CHAPTER III

«OSMOTIC PRESSURE AND THE THEORY OF SOLUTION II»

Accepting van 't Hoff's theory of solutions, then, as based on experimental evidence as well as on sound thermodynamic reasoning, we find a number of interesting questions still confronting us. Most insistent is the question as to the source of the remarkable agreement between the osmotic pressure of a solute and the gas pressure, which it would exert in the same volume, as a gas, at the same temperature, and as to the identity of the laws governing the two forms of pressure. Then, we may also ask, what is the mechanism of the process by which osmotic pressure reveals itself, especially in the case of cells with semipermeable membranes. And, finally, we may ask what is the cause of the semipermeability of the membranes.

«Semipermeability.»—Taking up the last question first, as the simplest one, we find that it was long ago recognized that permeability depends on the power of membranes to dissolve certain substances, or to form unstable combinations with them. A membrane is semipermeable if it will dissolve one component only of a solution, the solute or the solvent, and not the other.[29]

We find the simplest evidence of the cause of semipermeability in the case of gases. Palladium, especially when heated, dissolves hydrogen readily, but not nitrogen or oxygen, and a wall of palladium may be used as a semipermeable membrane to separate a mixture of hydrogen and nitrogen from pure hydrogen, just as copper ferrocyanide membranes are used with aqueous sugar solutions and water. The results with the gases duplicate in every particular the observations made on the solutions (see below, p. 24). Certain gases, such as ammonia and hydrogen chloride, are easily soluble in water, while others, like oxygen, nitrogen and hydrogen, are very difficultly soluble, and a film of [p022] water may be used as a semipermeable membrane for such gases.[30]

EXP. If the moist membrane of a cell (Fig. 4), containing air, is covered with an atmosphere of hydrogen, there is no increase of pressure produced in the cell, as indicated by the column of colored oil in the manometer in which the cell ends: hydrogen, being very little soluble in water, cannot pass through the film of water in the few minutes it is allowed to act. If now an atmosphere of ammonia is substituted for the hydrogen, the gas passes through the film into the cell. It turns the color of a piece of litmus paper placed in the cell and produces an increased pressure in the cell, the air remaining in the latter, because oxygen and nitrogen are very little soluble in water.

Membranes will be, similarly, semipermeable to solvent or solute, when only one of these is soluble in the membrane, or is capable of forming an unstable compound with it. For instance, salts, holding water of crystallization which is readily lost and recovered, may easily be conceived of as assuming the rôle of semipermeable membranes, allowing the passage of water say from a wet atmosphere to a dry one, or from pure water to a solution; and Tammann[31] has realized such membranes by the use of zeolites—silicates, which hold water of crystallization but are insoluble in water. Kahlenberg[32] has recently used rubber membranes, that are permeable for solvents like benzene, pyridine, etc., which are soluble in rubber, but not permeable for water, which is insoluble in rubber.

«Osmosis.»—The recognition of this rôle of the semipermeable membrane leads to the second question raised, namely as to the mechanism of the process by means of which osmotic pressures are measured directly with the aid of such membranes (p. 11). [p023] The answer hinges on the question of the mechanism of the diffusion of the solvent into the cell, a diffusion which is called its osmosis.[33] If we consider the pure solvent, say water, on one side of a semipermeable membrane, and a solution (‹e.g.› of sugar in water) on the other side, it is obvious that the ‹solvent› itself has a ‹higher concentration› on the side where it is pure, than on the side of the solution, where it is diluted—distended by the solute in it. The solvent is soluble in the membrane, and its solubility will be proportional[34] to its own (the solvent's) concentration; it will, consequently, be more soluble in the membrane on the side of the pure solvent than on the side of the solution. If we bring such a membrane first into contact with the pure solvent (Fig. 5), the membrane will take up the solvent (from the side ‹A›) until it is ‹saturated› with it. Let the solubility, which represents the concentration of the solvent in the membrane at this stage, be called ‹c›. The membrane may then be considered to be taking up in unit time just as many molecules from the solvent as it gives up to it (dynamic equilibrium), exactly as, when water is in equilibrium with water vapor, we consider the water to be vaporizing just as fast as vapor is condensing to form water. Now, if a solution of sugar is placed on the other side of the membrane, the solvent will pass out of the membrane into the solution just as fast as it passes back into the pure solvent. At first the concentration of the solvent on the surface ‹B› of the membrane is just as great (‹c›) as on the surface ‹A›; but ‹the membrane will here receive the solvent more slowly from the solution, which is less concentrated as to the solvent›; and consequently the membrane ‹will lose water to the solution›. The solubility (‹c′›) of the solvent at this surface ‹B› of the membrane, corresponding to the smaller concentration of the solvent in the solution, will be less than the solubility (‹c›) at ‹A›, where the membrane is in contact with the pure solvent, and water will pass into the solution at ‹B›, until the concentration of the water in the membrane at ‹B› has fallen [p024] from (‹c›) to (‹c′›). In such a membrane, as in every solution or gas, there must be a tendency towards the establishing of uniform concentration by diffusion from points of higher concentration to those of lower, and the solvent will, therefore, ‹diffuse from points along the surface A of the membrane to points along the surface B›; the surface ‹A› will become ‹unsaturated› and will take up solvent from the pure liquid bathing it, and the surface ‹B› will be kept continuously supersaturated and will lose solvent continually to the solution. Consequently, the solvent will pass continuously through the membrane from the pure solvent to the solution. Equilibrium will be reached, and the flow will cease, only ‹when the solution has become infinitely dilute›, equal hydrostatic pressure obtaining on solution and solvent, or ‹when the disturbing influence of the solute, which dilutes the solvent in the solution, is exactly counterbalanced by an external hydrostatic pressure, exerted on the solution›. When such a pressure on the surface of the solution balances the force exerted against the solvent by the solute we shall have equilibrium. It is clear, then, that the ‹osmosis›, or passage of the solvent through the membrane, is brought about by the unequal concentrations (or, more exactly, the resulting ‹unequal partial pressures›) ‹of the solvent itself. But this inequality is produced by the presence of the solute, and it is a characteristic and significant fact, that the effect of the latter, in dilute solutions, may be overcome by a hydrostatic pressure, corresponding to the gas pressure which the same number of molecules of a gas in the same volume at the same temperature would exert against this hydrostatic pressure›.

«Osmosis and Gas Pressure.»—The legitimacy of the interpretation given is most strikingly shown by experiments with a membrane, semipermeable for gases, which enables us to measure gas pressures, that may be unknown, by exactly the same process as is used to measure the unknown osmotic pressure of a solute in solution. Van 't Hoff[35] and Arrhenius[36] predicted such a result, and Ramsay[37] proved by experiment the correctness of their assumptions. A mixture of nitrogen and hydrogen may be enclosed in a palladium vessel connected with a manometer (see Fig. 6).[38] The partial pressure ‹P›_{‹N›} of the nitrogen may be [p025] determined by surrounding the palladium vessel with pure hydrogen, at a pressure which is known and is greater than the partial pressure of the hydrogen in the vessel, and by observing the final total gas pressure which is obtained in the vessel. The hydrogen diffuses from the point of higher concentration, outside of the vessel, through the palladium, into the interior where the concentration of the hydrogen is lower. The experiment may be carried out at 280°, a temperature at which palladium readily dissolves hydrogen and is permeable to it. The metal does not dissolve nitrogen and is not permeable to it. The volume of the enclosed gas is kept constant by raising the mercury level in the outside arm of the manometer, and the total pressure of the enclosed gas is measured when equilibrium is reached. If this total pressure is ‹P›_{final} and the known pressure of the hydrogen outside of the vessel is ‹P›_{‹H›}, then, if equilibrium is reached when the hydrogen on both sides of the semipermeable palladium membrane has the same concentration (pressure), ‹P›_{‹H›} + ‹P›_{‹N›} = ‹P›_{final} and ‹P›_{‹N›} = ‹P›_{final} − ‹P›_{‹H›}. In other words, the excess of the final combined pressure inside, over the outside pressure of the hydrogen, ‹is equal to the pressure of the nitrogen in the vessel›. Ramsay's results showed that the amount of hydrogen actually entering the vessel was 90–97% of the amount predicted by the theory on the basis of the assumption that equilibrium will be reached, when the hydrogen has the same concentration (pressure) on both sides of the palladium membrane.

The experiment is particularly instructive, in the first place, because it illustrates with a gas, subject to the laws of gases, why and how osmosis takes place through a semipermeable membrane—namely as a result of the solubility of the diffusing substance in the membrane, and through the flow of the diffusing substance [p026] from higher to lower concentrations. In the second place, while the increase in total pressure in the inner chamber undoubtedly is ‹brought about› by the ‹osmosis› of ‹hydrogen› into the chamber, the excess pressure when equilibrium has been reached, necessarily measures accurately the partial pressure of the ‹nitrogen›. In other words, the semipermeable membrane is merely a means or ‹device for measuring› the partial pressure of the nitrogen—the membrane is not the ‹cause› of the pressure; the latter is a definite one, whether we know what it is or not, and the osmosis of the hydrogen through the palladium merely gives us a means of ascertaining it. Similarly, it would be wrong to consider that the osmotic pressure of a solution is caused, or brought about, by the flow of the solvent through a semipermeable membrane (osmosis); the latter simply is a ‹device› which enables us to recognize and ‹measure› the pressure that exists in the solution, both in the presence and the absence of such a membrane.

We may consider, then, that the osmosis, or migration of the solvent through a semipermeable membrane into a solution, is the result of the reduced concentration (or ‹partial pressure›) of the solvent in the solution, resulting from the presence of the solute.

Inasmuch as the ‹effect› of the ‹solute› on the solvent can be overcome by a ‹pressure› on the surface of the solution, one is led to the conclusion that the solute acts by exerting, in turn, ‹a force or pressure› against the surfaces of the solvent, in the directions opposite to the hydrostatic pressure required to overcome it. The significant identity of the value of this pressure, as thus measured, with the gas pressure that would be exerted by a gas of the same number of molecules, in the same volume and at the same temperature, leads us to the last of the three questions which have been raised, namely, the question concerning the theory of the intimate relations between gas and osmotic pressures (p. 21).

«The Kinetic Theory and Osmotic Pressure.»—For an answer to this fundamentally interesting theoretical question one turns, naturally, to the kinetic theory, which, in the hands of Clausius, Joule, van der Waals and others, has given us a very satisfactory and essentially complete theoretical interpretation of the behavior of gases, and of the liquids to which they may be compressed.

The laws of gases, it is known, are in accord with the two simple assumptions of the kinetic theory. The first assumption is that [p027] gases consist of ultimate discrete particles (molecules), which move in all directions through the space filled by the gas and, at ordinary pressures, are so far apart, that the forces of molecular attraction between them are negligible; the ‹pressure› of the gas is simply the net result of the impacts of these flying particles upon the walls of the containing vessel. The second assumption of the kinetic theory is that ‹temperature› is a function of the mean kinetic energy of the moving molecules, and that the molecules of gases of the same temperature have the same mean kinetic energy. The kinetic energy of particles is a function of their mass ‹m› and their velocity ‹u› (K.E. = ½ ‹m› ‹u›^2). When a gas is heated, the kinetic energy of its molecules is increased, and, since their masses remain unchanged, their velocity must increase. As a result, the number and the force of their impacts against the walls of a given space increase, and thus the pressure is increased.

We may ask, whether this theory cannot be used to explain the connection between osmotic and gaseous pressure. If temperature is a function of the kinetic energy of the molecules of which a substance consists,—and the whole behavior of gases confirms such a conception,—then one must conclude, that the mean kinetic energy of molecules, at a given temperature, must always be the same, irrespective of whether they are present in gaseous, or liquid, or solid form, or even in solution.[39] The tendency of the molecules to move, resulting from the kinetic energy inherent at a given temperature, may be largely balanced (liquids), or overcome (solids), by molecular attractions of surrounding particles, but such conditions are altogether in harmony with the conception of a definite mean molecular kinetic energy, persisting at a given temperature, irrespective of the physical surroundings of the molecule. According to the kinetic theory, then, when we have a dilute solution, say of alcohol in water, the molecules of alcohol, at a given temperature, would have a given mean kinetic energy, [p028] and would be tending to move in all directions with a mass[40] and velocity, the same as if the alcohol were present as a gas or vapor at the same temperature. If the solution is sufficiently dilute, the dissolved alcohol molecules are sufficiently far apart, for average time, to make the molecular attractions between them negligible, just as is assumed for gases. As far as the alcohol (solute) molecules alone are concerned, they may, evidently, be assumed to be present in the solution, in the same condition, as to number, mean kinetic energy and mean velocity, as they would be in alcohol vapor of the same concentration and temperature. We may ask, now, whether the osmotic pressure of the solution may not result from the pressure on the solvent, growing out of its bombardment by the solute molecules. And we may ask, further, what numerical relation would subsist between such a pressure and the pressure of the solute, if the latter were present as a gas, under the same conditions of temperature and concentration. In order to be prepared to answer these questions, we must consider, in what way the presence of the solvent must modify the motions and the forces of impact of solute molecules.

One great difference between the dissolved substance and the gas would be, that, in the solution, the solute is in intimate contact with the ‹solvent›. A decided attraction must exist between the solute molecules and the solvent molecules, since we could not otherwise understand how a solvent, like water, in dissolving a nonvolatile substance like sugar, could overcome those molecular attractions between the sugar molecules, which make sugar a solid. But we note, that all the solute molecules in a solution, except those at the surface, are surrounded on all sides equally by the solvent. The attractive forces, exerted upon the single molecules of the solute by the solvent molecules, thus sum up to ‹zero›, and need not be considered further. Only the small number of solute molecules, which are at the surface of the liquid, would involve a minor correction in the application of the kinetic theory, and this need not be considered here.

A second point of difference between a substance in solution, and the same substance as a gas or vapor at the same temperature [p029] in the same volume, lies in the fact that a gas molecule will go a much greater distance without colliding with some second molecule and changing its path, than would a solute molecule, the latter molecule being closely surrounded by the molecules of the solvent. The mean ‹free path›, as it is called, will be very much shorter for a solute molecule than for a gas molecule, and we note, as a matter of fact, how slow is the diffusion through a solvent (see ‹exp.› p. 8). But the shortness of the previous path ‹does not affect the force of a blow resulting from the impact of a moving mass›, the force of the impact being dependent only on the mass and the change in speed of the striking particle, at the moment of impact. Thus the short free mean path of a dissolved molecule does not affect the mean ‹force› of the blow, ‹delivered when it strikes the resisting medium›.

The slow diffusion of a dissolved substance represents a difference in degree, not in kind, between gases and dissolved substances. Even in gases, we have such frequent collisions that the mean free path of an oxygen molecule at 0° and atmospheric pressure is only 0.00001 cm., whereas the velocity, the total path covered in one second, is 42,500 cm.

EXP. If a bulb containing a few drops of bromine is broken at the bottom of a tall cylinder, the bromine vapor is seen to diffuse rather slowly into the upper part of the cylinder, the bromine molecules, in their passage upward, rebounding from the air molecules, with which the cylinder is filled. If a second cylinder is first evacuated, and the bromine bulb is broken ‹in vacuo›, the vapor is seen to fill the cylinder instantly, the high velocity of the bromine molecules being thus revealed.

But a third question, of fundamental importance in the comparison of the condition of a substance existing as a gas and its condition in a solution of the same concentration and temperature, results from a consideration of the ‹frequency of the impacts› of the solute molecules against the solvent, growing out of the reduction of the lengths of the mean free paths of the solute molecules.[41] In order to be able to take this fact properly into account, it will be necessary to consider somewhat more precisely the manner in which, according to the kinetic theory, gas pressure is produced.

We may consider that we have in a cube of unit volume (1 c.c.) ‹n› molecules of a gas, each of mass ‹m› and average velocity ‹u› cm. per second. We may assume that one-third of the total number [p030] of molecules moves in each of the three dimensional directions.[42] A single molecule of mass ‹m›, striking the surface with a velocity ‹u› and rebounding with the same velocity in the opposite direction, will exert on the surface a force of 2 ‹m› ‹u› units. But, with a velocity of ‹u› cm. per second, it will reach the opposite wall and return to the surface we are considering, ‹u› / 2 times in one second. A single molecule will consequently exert a force 2 ‹m› ‹u› × ‹u› / 2 or ‹m› ‹u›^2 on the surface, and the ‹n› / 3 molecules moving in the same direction will exert a force ‹n› / 3 × ‹m› ‹u›^2 on the unit surface. This represents, therefore, the pressure of such a gas, as calculated on the basis of the assumptions of the kinetic theory. Now, when a gas is so strongly compressed, that the bulk of the molecules is not negligible in comparison with the total volume of the gas, the number of impacts on unit surface in unit time becomes sensibly greater than ‹n› / 3 × ‹u› / 2, since the distance to be covered between successive blows on the surface will be sensibly less than 2 cm., in a cube of unit volume. If we imagine, for the sake of a rough illustration, that one-third of the molecules in 1 c.c. are united into one spherical mass (indicated by A in Fig. 7), moving upwards and downwards, it is obvious that the distance covered between two successive blows on a surface is not 2 cm., but that distance diminished by twice the diameter of the sphere. For strongly compressed gases, the total number of impacts on unit surface is therefore sensibly greater than ‹n› / 3 × ‹u› / 2, and the pressure is proportionately greater. According to van der Waal's correction for this effect, ‹P› = ‹n› / 3 × ‹m› ‹u›^2 / (1 − ‹b›), where ‹b› represents the volume actually occupied by the molecules in 1 c.c. of the gas.[43]

Now, for solute molecules, the "free space" of movement, as we may call it, is, similarly, very considerably reduced by the presence [p031] of the solvent, and the reduction of this free space, as Nernst has shown, will have the same effect on the pressure produced against unit surface of the solvent by the bombardment of the solvent by the solute, as the reduction of the free space has on the gas pressure when a gas is strongly compressed. The resulting pressure on unit surface of the solution must thus be increased, from the pressure ‹P›_{gas}, which would be exerted by the solute against the walls of a vessel, if it were present as a gas of the same concentration, at the same temperature, to ‹P›_{gas} / (1 − ‹v›), where ‹v› represents the real volume occupied by the solvent and (1 − ‹v›) the ‹free space› for the solute molecules ‹in unit volume› of solution.[44] If osmotic pressure is the result of such a bombardment of the solvent by the molecules of the solute, one might, therefore, expect to find the osmotic pressure ‹very much greater› than the gas pressure of the same substance in the same volume at the same temperature. However, in all the ‹experimental determinations› (by means of semipermeable membrane, vapor pressure, boiling-point and freezing-point measurements) of the osmotic pressure as defined on p. 10, this ‹corrective factor cancels› out again.[45] ‹According to the kinetic theory›, the osmotic pressure of a substance in ‹dilute solution should, consequently, be found by experiment to be equal to the gas pressure which a gas, of the same molecular concentration, would exert at the same temperature›.[46]

We find thus that the significant coincidence between the osmotic pressure of a substance in dilute solution, as defined and measured according to van 't Hoff, and the gas pressure which the substance would exert, if it were present as a gas in the same volume and at the same temperature, is in agreement with the fundamental assumptions of the kinetic theory. This theory, consequently, gives us an adequate theoretical explanation of [p032] osmotic pressure, as it does of gas pressure. As van 't Hoff says,[47] "if the osmotic pressure follows Gay-Lussac's law and is proportional to the absolute temperature, then, like gas pressure, it will become zero at 0° absolute temperature and will vanish when molecular movements come to rest. It is therefore natural to look for the cause of osmotic pressure in kinetic phenomena and not in attractions."[48]

FOOTNOTES:

[29] L'Hermite, ‹Compt. rend.›, «39», 1177 (1854); van 't Hoff, ‹Lectures on Physical Chemistry›, ‹Part II›, p. 37.

[30] Nernst, ‹Theoretical Chemistry›, p. 103.

[31] Van 't Hoff, ‹Lectures on Physical Chemistry, Part II›, p. 37.

[32] ‹J. Phys. Chem.›, «10», 141 (1906).

[33] This term ‹must not be confounded with the term osmotic pressure›, which has been defined on p. 10.

[34] See Chapter VII on the law of physical or heterogeneous equilibrium, where the relations are discussed in detail.

[35] ‹Z. phys. Chem.›, «5», 175 (1890).

[36] ‹Ibid.›, «3», 119 (1889).

[37] ‹Phil. Mag.›, «38», 206 (1894).

[38] ‹Cf.› van 't Hoff's ‹Lectures on Physical Chemistry›, Vol. II, 40 (1899).

[39] The molecules may have different masses in the different conditions, and the principle of the mean kinetic energy would always apply to them as ‹they are›, in the condition under observation, and not as they are in some other condition; any change in mass, in solution, for instance, would show itself in the osmotic pressure measurements (see p. 18), just as it is shown in the measurements of gases, when the gas molecules show a change in composition, as is the case with hydrogen fluoride (H_{2}F_{2} ⇄ 2 HF), nitrogen tetroxide (N_{2}O_{4} ⇄ 2 NO_{2}), phosphorus pentachloride (PCl_{5} ⇄ PCl_{3} + Cl_{2}) and other compounds.

[40] The molecular weight of alcohol in dilute aqueous solution is the same (46) as in vapor form. Raoult, ‹Z. phys. Chem.›, «27», 656; Loomis, ‹ibid.›, «32», 592.

[41] Nernst, ‹Theoretical Chemistry›, p. 245.

[42] This assumption is not made in the rigorous development of the above relations on the basis of the kinetic theory, but it leads to the same net result.

[43] Even for gases of ordinary concentration, the introduction of the same correction gives an expression for the relation of pressure and volume, which is more exact than Boyle's law and is used in all exact calculations with gases.

[44] One may imagine, first, ‹n› molecules of the solute as a ‹gas›, with the pressure ‹P›_{gas}, in 1 c.c. Then, one may imagine, crudely, the ‹n› molecules of solute, in a free (gas) space of (1 − ‹v›) c.c., in the center of 1 c.c. of the solvent, and exerting by their impacts a pressure ‹P›_{osm.}, against the solvent. According to Boyle's law, we should then have, ‹P›_{gas} × 1 = ‹P›_{osm.} × (1 − ‹v›), and therefore ‹P›_{osm.} = ‹P›_{gas} / (1 − ‹v›).

[45] ‹Vide› Nernst, ‹Theoretical Chemistry›, p. 245, for the detailed discussion of this relation.

[46] This conclusion is reached more rigorously and more simply by thermodynamic analysis.

[47] ‹Lectures on Physical Chemistry›, Part II, p. 35.

[48] Rigorous developments of the relations between solute and solvent, for dilute and concentrated solutions, have been made by van der Waals, ‹Z. phys. Chem.›, «5», 133 (1890); van Laar, ‹ibid.›, «15», 457 (1894); G. N. Lewis, ‹J. Am. Chem. Soc.›, «30», 675 (1908), and Washburn, ‹ibid.›, «32», 653 (1910). An admirable review of the theories of osmotic pressure, by Lovelace, will be found in the ‹Am. Chem. J.›, «39», 546 (1908) («Stud.»).

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