. The application of Nernst’s frictional coefficients assures a consistent theory. According to eqn (22), j , is inversely dependent on the frictional coefficients of the counter-ions if j , and can be neglected. Replacing Li* by K in the membrane and consideringf,:f, = 27.5: 13.5,14 the theoretical ratiojK:jLi = 2.04 should be obtained.However, we have found the value 1.80 in the point of inversion, being approximately in accordance with our frictional model. It is possible that this deviation results from the influence of the matrix. However, a proof will not be possible. We prefer to conceive the matrix of the membrane as the container of an electrolytic solution. As was demonstrated by measurements, the narrow interspace between the discrete sections of the matrix strongly reduces the mobility of the ions. Thus the frictional coefficient of the chloride ion increases from the value f3c1 = 1.22 x los (J s cm-2 mol-l) in a solution tofg = 16.0 x lo8 in a membrane.14 The ratio between the mobility of cations and anions, however, was found to be nearly equal in both phases.Concluding Remarks Constraints Let us compare our concept with the theory based on the application of the Onsager relations to a suitable flux scheme. This can be obtained by splitting off the flux of the solution into a flux of the solventj, and a flux of the dissolved particlesj, and choosing two suitable forces. The following dilemma results therefrom : if no constraint arises from the membrane, the solution permeates unchanged and a discrimination between j , andj, is irrelevant. As the fixed ratioj,/j, of the solution reduces both the equations to a single one, an Onsager relation is not realizable. If constraints resulting from the membrane are controlling the permeation through the membrane, these can give rise to3302 The Separation Factor of Reverse Osmosis a breakdown of the Onsager relations, as stated by G~armati.~ This statement calls for an experimental decision. The introduction of the reflection coefficient Q by Stavermanls controlling osmosis as well as reverse osmosis will be helpful.As shown by SchlOgl,l7 Q = 1 means a semipermeable membrane where the theoretical osmotic pressure occurs; concerning reverse osmosis, the solute is held back totally in this case. If Q = 0, the solution passes unchanged through the membrane and simultaneously the apparent osmotic pressure should vanish in osmosis. From an imaginary experiment we conclude that the general validity of the latter relation is realized only in a trivial case. Taking a paper membrane, in reverse osmosis the solution flows unchanged through the paper over the whole concentration range; simultaneously no osmotic pressure occurs if concentration differences are applied to this membrane.However, replacing the paper by an ion-exchange membrane, the solution passes unchanged through the membrane only if cD = cF. At this point, Q changes from a positive to a negative value. Concerning the osmotic pressure, however, at this point no inversion of the apparent osmotic pressure could be found in investigations dealing with univalent electrolytes.l*~ l9 This is a consequence of the influence of the constraints as stated in the dilemma. However, considering the electro-osmotic pressure in isotonic osmosis, ‘negative’ osmosis takes place, if cD > cF. Inverse Functions of Transcendentals In order to understand the connection between osmosis and reverse osmosis, let us consider the mathematical correlations between both effects.Concerning osmosis this phenomenon is defined by the boundary conditions of the adjacent solutions. This becomes clear by considering in a simple example that the free energy F is dependent on the concentrations c, and c, ~ ~ ( c o , ce ) = df[~(c)l. (24) co Turning to reverse osmosis and taking c, as the concentration of the original solution, the end concentration c, is an unknown variable in this case. Therefore we have to replace the limit of integration ce by a variable c. Solving eqn (24) for c we obtain c = Y(F) representing the inverse of the transcendental function (24). Men of mathematics succeeded in solving such problems in cases of transcendental functions.Excellent examples are the elliptic integrals and their inverse functions, the elliptic transcendentals. More than a hundred years were necessary to solve and understand this problem to its full extent. However, such general methods cannot be applied in physics. Variation of the End-point Fortunately a simplifed method appropriate for dealing with physical problems exists. Instead of solving the integral (24) for the integration variable c, Hamilton showed that the variation of the end-point leads to the goal if an extremum principle (in our case the principle of least constraint) controls the path through the membrane. Pressing a solution through a membrane and keeping in mind that the drag coefficient of the Donnan-ion is found to be Q = 1, no constraint (the total minimum) is acting on this i6n.Therefore the variation of the end-point represented by eqn (14a) vanishes along the path through the membrane. Concerning the counter-ions, the restricted dragging, resulting from a = &, gives rise to an increased electric field [see eqn (1 5)] in order to fulfilG, Dickel and A . Chabor 3303 the condition j , = j D . With increasing concentration of the Donnan-ion an increasingly concentrated solution passes the membrane and finally the desalination turns over into a concentration. We note that Hamilton’s method plays a dominant role in any separation theory, especially in the multistage separation theory, because the concentration is the unknown variable in any separation device.The minimum value of the energy consumption of a separation cascade follows from Hamilton’s eiconal. This yields Cohen’s value function3, V = Y [sin(F), cos (F), Ac] representing a rational function of the transcendentals sin(F) and cos(F). Theory of Fields of Extremals The difficulties arising from the restrictions of the application of the Onsager relation to osmosis and reverse osmosis and the unconvincing experimental results concerning the osmotic phenomena have suggested a complete reversal by taking Hamilton’s theory in the modern form of the theory of the fields of extremals as basis of our investigations. As shown in a preceding paper,* this way we went over to a potential representation. According to Weierstrass all terms of higher order of a Taylor series vanish and linear sets of forces satisfying the relation F(7tl, x,,.. . , 7tL) = 0 (26) can be obtained. In eqn (26) nB denote potential gradient^.^ An example is the Gibbs-Duhem relation. It must be emphasized that such relations exist only if extremals are controlling the process. This follows from Hamilton’s theory as well as from Hilbert’s independence theorem. Only in such cases Onsager’s principle of least dissipation of energy is applicable. The investigations concerning this concept were the object of our first publications in the field of membrane transport. Because the strong coupling of the solvent and the dissolved particles gave rise to difficulties, we have taken 7tL = 0 in eqn (26) by restricting to isotonic states. The results of our first paper20 can be summarized as follows: (a) the flux of water is nearly linear dependent on the pressure; (b) in the range of low concentrations of the Donnan-ions cD, the electro-osmotic pressure depends on the concentration of the fixed ions according to 4 c,; (c) because the electro-osmotic pressure decreases with increasing cD, the Donnan-ion must be conceived as a parameter of the electro-osmotic pressure. In order to cover the full concentration range, the isotonic system HCl/(HCl-LiCl) was used.The result is represented in fig. 4.,l In a further paper7 we showed that the inversion results from the application of an electromechanical equilibrium to eqn (13). In this way we could demonstrate that linear relations between potential forces exist in isotonic osmosis.22 Eqn (14) is a further example.This, however, cannot be postulated in non-isotonic osmosis. To understand this, let us consider fig. 4. Starting from point PI on the left-hand side of an ion-exchange membrane, where a 1.5 mol dm-3 solution of HCl adjoins the membrane, the path Pl-Pi leads to an isotonic LiCl-HCl mixture of mole fraction y = 0.8 on the other boundary. Because the path P1-P; is taken along an extremal of the variational problem (1 l), the principle of least dissipation is fulfilled. Going over to non-isotonic conditions and regarding instead of Pi the point P,, where a,, > a,, the point P, is inaccessible along an extremal and so the principle of least dissipation of energy is not fulfilled along the path PI-P,. Moreover, in the neighbourhood of any extremal there is an arbitrary manifold of points being inaccessible from any point along an extremal. Nevertheless, any arbitrary point is attainable by using Weierstrass’s excess function representing the remainder of a Taylor series if we accept an increased dissipation of3304 The Separation Factor of Reverse Osmosis t 1 .o -41 -+\ Fig.4. System of fluxes of water in the isotonic system HCl/(HCl-LiC1). Concentration of the HCl solution and/or the activity of water of the adjoining HCl and HCl-LiCl solutions us. ratio of LiC1:HCl of the solutions adjoining the membrane. Isotones would be presented by lines parallel to the ordinate. energy. An example of this important theorem of the theory of fields of extremals should demonstrate this.The orbit of a space shuttle represents an extremal, because no dissipation of energy takes place along this path. Any other point in the neighbourhood of this orbit, however, is inaccessible along any extremal. So the return path to the earth leads to the total dissipation of the potential energy. No extremal exists in this case. This, however, does not mean that a calculation of a path involving essential dissipation effects cannot be performed, rather it means that in such cases linear relations between potential forces do not exist. Nevertheless, we could calculate the flux ratio jL/j6 = Add) in non-isotonic osmosis from the mechanical equilibri~m~~ represented by a relation between a potential force and a frictional force. Only forces resulting from extremals according to Hamilton’s theory are potential forces. It was not disappointing that no extremal involving non-isotonic osmosis could be found.There is no extremal connecting the different orbits of an electron in an atom. Any change of an electron to a deeper orbit is connected with a total dissipation (radiation) of energy. Concerning a permeable ion-exchange membrane, ca. 95-97 % of the osmotic energyl8, lB will be dissipated in non-isotonic cases. Reciprocal Relations Whilst no comparison between isotonic and non-isotonic osmosis is possible, a comparison between isotonic osmosis and the electrokinetic effects is. The latter point out how to get the electro-osmotic pressure dP/d& and its reciprocal effect d#/dP, the streaming potential, from a linear set.Taking into account that in isotonic osmosis the electrochemical potential brings about the electro-osmotic pressure, we have to regard dP/dji instead of dP/d&; analogously, in reverse osmosis, the pressure brings about the ‘electrochemical’ effect dji/dP. Keeping in mind that ji is a transcendental function, noG. Dickel and A . Chabor 3305 linear correlation must exist between osmosis and reverse osmosis. From this we conclude : reverse osmosis represents the transcendental inverse function of osmosis. The intensive investigations of mathematicians lead us to assume that in the case of transcendental functions reciprocal relations can be found in another way. To find such relations let us consider eqn (13), from which it follows immediately that in reverse osmosis the action of the flux of solvent on the counter-ions equals half the action on the co-ions.Regarding the relation cF + cD = cc, the equation of electro-osmotic pressure’ can be written It follows that in osmosis the action of the counter-ions on the electro-osmotic pressure equals half the action of the co-ions. dp = - (+ CC - c,) Fd4. (27) The experimental investigations were performed in the years 1977-79, financially supported by the Deutschen Forschungsgemeinschaft. The inversion of reverse osmosis was found in the first months, but an explanation of this effect could not be given before Professor Meares9,10 had given a proof of the ‘anomalous’ behaviour of the drag coefficient of the counter-ions. We should like to express our thanks. References 1 S. Sourirajan, Reverse Osmosis (Logos, London, 1970). 2 W. H. Furry, R. C. Jones and L. Onsager, Phys. Rev., 1939,55, 1003. 3 K. Cohen, The Theory of Isotope Separation (McGraw-Hill, New York, 1951). 4 I. Gyarmati, Non-equilibrium Thermodynamics (Springer-Verlag, Berlin, 1970). 5 S. R. De Groot and P. Mazur, Non-equilibrium Thermodynamics (North-Holland, Amsterdam, 1962). 6 G. Dickel and E. Triitsch, Isotopenpraxis, 1966,2,429. 7 G. Dickel and G. Backhaus, J. Chem. Soc., Faraday Trans. 2, 1978,74, 115. 8 G. Dickel, 2. Phys. Chem. (Munich), 1985,144, 33. 9 P. Meares, J. Membrane Sci., 1981,8, 295. 10 P. Meares, Faraday Discuss. Chem. SOC., 1984,77, 217. 11 R. Kretner and G. Dickel, 2. Phys. Chem. (Frankfurt am Main), 1977,105, 221. 12 R. Schliigl, Farahy Disms. Chem. SOC., 1956, 21, 46. 13 G. Dickel, in Topics in Bioelectrochemistry and Bioenergetics, ed. G. Milazzo (John Wiley, New York, 14 G. Dickel and R. Kretner, 2. Phys. Chem., 1979,118, 161. 15 G. Dickel and R. Kretner, J. Chem. SOC., Farahy Trans. 2, 1978,74, 2225, 16 A. I. Staverman, Red. Trav. Chim. Pays-Bas, 1951,70, 344; 1952,71,623. 17 R. Schlogl, Stoftransport durch Membranen (Steinkopf-Verlag, Darmstadt, 1964). 18 H. Kramer, Dipl. Arbeit (University of Giittingen, 1963). 19 U. Balz, Dissertation (University of Munich, 1972). 20 G. Dickel and W. Franke, 2. Phys. Chem. (Frankfurt am Main), 1972,80, 190. 21 H. Honig, Z . Phys. Chem. (Frankfurt am Main), 1973, 87, 278; Dissertation (University of Munich, 22 R. Kretner, H. Honig and G. Dickel, 2. Phys. Chem. (Frankfurt am Main), 1977, 106, 330. 23 G. Dickel, U. Balz and B. Pitesa, J. Chem. Soc., Faraday Trans. 2, 1981, 77, 451. 1981), V O ~ . 4, pp. 271-340. 1973). Paper 61696; Received 9th April, 1986