CHAPTER II

«OSMOTIC PRESSURE AND THE THEORY OF SOLUTION I»

If a concentrated solution of a substance like sugar or cupric nitrate is allowed to flow into a cylinder of water (‹exp.› with cupric nitrate), we find that the outside forces—gravity—tend to draw the solution, whose specific gravity is greater than that of water, to the bottom of the cylinder. In this method of proceeding there is some inevitable mixing of the solution with the solvent, as the result of friction, but the main portion of the deep blue solution is drawn to the bottom of the vessel and forms a blue layer under the colorless water. A much sharper line of separation may be obtained by allowing the cupric nitrate solution to enter a cylinder of water from under the water (see Fig. 1), the great density of the nitrate solution causing it to displace the water without perceptible mixing of the two liquids (‹exp.›). If these vessels are left at rest, it may be noted, from day to day, that the copper nitrate diffuses further and further into the pure solvent, and careful examination would show that the solvent, in turn, diffuses also into the copper nitrate solution. The process is a very slow one, but it will continue until a solution of ‹uniform concentration› is reached, and this will be the case, whatever the shape of the vessel may be. If, for the moment, only the copper nitrate be considered—and what we are developing for the copper nitrate may be applied equally well, ‹mutatis mutandis›, to the diffusion of water into the copper nitrate solution—it is obvious then that copper nitrate in solution ‹diffuses in all directions›, even against the force of gravity, and it is plain also, that any object, resisting or arresting such a motion of material particles, must have a force or pressure exerted upon it. Whatever the ultimate [p009] cause of the motion, whether it is the result of inherent molecular velocities of the dissolved copper nitrate, or of an attraction between the solute and the solvent, or both, it is inevitable that a pressure must result from the impact of the moving solute against the solvent.[3]

We have thus phenomena of diffusion of solutes through solvents, exactly as we have the well-known diffusion of gases, and the two phenomena are unquestionably very much alike, the solute, like the gas, tending to diffuse from the place of higher, to that of lower concentration.[4] Likewise, if a solution of uniform concentration is heated in one part and not in another, the solute,[4] like a gas under similar conditions, will move from the warmer to the colder part of the solution, as was demonstrated by Soret.[5] Without committing ourselves for the present to any given reason for the diffusion, we note that the tendency to diffusion is a fact, and we must accept the conclusion that every obstacle to such diffusion must have a ‹pressure› exerted upon it.

Now, if a solution is separated from the pure solvent by means of a so-called ‹semipermeable membrane›, some of the results of this tendency to diffusion may be demonstrated.

EXP. A concentrated solution of cane sugar in water, colored with some aniline dye, is enclosed in a thimble of parchment paper firmly fastened to a long narrow glass tube (see Fig. 2) and the cell is placed in a vessel of pure water. The parchment is not absolutely semipermeable, but it is approximately so, allowing the solvent, water, to pass, but being practically impervious to the solute sugar. A Schleicher and Schüll diffusion-thimble, No. 579, may be used, with advantage, as the thimble. (‹Cf.› Smith's ‹Introduction to Inorganic Chemistry›, p. 284.) [p010]

We observe, presently, that the system is not in a condition of equilibrium; water passes through the thimble into the sugar solution and the latter expands, producing a decided difference of level, and consequently a hydrostatic pressure, between the liquid in the cell and the solvent outside of it. We may note two facts: first, that the change includes an expansion of the solute,[6] the sugar, in the solution—that is, the tendency of the solute to expand into larger volumes of the solvent is satisfied exactly as in the experiment (Fig. 1) described above. In the second place, like all natural phenomena which proceed spontaneously, ‹the change is in the direction of equilibrium›; for when the hydrostatic pressure on the solution in the cell becomes sufficiently great, or if it is made sufficiently great at once by the application of some outside pressure, ‹a point of equilibrium is reached, at which water will pass neither into the cell nor out of it›. At that point, the tendency to expansion, both of the solute and of the solvent in the solution, is just overcome by the pressure on the solution.

«Definition of Osmotic Pressure.»—‹The hydrostatic pressure which is necessary to bring the solution into equilibrium with the pure solvent, when the two are separated by a semipermeable membrane, may be defined, according to van 't Hoff, as the measure of› what is called the ‹osmotic pressure of the solution›. We note that this definition still does not commit us to any theory as to the origin of the pressure, but merely ‹formulates› an ‹experimental relation›.

«Measurement of Osmotic Pressure.»—More perfect semipermeable membranes can be produced. These make possible quantitative measurements of the hydrostatic pressure on a solution, when equilibrium between the solution and the pure solvent [p011] has been reached. Such membranes were first used by Pfeffer. They consist of certain gelatinous precipitates, notably copper ferrocyanide. Films of these precipitates may be formed, under proper conditions, which are permeable to water but not to certain solutes, such as cane sugar, glucose and galactose.

By precipitating these membranes in the pores of unglazed clay cells, especially by the process devised by Morse,[7] we may make them sufficiently strong to resist enormous pressures—some used by the Earl of Berkeley were found to withstand a pressure of 130 atmospheres. The hydrostatic pressure required to produce equilibrium may then be measured in either of two ways. The first method, used originally by Pfeffer and more recently by Morse and Frazer[8] and their collaborators in a wonderfully conscientious study of osmotic pressures, consists in allowing the hydrostatic pressure to establish itself by the passage of very small quantities of the solvent, through the membrane, into the tightly closed cell containing the solution. When the resulting pressure produces a condition of equilibrium, it is measured[9] by a manometer connected with the solution, much as a gas pressure may be measured (Fig. 3).[10] This process requires considerable time for exact measurements—weeks, during which the cell must be kept at a constant temperature. The second method, which has been used by Berkeley and Hartley,[11] is very much more rapid and requires only a few hours for the measurement. It consists in having the pure solvent within the cell, instead of outside of it, and in [p012] exerting an external pressure on the solution outside of the cell, until a delicate manometer, communicating with the pure solvent, shows that water does not pass through the membrane in either direction—equilibrium having been reached.

«Osmotic Pressure and the Laws of Gases.»—The work of van 't Hoff, which has proved of inestimable value to the development of chemistry, succeeded in demonstrating that, ‹for dilute solutions, the osmotic pressure›, as defined above, ‹obeys the common laws of gases›,[12]—‹that, in fact, a substance in a dilute solution has an osmotic pressure equal to the gas pressure which it would exert if it were a gas of the same volume and at the same temperature›.[13]

Space does not permit the presentation of all the details of the evidence confirming this conclusion, but some of the most direct experimental proofs[14] will be considered. [p013]

«Boyle's Law.»—Boyle's law for gases states that, at a constant temperature, the pressure of a gas changes inversely as its volume, or directly as its concentration. Mathematically we have ‹P› : ‹P′› = ‹V′› : ‹V› or ‹P› ‹V› = ‹P′› ‹V′› = a constant, and ‹P› : ‹P′› = ‹C› : ‹C′› or ‹P› : ‹C› = ‹P′› : ‹C′› = a constant. When van 't Hoff published his first paper on the subject, Pfeffer's results from the direct measurement of the osmotic pressures of cane-sugar solutions were available, and even these, although experimentally not as exact as more recent determinations, showed plainly that, at a given temperature, the osmotic pressure of a sugar solution varies directly as the concentration, or inversely as the volume containing a given weight of the sugar. At 13–16° we have:

Concentration. Osmotic Pressure. Pressure/ mm. Mercury. Concentration. 1.00% 535 535 2.00% 1016 508 2.74% 1518 554 4.00% 2082 521 6.00% 3075 513

The ratio of pressure to concentration varies irregularly round a mean value of 526, and is approximately constant. The more recent, exceedingly careful measurements of Morse and Frazer confirm the conclusion, that Boyle's law holds for the osmotic [p014] pressures of dilute solutions; they find that the osmotic pressures of glucose and of cane-sugar solutions vary directly as the concentrations of the solutions, at a constant temperature.[15]

«Gay-Lussac's Law.»—Gay-Lussac's law for gases states that, if the volume of gas is kept constant, its pressure increases by 1 / 273 of its value for every degree above 0° C., or ‹P›_{‹t›} = ‹P›_{0} (1 + ‹t› / 273).

Expressing the temperature in absolute degrees, we have more simply:

‹P›_{‹t›} = ‹P›_{0} (‹T› / 273) or P_{t} / T = ‹P›_{0} / 273 = a constant.[16]

That is, the pressure of a gas varies directly as its absolute temperature, if the volume is kept constant.

Pfeffer's results, on the osmotic pressure of sugar solutions at different temperatures, were not sufficiently accurate to enable van 't Hoff to use them to confirm positively the rigorous thermodynamic proof (footnote 3, p. 12), that the osmotic pressure must increase proportionally to the absolute temperature, as required by Gay-Lussac's law. But the data did show, uniformly, a marked increase of the osmotic pressure with the temperature and, frequently, excellent agreement between theory and experiment. More striking were the results obtained by van 't Hoff in testing the correctness of this extension of Gay-Lussac's law by means of Soret's results on the diffusion of a solute from a warmer to a colder place. It was found that the concentrations, obtained by Soret when equilibrium was reached, agreed closely with the demand that the osmotic pressures in the colder and the warmer parts of the solution should be equal, and that the osmotic pressure of a given weight of solute in a given volume should increase proportionally to the absolute temperature. An elevation of temperature, in a portion of a uniform solution, will increase the osmotic pressure of this part. Diffusion will follow, until the loss in concentration of the solute, and therefore the loss of osmotic pressure (Boyle's law), of the warmer part, and the increased concentration and increased pressure of the colder portion result in all parts of the solution having the same osmotic pressure. [p015] As an example, a concentration of 17.33% copper sulphate at 20° was found to be in equilibrium with a concentration of 14.03% at 80°. Now, if the 17.33% solution had an osmotic pressure of ‹P› mm. at 20°, a 14.03% solution at the same temperature would have a pressure of (14.03 / 17.33) × ‹P› mm. (Boyle's law), and this would increase to (14.03 / 17.33) × ‹P› × (353 / 293) mm. at 80° C., or 0.975 ‹P› mm.—a result showing that the osmotic pressure in the hot part was practically the same as that, (‹P›), in the cold part of the solution.[17]

It is a source of great satisfaction, that the recent very exact and painstaking work of Morse and Frazer,[18] in measuring osmotic pressures directly, completely confirms this fundamentally important conclusion, that the osmotic pressure of a solution does increase proportionally to its absolute temperature.

«The Avogadro-van 't Hoff Hypothesis.»—For chemists, the most important part of van 't Hoff's work lies in the extension of ‹the Avogadro Hypothesis› to solutions. As van 't Hoff expresses it, "equal volumes of the most different solutions, having the same osmotic pressure and the same temperature, contain the same number of dissolved molecules,—that number, namely, which would be found in the same volume of a gas at the same gas pressure and temperature."[19] [p016]

Pfeffer's measurements, with solutions of 1 g. of sugar in 100 c.c. of water (the volume of the solution is 100.6 c.c.), were shown to prove, that the observed osmotic pressures agreed excellently with the gas pressures, calculated for the equimolar weight of hydrogen, in the same volume and at the same temperature:

Osmotic Pressure Temperature. ──────────────────────────

Found. Calculated.[20] Atmosphere. Atmosphere. 6.8 0.664 0.665 13.7 0.691 0.681 14.2 0.671 0.682 15.5 0.684 0.686 22 0.721 0.701 32 0.716 0.725 36 0.746 0.735

Morse's more recent and more exact results show, that the osmotic pressure of solutions of cane sugar and of glucose (corrected for the volume occupied by the sugar, see footnote, p. 15) agrees within 6% with the values demanded by van 't Hoff's theory, being about 6% larger for concentrations ranging from 0.1 to 1.0 molar. The difference of 6% is noteworthy and is probably due to secondary causes, but suggests extended investigation of its source.

«Indirect Determinations of Osmotic Pressure.»—The experimental results given have been obtained by direct measurements of osmotic pressures with the aid of semipermeable membranes. [p017] Perfect membranes are very difficult to prepare, and membranes of this kind can be used only with a few solutes. Nature offers us, however, forms of semipermeable "walls" between solutions and pure solvents, which in many instances are perfect. The atmosphere, above a volatile pure solvent and a solution of a nonvolatile substance in that solvent, when both liquids are placed side by side in a closed space, would serve as a semipermeable wall: the solvent vaporizes and may pass freely from solvent to solution and ‹vice versa›, but the solute, in the case under consideration, is nonvolatile and therefore cannot pass through the atmosphere. The vapor pressure of a pure solvent being always found to be higher than that of a solution in this solvent, at the same temperature, the solvent would pass in such a closed space as vapor ‹from the pure solvent› and would ‹condense› in the solution; it thereby dilutes the solution and the solute, and the solvent in the solution, expand, exactly as in the absorption of a solvent by a solution through a semipermeable membrane. Again, the vapor pressure of a solution being lower than that of the pure solvent, the solution (of a nonvolatile solute) must be heated higher than the pure solvent, to bring both to the boiling-point; that is, there is an ‹elevation of the boiling-point›, when a nonvolatile solute is dissolved in a solvent. The solute being nonvolatile, only the solvent passes off in the process of boiling, the solute becomes ‹more concentrated›, and, according to van 't Hoff's extension of Boyle's law, the ‹osmotic pressure of the solution increases›. Similarly, when a solution is cooled until freezing occurs, provided the solute does not crystallize out with the solvent, the concentration of the solute is again increased, and therefore the ‹osmotic pressure› of the solution is also increased. Van 't Hoff recognized the relations existing between the freezing, boiling and vaporization of solutions, on the one hand, and the changes of their osmotic pressures on the other. By developing rigorously the ‹relations between the lowering of the vapor tension, the raising of the boiling-point, the lowering of the freezing-point› of a solvent by a solute ‹and the osmotic pressure of the solution›, he made it possible[21] to use [p018] extensive experimental material,[22] on the elevation of boiling-points and the lowering of freezing-points and of vapor tensions, to determine the osmotic pressures of solutions. The theory of the relation of osmotic pressure to gas pressure is fully confirmed by these measurements, for those cases to which it may properly be applied, namely, to sufficiently dilute solutions and such as have only negligible heats of dilution, ‹i.e.› in which dilution does not involve chemical changes.

«Apparent Exceptions.»—Instead of discussing the vast amount of material of this kind, which agrees with van 't Hoff's theory, we may consider, more profitably, typical cases of ‹apparent exceptions›. The most important instance of this kind, the case of solutions of compounds which undergo ‹electrolytic dissociation or ionization›, will be separately discussed in the next chapter, and we shall find that van 't Hoff's great generalization is a vital element in the evidence of this important form of dissociation. Of other apparent exceptions, we may note the fact that some solutes seem to give "abnormally" ‹low› osmotic pressures[23] in certain solutions. For instance, benzoic acid, in benzene solutions, gives only a little more than half as great an osmotic pressure as it does in aqueous solutions of the same concentration and temperature, and as would be calculated on the basis of the Avogadro-van 't Hoff Hypothesis for a compound of the formula C_{6}H_{5}COOH and the molecular weight 122. But a rigorous study[24] of the distribution of benzoic acid between water and benzene, when solutions of the acid in the two solvents are shaken together until equilibrium is established (Chapter VIII), has proved that the distribution is strictly in accord with the assumption that benzoic acid, in aqueous solution, has the molecular weight 122 and the composition C_{6}H_{5}COOH, and that, in benzene solution, it has the molecular weight 244 and the composition (C_{6}H_{5}COOH)_{2}; only a ‹small part› of the acid (C_{6}H_{5}COOH)_{2} is decomposed in benzene solution into the simpler molecules, of the composition C_{6}H_{5}COOH. In other words, the simpler molecules C_{6}H_{5}COOH are ‹polymerized› or ‹associated› to form larger molecules in benzene solution, much as the gas nitrogen dioxide NO_{2} goes over more or less into the gas N_{2}O_{4}, especially at low temperatures, and as hydrogen fluoride at low temperatures has the composition H_{2}F_{2}, while at higher temperatures it is HF. The divergence of the benzene solutions of benzoic acid from the Avogadro-van 't Hoff principle is therefore only an ‹apparent one›, not a real one, inasmuch as the osmotic pressure of the solutions agrees perfectly with that calculated for solutions of a substance (C_{6}H_{5}COOH)_{2}, of molecular weight 244. Such associated molecules (of organic acids, alcohols, phenols, etc.) occur [p019] particularly readily in liquids of small dissociating power, like benzene, and such solutions show marked ‹absorption of heat on dilution›,[25] the dilution being accompanied by a ‹chemical change›. The associated molecules are dissociated more and more completely [(C_{6}H_{5}COOH)_{2} ⇄ 2 C_{6}H_{5}COOH], even in these solvents, as the solutions are diluted. Since dilution results in a ‹chemical increase› in the number of molecules, the osmotic pressure cannot decrease proportionally with the increase of volume in such a case as this. Nor does gas pressure, it must be remembered, decrease proportionally to the volume in the case of gases which show ‹chemical changes› with change of volume, ‹e.g.› in the case of nitrogen tetroxide, for which we have N_{2}O_{4} ⇄ 2 NO_{2}.

In still other instances, apparently too high osmotic pressures, or too low molecular weights, have been found by the application of the Avogadro-van 't Hoff Hypothesis to solutions: for instance, the molecular weight of sodium, when dissolved in mercury, was found by Ramsay to vary from 21.6, in dilute, to 15.1 in concentrated solutions. But Cady found that the heat of dilution of sodium in mercury solution is considerable, and by taking this properly into account, Bancroft was able to show that the molecular weight, correctly calculated in a given experiment, is 22.7 (agreeing well with the theoretical weight 23), in place of 16.5, as calculated without making the required allowance for the heat of dilution.[26] These determinations are most instructive in showing that the sources of some of the most important deviations from the van 't Hoff-Avogadro principle, deviations which have been brought forward as arguments against its assumptions, are due, not to any untrustworthiness of the general principle, but to the error of neglecting to observe the limiting conditions of the formulation, or of neglecting to make corresponding corrections for the non-observance thereof.

«Summary.»—Van 't Hoff's theory of solution—that the osmotic pressure of substances in solution obeys the laws of gases, and that equal volumes of the most varied dilute solutions, having the same temperature and osmotic pressure, contain the same number of dissolved molecules, that number, namely, which would be found in the same volume of a gas at the same temperature and gas pressure,—accords thus, not only with the demands of thermodynamics,[27] but is also, within the limits demanded by the theory itself, in agreement with the best experimental measurements of osmotic pressures that have been made in recent years. The apparent exceptions, as in the cases just described and, as we shall find, in the case of electrolytic dissociation, are found to be no exceptions, when the conclusions, reached on the assumption that [p020] the theory is correct, are tested rigorously by independent methods of investigation.[28]

The fundamental laws of gases and the Avogadro Hypothesis may be condensed into the following general equation, expressing all of the laws, viz.: ‹P› ‹V› = ‹n› ‹R› ‹T›. This equation applies equally to the osmotic pressures of dilute solutions, the osmotic pressure being substituted for the gas pressure. In the equation, ‹T› is the absolute temperature of the gas or solution, ‹P› the gaseous or osmotic pressure, ‹V› the free space of the gas volume, ‹i.e.› the volume of the gas less the volume occupied by the gas molecules, or the volume of the pure solvent in the solution used, ‹i.e.› the volume of the solution less the volume of the solute. ‹R› is the so-called ‹gas-constant›, and represents ‹the work› done against the external pressure when one gram molecule, or mole, of the gas is heated one degree and allowed to expand, say at constant pressure ‹P›, against an external pressure ‹P›; ‹n› represents the number of gram molecules or moles of gas or solute used (the total weight of solute or gas, divided by the average weight of a mole in the gas or solute). If a given weight of a gas or solute is taken, and no dissociation or association occurs (such as would involve appreciable heats of dilution), then ‹n› is a given number; and, therefore, at a given temperature ‹T›, all the factors on the right side of the general equation being given numbers, ‹P› ‹V› ‹is a constant› (Boyle's law). For a given quantity of gas or solute (‹n› is a given number), kept at ‹constant volume› ‹V›, the pressure must vary as the absolute temperature (Gay-Lussac's law); ‹P› / ‹T› = ‹n› ‹R› / ‹V› = ‹a constant›. When the pressure, volume and temperature of two gases, or two dilute solutions, are equal, ‹n›, the number of gas or solute molecules present, must be the same (Avogadro-van 't Hoff Hypothesis); ‹n› = ‹P› ‹V› / (‹R› ‹T›), and all the factors of the right side are the same for the gases and solutions which we are comparing. Finally, if the pressure is expressed in atmospheres, the volume in litres, and the temperature in absolute degrees, the ‹gas-constant› ‹R› = ‹P› ‹V› / ‹T› = 1 × 22.4 / 273 = 0.082.

FOOTNOTES:

[3] Even after a solution of uniform concentration of the solute is formed, the tendency toward diffusion, and the diffusion itself, and the resulting pressure must still persist. But a state of ‹dynamic› (or flowing) ‹equilibrium› must be considered now to exist, the loss caused by the moving away of the solute, from a given part of the solution, being balanced by the diffusion (into that part) of the solute from the neighboring parts. Whether one ascribes the diffusion to inherent molecular velocities of the solute, or to an attraction between solvent and solute, the discrete particles of the solute in a solution of uniform concentration will continue to have such inherent velocities (Chap. III), and will also continue to be surrounded by pure solvent, exactly as in solutions of unequal concentrations, where the diffusion may be observed, because the net result, in such a case, is a one-sided action.

[4] This again holds equally for the solvent.

[5] See below.

[6] At the same time, the change is also in the direction of an expansion of the ‹solvent in the solution›. The two changes are not opposed to each other, but supplementary.

[7] ‹Am. Chem. J.›, «28», 1 (1902); «40», 266, 325 (1908) («Stud.»).

[8] ‹Am. Chem. J.›, «34», 1 (1905); «36», 39 (1906); «37», 324, 425, 558 (1907); «38», 175 (1907).

[9] The exact concentration of the solution at the point of equilibrium is determined by subsequent analysis.

[10] ‹Cf.› Smith's ‹Inorganic Chemistry›, p. 287.

[11] Berkeley and Hartley, ‹Phil. Trans. Roy. Soc.› A, «206», 481 (1906).

[12] When appreciable ‹heat of dilution› is shown by a solution, some chemical change, resulting from dilution, is indicated (such as, dissociation of the solute, hydration, hydrolysis, etc.). In such a case, the Avogadro-van 't Hoff principle holds for each concentration for its actual composition, and the principle may often be used to determine the extent of the chemical change produced by dilution. But then the osmotic pressure will not obey Boyle's and Gay-Lussac's laws. The same exception applies also to gases which undergo chemical changes, as the result of dilution or change of temperature. In the case, for instance, of nitrogen tetroxide, which dissociates according to N_{2}O_{4} ⇄ 2 NO_{2}, the extent of the dissociation varies with changes of concentration (pressure) and of temperature, and the gas does not obey the laws of Gay-Lussac and of Boyle. In regard to the rôle of heat of dilution in connection with osmotic pressure, see Bancroft, ‹J. Phys. Chem.›, «10», 319 (1906).

[13] See p. «15» for a more rigorous statement concerning the volume. ‹Cf.› Morse and Frazer, ‹Am. Chem. J.›, «34», 1 (1905).

[14] As a result of numerous vain endeavors, as well as of much direct evidence of a positive character, the scientific world has, for many years, held the opinion that any sort of "perpetual motion machine" is impossible. Every one now admits that a machine which would be able to work continuously, without consuming energy, is an impossibility—that is, that a "‹perpetuum mobile of the first class›," as it is called, is impossible (law of the conservation of energy or ‹first law› of thermodynamics). From this law it does not of necessity follow, however, that it would be impossible to make a machine or device that would convert ‹continuously› into available energy or work, say, the enormous amounts of heat energy of the earth or of large bodies of water ("dissipated energy") which would thereby be ‹cooled below› the temperatures of their surroundings. Such a hypothetical process has been termed a "‹perpetuum mobile of the second class›"; it has never been realized and is universally conceded to be an impossibility; the so-called "‹second law› of thermodynamics" gives expression to this fact.

Now van 't Hoff [‹Z. phys. Chem.›, «1», 481 (1887)] showed, first, that a gas like oxygen, nitrogen, hydrogen, etc., which is soluble in proportion to its gas pressure (Henry's law), must exert, in solution, an osmotic pressure equal to the gas pressure, which it would have, if present in the same quantity as a gas in the same volume at the same temperature; for, if such were not the case, the solution and gas could be used to produce a ‹perpetuum mobile of the second class›, which, according to the above law, is an impossibility. Similar proofs were given by Rayleigh [‹Nature›, «55», 253 (1897)] and by Larmor [‹Phil. Trans.›, «190», 266 (1897), ‹Nature›, «55», 545 (1897)] that the principle applies to solutions of other solutes.

Provided, then, that we have (1) perfect semipermeable membranes, (2) sufficiently dilute solutions, and (3) none but negligible heats of dilution (p. 12), van 't Hoff's generalization, concerning the relation of osmotic pressure and the laws of gases, must hold, if the ‹perpetuum mobile of the second class› is impossible, as is demanded by the second law of thermodynamics.

[15] See p. 15 in regard to the relation for concentrated solutions.

[16] The pressure ‹P›_{0} of a given quantity (weight) of a gas at 0° C., in a given constant volume, is also a given number and consequently ‹P›_{0}/273 is a constant under these conditions.

[17] The slight differences in the ionization of copper sulphate solutions of 14% and 17% and at 20° and 80° are not included in the calculation, ionization being unknown, when van 't Hoff made his calculations.

[18] ‹Am. Chem. J.›, 41, 258 (1909).

[19] In the light of recent work, especially by Morse and Frazer, the law would state, more exactly, that a substance in solution produces the osmotic pressure, at a given temperature, which it would exert, if it were contained as a gas, at the same temperature, ‹in the volume occupied by the pure solvent› of the solution. For sufficiently dilute solutions, the volume of the solution and the volume of the solvent may be considered identical; for more concentrated solutions, there is a decided difference, and the correct volume to use in calculation is the volume of the solvent alone, ‹i.e.› the volume of the solution reduced by the volume of the pure solute. This corresponds to the correction of the volume in the more accurate expression for the behavior of gases, developed by van der Waals; in place of ‹v›, the total gas volume, (‹v› − ‹b›), the total volume of the gas less the volume of the spheres of action of the gas particles, is used, especially for strongly compressed or concentrated gases. It may be added that van 't Hoff's thermodynamic proof involves the same correct definition of the volume that Morse and Frazer subsequently developed experimentally. ‹Cf.› Bancroft, ‹J. Phys. Chem.›, «10», 319 (1906).

[20] One gram of cane sugar, C_{12}H_{22}O_{11} (the mol. wt. is 342) corresponds to 1 / 342 gram molecule or mole and, therefore, to 2.02 / 342 gram of hydrogen. The volume containing this quantity of hydrogen is 100.6 c.c.; a liter would contain 2.02 / 342 × 1000 / 100.6 gram of hydrogen. The pressure of a mole or 2.02 grams of hydrogen, contained in a liter at 0°, is 22.4 atmospheres, and the pressure of the quantity of hydrogen given above, in a liter, would be (2.02 × 1000) / (342 × 100.6) × (22.4 / 2.02) at 0°. At 36° C., for instance, the pressure would be 309 / 273 times as great, or ‹P›_{calculated} = (2.02 × 1000 × 22.4 × 309) / (342 × 100.6 × 2.02 × 273) = 0.735 atmosphere.

[21] The exact relations are discussed in van 't Hoff's ‹Lectures on Physical Chemistry›, Part II, pp. 42–59, Nernst's ‹Theoretical Chemistry› (1904), pp. 142 and 148, and H. C. Jones's ‹The Elements of Physical Chemistry› (1909), pp. 252, 271.

[22] ‹Vide› Raoult, ‹Scientific Memoir Series›, «4», 71, 127.

[23] ‹I.e.› abnormally small depressions of freezing-points or elevations of boiling-points.

[24] Nernst, ‹Theoretical Chemistry›, p. 486; Hendrixson, ‹Z. anorg. Chem.›, «13», 73 (1897).

[25] ‹Cf.› Bancroft, ‹J. Phys. Chem.›, «10», 319 (1906).

[26] For the discussion of other instances, ‹vide› Bancroft, ‹loc. cit.›

[27] Footnote 3, p. 12.

[28] For example, determinations of distribution coefficients (p. 18), heats of dilution (p. 19), conductivities and chemical activity (Chapters IV–VI).

[p021]