I Raymond Chang and Lawrence J. Kaplan Williams College Williamstown. Massachusens 01267
II
Textbook Errors. 129
The Donnan Equilibrium and Osmotic Pressure
In addition to the important role that semipermeahle memhranes play in uiuo, they have been widely employed in biochemical research. Thev are used in dialwis and membrane filtration to separate molecules of different sizes, in equilibrium dialvsis to measure the bindine of small molecules to macromoiecules, and to determine &e molecular weight of relativelv laree molecules. Due to the semipermeable nature of the membranes a pressure difference is established between the two sides of the membrane. This pressure difference is called the osmotic pressure and arises as a result of the inequality of the chemical potential of the solvent in the solution and the pure solvent itself. Another phenomenon that arises when the macromolecular solute has a net charge is called the Donnan equilibrium. This causes an increase in the observed osmotic pressure. While most biochemistry and physical chemistry L x t books discuss the osmotic pressure situation and define its origin due to the chemical potential of the solvent, a number of them present a t best a very confusing treatment of the Donnan phenomenon. Many books contain errors frequently arising from the lack of a clear definition of the Donnan effect and from neglect in considering that the osmotic pressure is due to the difference in solute concentration between the two sides of the membrane. The origin of "simple" osmotic pressure, the Donnan effect which adds to the "sim~le"osmotic Dressure and the effect of added salt which sippresses the ~ o n n a neffect, can be understood by considering the three cases shown in the figure. Case 1 If a solution of a macromolecule (such as a rotei in) is seaarated from pure solvent by a semipermeable membrane, the overall svstem will not he at eouilibrium initiallv hecause the chemicai potential of the solvent in the soluti& will be less Sueeestions of material suitable for this column and columns suit& for nublieation directlv, should he sent with as man" details as possible. and particularly with reference to modern textbooks, to W. H. Eberhsrdt, School of Chemistry, Georgia Institute of Technology, Atlanta, Georgia 30332. Since the purpose of this column is to prevent the spread and continuation of errors and not the evaluation of individualtexts, the scores of errors discwed will not be cited. In order to be presented, an error must occur in at least two independent recent standard hooks. 1 See any standard physical chemistry text. ~
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Three different arrangements lor measuring the osmotic pressure of a proteln solution. la). The .DrOtein molecule bears no net charae: water is ~laeedin thetight c m w m n m . lbl The potm mlecule bears z n.mber of negatwe chwgss and water is placed in lhs right companmenl. (c) Same as (bl except .not ally same NaCl is added to me solvenl n me right comparlment. ~
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than that in Dure solvent. Recall that the condition for eauilibrium at constant tempera'ture is that the chemical potential for a eiven comwnent be the same throughout the system. Of course, this condition does not apply G t h e protein since it cannot cross the membrane to the right side. T o attain equilibrium, the chemical potential on the solution side (left side) is increased due to the increase in pressure to compensate for the decrease in chemical potential caused by the solute. This pressure difference between the two sides iscalled the osmotic pressure, s,and can be shown to be related to the solute concentration, Cp (molarity) by1
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218 1 Journal of Chemical Education
T I = C&T
(1)
where R is the gas constant and T is the absolute temperature. If the concentration (cz) is expressed as gA, then the equation becomes
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where M2 is the molecular weight of the sdlute. This "simple" type of consideration applies to the osmotic pressure arising from a nonionic solute or from a protein a t its isoelectric pH where the molecule has no net charge. Case 2 For a protein a t a p H other than its isoelectric condition where i t will possess either a net positive or negative charge, an additional factor must he considered; namely, the counterions needed to maintain charge neutrality. In this situation the protein is present as Na,+PZ- which is assumed to be a strong electrolyte. While the Na+ ions would normally he freely diffusible, they are held on the left side to maintain the electrical neutrality. These ions add to the solute concentration and result in an increase in osmotic pressure. This phenomenon is called the Donnan equilibria2 and the resulting osmotic pressure can he calculated from r z = (z + 1)CzRT (3) where (z 1)is the number of ions and Cz(z 1)is the total molar concentration of solute including the macromolecule, C2, and the counterion, zC2. From eqn. (2), we see that the Donnan effect leads t o a molecular weight (actually a numher average molecule weight) which is much smaller than the true molecular weight of the macromolecule.
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Case 3
Since measurements on proteins are frequently made in a buffer or in the presence of added salt, it is worthwhile t o analyze this si&ation. Consider dissolving the protein Na,+Pz- in water and placing i t on the left side and putting a solution of NaCl on the right side of the membrane. Now the requirement that the chemical potential of a component he the same throughout the system applies to the NaCl as well as the water. T o attain equilibrium, NaCl will move from the right to left compartment and theactual amount transported can he calculated. The initial moltu concentration of Na,-I'zis C;!and that of the NaCl is b. At equilihrium, the concentrations are [Pz-1,. = (Cd; [ N a f ] = ~ (zC2 + x); [Cl-]L = (x) and = ( b - r) [Nat]n = ( b - x); [ a - ] ~ where x is the amount of NaCl that has diffused from right to left. = (PN~CI)R and for dilute solutions we m u m e Since (ILN~COL ([Nat][C1-])L = ([Na+][Cl-])R,then
The osmotic pressure, which is proportional to the difference in solute concentration between the two sides, is now given by ~~=((CZ+ZC~+X+T)-(~-X+~-X))RT
salute on the right solute on the left = (C2+ ZCZ- 26 + 4z)RT Substituting for x
Equation (5) was derived assuming no change in either p H or volume of the solutions. Two limiting cases can be considered. If b > z2C2 (the added salt concentration is much greater than the protein c~ncentration)~ 2C2b ,r3--RT=CzRT=n 2b Notice that in the first limiting case (at low salt concentration) the osmotic pressure approached that of Case 2 (no added salt present). In the second limiting case (at high salt concentration) the osmotic oressure approaches that of the pure isoelectri'c protein. In effect, the added salt reduces (and a t hieh enoueh salt concentrations, eliminates) the Donnan effect. The effect of these osmotic nressures on the observed molecular weight can be quite iarge. As already stated, the Donnan effect reduces the molecular weieht sienificantlv. Bv reducing the Donnan effect, the presenee of ;he added sait reduces the osmotic pressure and increases the observed molecular weighL4 One must also consider the consequences of the Donnan equilibrium in other experiments. ~oLexample,in measuring ion binding by proteins the Donnan effect can lead to erroto the Donnan effect are observed neous r e s u k s ~ i m i l acases r when a solution containing macromolecules is separated from a solution without macromolecules. This is true even if there is no membrane present a t the interface. The experimental procedures of sedimentation (in an ultracentrifuge), diffusion and electrophoresis may involve a separation of this kind. The Donnan effect mav be important in these situations unless a high salt concentpation i i employed to minimize it. For example, a molecular weight of 3000 was ohtained for plasma albumin, now well characterized as being 67,000, when the Donnan effect was ignored during ultracent~ifugation.~ There are anumber of important points to be emphasized from the above treatment. First, the added salt (our Case 3) reduces the osmotic pressure (and increases the molecular weight) as compared to the situation without added salt (our C& 2). second, theDonnan effect arises as a result of the net charge on the macromolecule and the presence of the counteriois. Third, in determining the osmotic pressure the difference in solute concentration between the two sides of the membrane must be considered and not simply the concentration on the side containing the macromolecule.
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2Donnan,F. G . , Z . Electroehem., 17,572 (1911). M,z 1< 30, so thatz2C2 0.1 M.Thus, In practice, Cz S 1X in order for this limiting case to be valid, the concentration of added salt should be about 1M. For a recent interesting paper which employs osmotic pressure measurements to determine the molecular weights of proteins in the native and dissociated states, see Castellino, F. J., and Barker,D. R., Biochemistry, 5 2201 (1968). See Van Holde, K. E., "Physical Chemistry," Prentice-Hall, New Jersev. 1971. D. 47.
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