Proc. Nat. Acad. Sci. USA Vol. 71, No. 8, pp. 3182-3184, August 1974

On the Interpretation of Experiments on Osmotic (biophysics/physiology/membranes)

RUSSELL K. HOBBIE School of Physics and Astronomy, University of Minnesota, Minneapolis, Minn. 55455 Communicated by Alfred 0. C. Nier, May 15, 1974

ABSTRACT Scholander and his coworkers have de- sure], followed by an equilibrium osmotic pressure that ap- scribed inTHESE PROCEEDINGS several experiments in which . . cannot be explained by supposed effects on the osmotic pressure exerted by colloidal particles appears proaches zero. to be altered when the particles are subject to an external the concentration of water at the membrane" (4). force; they interpret the experiments as showing that the The purpose of this note is to show that their results can be traditional explanation of osmotic pressure is wrong (Proc. explained quantitatively by making the following three Nat. Acad. Sci. USA 68, 1093-1094; 68, 1569-1571; 70, 124- assumptions, which amount to the "customary" interpreta- 128). (In their view the osmotic pressure is determined by the concentration of the colloidal particles at the free tion of osmotic pressure: surface rather than at the .) (1) The total pressure in a liquid is the sum of the partial This note shows that one must also consider the way in pressure of the , Pa, and any other constituents, each which the external force changes the total hydrostatic of which has a partial pressure Pi. pressure in the liquid. When that is done, the experiments The pressure of a constituent is a function of the of Scholander et al. are quantitatively consistent with (2) partial the traditional interpretation of osmotic pressure. local concentration of that constituent. (For a dilute solute, Pi = nikT, where ni is the number of solute molecules per Many different ways of looking at osmotic pressure have been unit volume, k is the Boltzmann constant, and T the absolute proposed. The author has pointed out recently that one of the temperature.) most appealing, because of its simplicity, is to assume that the (3) If the liquid is at rest, the total force on any volume total pressure in a fluid is the sum of the partial of element is zero. all its constituents (1). It can be shown (2) that the concept of Let us consider the magnetic experiments of refs. 3 and 4. partial pressure is valid for a dilute . If a membrane is We will ignore the variation of pressure with height due to permeable to some substance, the substance will then move gravity, since that effect was carefully removed from the from a region of higher partial pressure to one of lower partial experiments. The essentials of the system are shown in Fig. 1. pressure. This agrees with the customary interpretation in A container of water has two surfaces, x = 0 and x = T, at physiology texts. which the respective total hydrostatic pressures are P(O) and Scholander and his coworkers, in a series of ingenious experi- P(T). Someplace in between, at x = .I, is a membrane that is ments reported in THESE PROCEEDINGS, have measured the permeable to water but not to colloidal particles which are in pressure changes due to magnetic or gravitational forces act- the upper compartment. ing on particles in solution in an osmometer. They have inter- In the absence of any external forces, the total pressure preted their first experiment by concluding, "We see how the P(x) throughout the upper compartment is P(T), while osmotic pressure produced by the magnetic colloidal sub- throughout the lower compartment it is P(O). In the upper stance relates to the free surface and not to the membrane.... region the partial pressure of the colloidal particles is Pc The idea of osmotic force being driven by a difference in water (= nkT), and we have 'concentration' across the membrane must be abandoned" (3). In a later paper, they realized that the liquid had not P(T) = Pw + PC. [1] come to equilibrium under the external force, and they re- In the lower compartment peated the experiment to observe the decay of transient pres- sure changes. In that case, they concluded, "An explanation of P(0) = PW. [2] found in in physiology...is in- current textbooks the at validated by these experiments. It suggests that the presence Since water can pass through membrane, equilibrium of the solute or colloidal particles lowers the concentration of P20 is the same on both sides of the membrane. Combining we have the water at the membrane that results in the diffusion [sic]* these two equations, of water from the region of higher water concentration beyond P(O) = P(T) - PC. [3] the membrane to the region of lower water concentration on the solution side of the membrane. The fact that when the Thus, in order for the system to be in equilibrium, P(O) must colloidal particles are forced to migrate toward the membrane be maintained at a lower value than P(T). This pressure dif- generates a strongly negative COP [colloidal osmotic pres- ference, P(0) - P(T), is labeled the "osmotic effect" in ref. 3 and the "colloidal osmotic pressure (COP)" in ref. 4. * The flow of water under an osmotic pressure difference is bulk Now apply a magnetic field that exerts a force with vertical flow, not diffusive flow (ref. 1). component F(x) on each colloidal particle at height x. The 3182 Downloaded by guest on September 25, 2021 Proc. Nat. Acad. Sci. USA 71 (1974) Interpreting Osmotic Pressure Experiments 3183

x=T P(T) Ar kX^ Ff x) n (W Adx

x + dx E i

x=M

AP(x+dx) FIG. 2. A slab of the fluid of Fig. 1. The cross-sectional area is A. An external force F(x) acts on each solute particle. The hydro- static pressure is P(x). Gravity is neglected [or included in F(x)]. x =O r P(O) negative and causes P(O) to increase. This increase can result FIG. 1. A vessel of water, with pressure P(O) at x = 0 and in P(O) being larger than P(T), as shown in Fig. 2A of ref. 4. P(T) at x = T. A semipermeable membrane is located at x = M. If the upward magnetic force is allowed to continue, there Solute is confined to M < x < T. will be an upward migration of the magnetic particles until a Boltzmann distribution of particle density occurs. Since the particles will begin to move, quickly attaining a constant magnetic field in Scholander's experiments was not uniform, speed when the viscous drag on them is equal in magnitude this migration brought the particles into a region of stronger but opposite in direction to the magnetic force. By Newton's F(x) and (P increased, as shown in Fig. 2B of ref. 4. The value Third Law of Motion, each will then exert a force on the water of P(O) fell as (P increased. This is also shown ip Fig. 2B of equal to this magnetic force. Let the number of particles per ref. 4. However, the migration caused a small decrease in the unit volume be n(x). Consider a slab of the solution between x concentration of the colloidal particles at the membrane, and, and x + dx (Fig. 2). If the slab is at rest, the vector sum of all therefore, in P,(M), which worked in the opposite direction, to the forces on the slab must be zero. Since the force F(x) acts on raise P(O). This can be seen in the small overshoot on the each of the n(x)Adx magnetic particles in the slab, we can value of P(O) in Fig. 2B when the magnetic field was re- write moved. F(x)n(x)Adx - P(x + dx)A + P(x)A = O0 If the particles migrate under a continuing downward mag- netic force, there will be an increase in the magnitude of (P and or an increase in P,(M), both of which will cause P(O) to de- crease. When the field is removed, (P vanishes, but the in- AF(x)n(x)dx = A dx. [4] creased value for P,(M) results in P(O) being lower than it was dx before the magnetic field was switched on. This can also be This can be integrated to give seenin Fig. 2A of ref. 4. It is not difficult to derive a quantitative expression for the AT P(T) - P(M) = fTF(x)n(x)dx = (' [5] pressure difference when the colloidal particles have reached thermal equilibrium in the magnetic field, even though the field not uniform. (P force or is The only major assumption necessary is where is the total magnetic per unit area, "magnetic that the dipole moment of each colloidal particle is the same. pressure." This can be rewritten as Let U(x) be the potential energy of a particle at height x. The P(M) = P(T) -( [6] magnetic force is F(x) = - (dU/dx), and the distribution of the particles at equilibrium is given by From this we see that if F is positive, the total pressure P(M) at the upper surface of the membrane is reduced by the "mag- n(x) = n(1)e-[U(x) - U(M) IlkT [8] netic pressure"; if F is negative, P(M) is increased by the same amount. The "magnetic pressure" defined in Eq. 5 is then rT The experiment in ref. 3 was carried out before the particles - had time to migrate appreciably. Therefore, at the upper sur- @ = fTF(x)n(M)e-[US) U(M)]IkTdX [9] face of the membrane the concentration was unchanged and Since F(x) = -(dU/dx), this can be rewritten as Ptv(M) = P(M) -Pc = P(T) - (P -P = (P = -8n(M)e+U(M)IkTf (T {) e-U(x)/kT dX. while below the membrane, PO P(O). Combining these gives Mdx P(0) = P(T) - P, -(P. [7] This can be directly integrated to give When the magnetic field was turned on, equilibrium could be (P = - 1), [10] maintained only by further reducing P(O) by an amount equal n(M)kT(e-U/kT to the "magnetic pressure." This linear relationship was where AU = U(T) - U(M) is the increase of magnetic shown in Fig. 2 of ref. 3. Consider now the case where the potential energy of a colloidal particle in going from the mem- direction of the magnetic field is reversed. This makes (P brane to the top surface. The quantity n(M)kT is just P,(M). Downloaded by guest on September 25, 2021 3184 Physics -. Hobbie Proc. Nat. Acad. Sci. USA 71 (1974) Hence we have the general relationship is replaced by the buoyant force on the particles. The analysis is the same, except that there was no noticeable change in the (P = PC(M) (e AU/kT - 1). [11] concentration. In the case that F points down, AU is positive. For AU/kT >> We see that the Scholander data can be explained quantita- 1, this gives (P =-P,(M) or tively by using the standard view of osmotic pressure. In- deed, the direction of those pressure changes that are due to P(O) = P(T). [12] concentration changes show that it is the concentration of solute at the membrane, and not at the free surface, that Inspection of Fig. 2A of ref. 4 shows that as equilibrium is determines the osmotic pressure. approached, the "COP" does approach zero. If the force points up, AU is negative. For AU/kT <<-1, P,(M) vanishes, 1. Hobbie, R. K. (1974) "Osmotic pressure in the physics while (P remains finite. From Eqs. 7 and 10, one obtains course for students of the life sciences," Amer. J. Physics 42, 188-197. 2. Fermi, E. (1956) Thermodynamic8 (Dover, New York), p. 113. P(O) = P(T) - (P. [13] 3. Scholander, P. F. & Perez, M. (1971) "Experiments on osmosis with magnetic field," Proc. Nat. Acad. Sci. USA 68, The data of Fig. 2B do not show this agreement, because the 1093-1094. authors did not wait 35 hr for equilibrium to be established. 4. Hammel, H. T. & Scholander, P. F. (1973) "Thermal motion They were apparently misled by the flattening off of the pres- and forced migration of colloidal particles that generate sure difference, which was actually a prelude to P(O) rising to hydrostatic pressure in solvent," Proc. Nat. Acad. Sci. USA - 70, 124-128. the value P(T) (P as P,(M) vanished. 5. Scholander, P. F. & Perez, M. (1971) "Effect of gravity on The data from ref. 5 are for various concentrations of oil osmotic equilibrium," Proc. Nat. Acad. Sci. USA 68, 1569- particles suspended in water. There the magnetic volume force 1571. Downloaded by guest on September 25, 2021