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Solution

Solid Solutions

In all the cases hitherto considered, the liquid phase alone has been capable of continuous variation in composition. The solid phases each have been of one definite substance. Crystals of ice may lie side by side with crystals of common salt, but each crystalline individual is either ice or salt; no one crystal contains both components in proportions which can be varied continuously. But, in other cases, crystals are known in which both components may enter. Such phenomena are well known in the alums - double sulphates of aluminium with another metal. Here the other metal may be one, such as potassium, or two, such as potassium and sodium, and, in the latter case, the proportion between the two may vary continuously throughout wide limits. Such structures are known as mixed crystals or solid solutions.

The theoretical form of the freezing point diagrams when solid solutions are present depends on the relation between the available energy and the composition in the two phases. This relation is known when the amount of either component present in the other is very small, for it is then the relation for a dilute system and can A B A B A B A FIG. 8. FIG. 9. FIG. 10. FIG.°I I.

be calculated. But at intermediate compositions we can only guess at the form of the energy-composition curve, and the freezing point composition curve, deduced from it, will vary according to the supposition which we make. With the most likely forms for the energy curves we get the accompanying diagrams for the relation between freezing point and concentration.

It will be noticed that in all these theoretical curves the points of initial fusion and solidification do not in general coincide; we reach a different curve first according as we approach the diagram from below, where all is solid, or from above, where all is liquid. Again, it will be seen that the addition of a small quantity of one component, say B, to the other, A, does not necessarily lower the melting point, as it does with systems with no solid solutions; it is quite as likely to cause it to rise. The second and third figures, too, show that the presence of solid solutions may simulate the phenomena of chemical combination, where the curve reaches a maximum, and of non-variant systems where we get a minimum. The fourth figure shows that, in some cases, it should be possible for solid solutions to be present in a limited part of the field only, being absent between the two nearly vertical lines in fig. II. Experiment has revealed the existence of systems in which these phenomena are displayed. As an example we may take the case of mixtures of naphthalene and 13-naphthol, substances which form solid solutions in each other. The freezing and melting point curves are exactly similar to theoretical curves of fig. 8, the point A representing pure naphthalene and B pure /3-naphthol. When the equilibria become more complex difficulties of interpretation of the experimental results often arise. It is often very difficult to distinguish between a chemical compound, for example, and the case of solid solution represented by fig. 9. All available evidence, from the freezing point curve and from other sources must be scrutinized before an opinion is pronounced. But the elucidation of the complicated phenomena of solid solutions would have been impossible without the theoretical knowledge deduced from the principle of available energy.

Supersaturation

When a crystal of the solid phase is present the equilibrium of a solution is given by the solubility curves we have studied. If, however, a solution be cooled slowly past its saturation point with no solid present, crystallization does not occur till some lower temperature is reached. Between the saturation point and this lower temperature, the liquid holds in solution more of the solute than corresponds with equilibrium, and is said to be supersaturated. A familiar example is to be found in solutions of sodium sulphate, which may be cooled much below their saturation point and kept in the liquid state till a crystal of the hydrate Na 2 SO 4 IoH 2 O is dropped in, when solidification occurs with a large evolution of latent heat. These phenomena are explicable if we consider the energy relations, ea -60 20 86 for the intrinsic energy of a system will contain terms depending on the area of contact between different phases, and, for a given mass of material, the area will be greater if the substance is finely divided. Hence the conditions necessary to secure equilibrium when the solid phase is present are not the same as those necessary to cause crystallization to start in a number of crystals at first excessively minute in size. The corresponding phenomenon in the case of vapours is well known. Dust-free air will remain supersaturated with water-vapour in conditions where a dense cloud would be formed in presence of solid dustnuclei or electric ions which serve the same purpose.

If a solution of a salt be stirred as it cools in an open vessel, a thin shower of crystals appears at or about the saturation temperature. These crystals grow steadily, but do not increase in number. When the temperature has fallen about 10° C. below this point of saturation, a dense shower of new crystals appear suddenly. This shower may be dense enough to make the liquid quite opaque. These phenomena have been studied by H. A. Miers and Miss F. Isaac. If the solution be confined in a sealed glass tube, the first thin shower is not formed, and the system remains liquid till the secondary dense shower comes down. From this and other evidence it has been shown that the first thin shower in open vessels is produced by the accidental presence of tiny crystals obtained from the dust of the air, while the second dense shower marks the point of spontaneous crystallization, where the decrease in total available energy caused by solidification becomes greater than the increase due to the large surface of contact between the liquid and the potentially existing multitudinous small crystals of the shower.

If the temperature at which this dense spontaneous shower of crystals is found be determined for different concentrations of solution, we can plot a "supersolubility curve," which is found generally to run roughly parallel to the "solubility curve" of steady equilibrium between liquid and already existing solid. When two substances are soluble in each other in all proportions, we get solubility curves like those of copper and silver shown in fig. 5. We should expect to find supersolubility curves lying below the solubility curves, and this result has been realized experimentally for the supersolubility curves of mixtures of salol (phenyl salicylate) and betol (/3-naphthol salicylate) represented by the dotted lines of fig. 12.

In practical cases of crystallization in nature, it is probable that these phenomena of supersaturation often occur. If a liquid mixture so of A and B (fig. 12) were inocu lated with crystals of A when its composition was that represented so by x, cooled very slowly and stirred, the conditions would be hose of equilibrium throughout. 6 When the temperature sank to a, on the freezing point curve, E40 crystals of pure A would appear. The residual liquid would thus become richer in B, and the tem perature and composition would pass along the curve till E, the eutectic point, was reached. The 20 40 60 80 100 liquid then becomes saturated with B also, and, if inoculated with B crystals, will deposit B alongside of A, till the whole mass is solid. But, if no solid be present initially, or if the cooling be rapid, the liquid of composition x becomes supersaturated and may cool till the supersaturation curve is reached at b, and a cloud of A crystals comes down. The temperature may then rise and the concentration of B increase in the liquid in a manner represented by some such line as b f. The conditions may then remain those of equilibrium along the curve f E, but before reaching f the solution may become supersaturated with B and deposit B crystals spontaneously. The eutectic point may never be reached. The possibility of these phenomena should be borne in mind when attempts are made to interpret the structure of crystalline bodies in terms of the theory of equilibrium.

Osmotic Pressure

The phase rule combined with the latent heat equation enables us to trace the general phenomena of equilibrium in solutions, and to elucidate and classify cases even of great complexity. But other relations between the different properties of solutions have been investigated by another series of conceptions which we shall proceed to develop. Some botanical experiments made about 1870 suggested the idea of semi-permeable membranes, i.e. membranes which allow a solvent to pass freely but are impervious to a solute when dissolved in that solvent. It was found, for instance, that a film of insoluble copper ferrocyanide, deposited in the walls of a porous vessel by the inward diffusion and meeting of solutions of copper sulphate and potassium ferrocyanide, would allow water to pass, but retained sugar dissolved in that liquid. It was found, too, when water was placed on one side of such a membrane, and a sugar solution in a confined space on the other, that water entered the solution till a certain pressure was set up when equilibrium resulted.

The importance of these experiments from the point of view of the theory of solution, lay in the fact that they suggested the conception of a perfect or ideal semi-permeable partition, and that of an equilibrium pressure representing the excess of hydrostatic pressure required to keep a solution in equilibrium with its pure solvent through such a partition. Artificial membranes are seldom or never perfectly semi-permeable - some leakage of solute nearly always occurs, but the imperfections of actual membranes need no more prevent our use of the ideal conception than the faults of real engines invalidate the theory of ideal thermodynamics founded on the conception of a perfect, reversible, frictionless, heat engine. Further, in the free surface the solutions of an involatile solute in a volatile solvent, through which surface the vapour of the solvent alone can pass, and in the boundary of a crystal of pure ice in a solution, we have actual surfaces which are in effect perfectly semipermeable. Thus the results of our investigations based on ideal conceptions are applicable to the real phenomena of evaporation and freezing.

Dilute Solutions

Before considering the more complicated case of a concentrated solution, we will deal with one which is very dilute, when the theoretical relations are much simplified. The vapour pressure of a solution may be Pressure. measured experimentally by two methods. It may be compared directly with that of the pure solvent, as the vapourpressure of a pure liquid is determined, by placing solvent and solution respectively above the mercury in two barometer tubes, and comparing the depressions of the mercury with the height of a dry barometer at the same temperature. This method was used by Raoult. On the other hand, a current of dry air may be passed through the series of weighed bulbs containing solution and solvent respectively, and the loss in weight of each determined. The loss in the solution bulbs gives the mass of solvent absorbed from the solution, and the loss in the solvent bulbs the additional mass required to raise the vapour pressure in the air-current to equilibrium with the pure solvent. The relative lowering of vapour pressure of the solution compared with that of the solvent is measured by the ratio of the extra mass absorbed from the solvent bulbs to the total mass absorbed from both series of bulbs. Experiments by this method have been made by W. Ostwald and J. Walker, and by Lord Berkeley and E. G. J. Hartley.

The vapour pressure of the solution of a non-volatile solute is less than the vapour pressure of the pure solvent. Hence if two vessels, one filled with solvent and one with solution, be placed side by side in an exhausted chamber, vapour will evaporate from the solvent and condense on the solution. The solution will thus gain solvent, and will grow more and more dilute. Its volume will also increase, and thus its upper surface will rise in the vessel. But as we ascend in an atmosphere the pressure diminishes; hence the pressure of the vapour in the chamber is less the higher we go, and thus eventually we reach a state of equilibrium where the column of vapour is in equilibrium at the appropriate level both with solvent and solution. Neglecting the very small buoyancy of the vapour, the hydrostatic pressure P at the foot of the column of solution is h g p where h is the height of the column and p the mean density of the solution. If the height be not too great, we may assume the density of the vapour to be uniform, and write the difference in vapour pressure at the surfaces of the solvent and of the solution as p - p' = hgo-. Hence we find that p - p' = Po-/p for a very dilute solution, where the difference p-p' is small and the height of the balancing column of solution small.

In practice the time required to reach these various conditions of equilibrium would be too great for experimental demonstration, but the theoretical consideration of vapour pressures is of fundamental importance. Let us suppose that we possess a partition such as that described above, which is permeable to the solvent but not to the solute when dissolved in it, and let us connect the solution and solvent of fig. 13 with each other through such a partition. If sol-' vent were to flow one way or the other through the partition, the a° A Percentage of Salo! in Mixture B FIG. 12.

height of the column of solution would rise or fall and the equilibrium with the vapour be disturbed. A continual circulation might thus be set up in an isothermal enclosure and maintained with the performance of an unlimited supply of work. This result would be contrary to all experience of the impossibility of "perpetual motion," and hence we may conclude that through such a semi-permeable wall, the solvent and the solution at the foot of the column would FIG. 13.

be in equilibrium under the excess of hydrostatic pressure represented when the solution is very dilute by P=(p - p')p/0. But such a pressure represents the equilibrium osmotic pressure discussed above. Therefore the equilibrium osmotic pressure of a solution is connected with the vapour pressure, arid, in a very dilute solution, is expressed by the simple relation just given.

Another relation becomes evident if we use as a semi-permeable partition a "vapour sieve" as suggested by G. F. Fitzgerald. If a number of small enough holes be drilled through a solid substance which is not wetted by the liquid, our knowledge of the phenomena of capillarity shows us that it needs pressure to force the liquid into the holes. A piston made of such a perforated substance, therefore, may be used to exert pressure on the liquid, while all the time the vapour is able to pass. By evaporation and condensation, then, the solvent can pass through this perforated partition, which thus acts as a perfect semi-permeable membrane. When the solution and solvent are in equilibrium across the partition, the vapour pressure of the solution has been increased by the application of pressure till it is equal to that of the solvent. In any solution, then, the osmotic pressure represents the excess of hydrostatic pressure which it is necessary to apply to the solution in order to increase its vapour pressure to an equality with that of the solvent in the given conditions.

Similar 'considerations show that, since at its freezing point the vapour pressure of a solution must be in equilibrium with that of ice, the depression of freezing point produced by dissolving a substance in water can be calculated from a knowledge of the vapour pressure of ice and water below the freezing point of pure water. But another method of investigation will illustrate new ways of treating our subject.

By imagining that a dilute solution is put through a thermodynamic cycle we may deduce directly relations between its osmotic pressure and its freezing point. Let us Freezing freeze out unit mass of solvent from a solution at its freezing point T - dT and remove the ice, which is assumed to be the ice of the pure solvent. Then let us heat both ice and solution through the infinitesimal temperature range dT to the freezing point T of the solvent, melt the ice by the application of an amount of heat L, which measures its latent heat of fusion, and allow the solvent so formed to enter the solution reversibly through a semi-permeable wall into an engine cylinder, doing an amount of work Pdv. By cooling the resultant solution through the range dT we recover the original state of the system. The well-known expression for the efficiency of the cycle of reversible operation gives us Pdv/L = dT /T or dT = TPdv/L as a value for the depression of the freezing point of the solution compared with that of the pure solvent.

The freezing point of a solution may be determined experimentally. The solution is contained in an inner tube, surrounding which is an air space. Then comes an outer vessel, in which a freezing mixture can be placed. This solution is stirred continuously and the temperature falls slowly below the freezing point, till the supersaturation point is reached, or until a crystal of ice is introduced. The solution then freezes, until the heat liberated is enough to raise the tern perature to the point of equilibrium given by the tendency of the solution taken in contact with ice to approach the true freezing point on one side and the temperature of the enclosure on the other. To get the true freezing point then, it is well to arrange that the temperature of the enclosure should finally be nearly that of the freezing point to be observed. One way in which this has been secured is by obtaining the under cooling by temporary cooling of the air space by a spiral tube in which ether may be evaporated, the outer vessel being filled with ice in contact with a solution of equivalent concentration to that within. Modifications of this method have been used by many observers, among others by Raoult, Loomis, H. C. Jones, and by E. H. Griffiths and T. G. Bedford, who compared directly the freezing points of dilute solutions with those of the pure solvent in similar conditions by the accurate methods of platinum thermometry.

Another application of the theory of energy enables us to coordinate the osmotic pressure of a dilute solution with the pressure of a gas occupying the same space. On the fundamental hypotheses of the molecular theory, Value we must regard a solution as composed of a number osmotic of separate particles of solute, scattered through- p out the solvent. Each particle may react in some way on the solvent in its neighbourhood, but if the solution be so dilute that each of these spheres of influence is unaffected by the rest, no further addition of solvent will change the connexion between one particle of solute and its associated solvent. The only effect of adding solvent will be to separate further from each other the systems composed of solute particle as nucleus and solvent as atmosphere; it will not affect the action of each nucleus on its atmosphere. Thus the result will be the same whatever the nature of the interaction may be. If solvent be allowed to enter through a semipermeable wall into an engine cylinder, the work done when the solution within is already dilute will be the same whatever the nature of the interaction between solute and solvent, that is, whatever be the nature of the solvent itself. It will even be the same in those cases where, with a volatile solute, the presence of a solvent may be dispensed with, and the solute exist in the same volume as a gas. Now the work done by allowing a small quantity of solvent to enter reversibly into an osmotic cylinder is measured by the product of the osmotic pressure into the change in volume. Hence the osmotic pressure is measured by the work done per unit change of volume of the solution. The result of our consideration, therefore, is that the osmotic pressure of a dilute solution of a volatile solute must have the same value as the gaseous pressure the same number of solute particles would exert if they occupied as gas a volume equal to that of the solution.

The reasoning given above is independent of the temperature, so that the variation with temperature of the osmotic pressure of a dilute solution must be the same as that of a gas, while Boyle's law must equally apply to both systems. Experimental evidence confirms these results, and extends them to the cases of non-volatile solutes - as is, indeed, to be expected, since volatility is merely a matter of degree. When the solution ceases to be dilute in the thermodynamic sense of the word, that is, when the spheres of influence of the solute particles intersect each other, this reasoning ceases to apply, and the resulting modification of the gas laws as applied to solutions becomes a matter for further investigation, theoretical or experimental. In the limit then, when the concentration of the solution becomes vanishingly small, theory shows that the osmotic pressure is equal to the pressure of a gas filling the same space. Experiments with membranes of copper ferrocyanide have verified this result for solutions of cane-sugar of moderate dilutions. But the most accurate test of the theory depends on measurements of freezing points.

A quantity of gas measured by its molecular weight in grammes when confined in a volume of one litre exerts a pressure of 22.2 atmospheres, and thus the osmotic pressure of a dilute solution divided by its concentration in gramme-molecules per litre has a corresponding value. But we have seen that the depression of dT of the freezing point of a dilute solution is measured by TPdv/L. Putting the absolute temperature of the freezing point of water as 273°, the osmotic pressure P as 22.2 atmospheres or 22.4X106, C.G.S. units per unit concentration, L the latent heat as 794X 4.184 X Io 7 in the corresponding units, and dv the volume change in the solution for unit mass of solvent added we get for the quantity dT/c, where is the concentration of the solution, the value 1.857° C. per unit concentration. Experimental measurements of freezing points of various non-electrolytic solutions have been made by Raoult, Loomis, Griffiths, Bedford and others and numbers ranging round 1.85 found for this concentration. Equally good comparisons have been obtained for solutions in other solvents such as acetic acid 3.88, formic acid 2.84, benzene 5.30, and nitrobenzene 6.95. Such, a concordance between theory and experiment not only verifies the accuracy of thermodynamic reasoning as applied to dilute solutions, but gives perhaps one of the most convincing experimental verifications of the general validity of thermodynamic theory which we possess.

Another verification may be obtained from the phenomena of vapour pressure. Since, in dilute solutions, the osmotic pressure has the gas value, we may apply the gas equation PV=nRT =npvi to osmotic relations. Here n is the number of gramme-molecules of solute, T the absolute temperature, R the gas constant with its usual "gas" value, p the vapour pressure of the solvent and v1 the volume in which one gramme-molecule of the vapour is confined.

In the vapour pressure equation p - p' = Pa/p, we have the vapour density equal to M/v 1, where M is the molecular weight of the solvent. The density of the liquid is MN/V, where N is the number of solvent molecules, and V the total volume of the liquid. Substituting these values, we find that the relative lowering of vapour pressure in a very dilute solution is equal to the ratio of the numbers of solute and solvent molecules, or (p - p')/p = n/N.

The experiments of Raoult on solutions of organic bodies in water and on solutions of many substances in some dozen organic solvents have confirmed this result, and therefore the theoretical value of the osmotic pressure from which it was deduced.

Although even good membranes of copper ferrocyanide are rarely perfectly semi-permeable, and in other membranes such as indiarubber, &c., which have been used, the defects from the theoretical values of the equilibrium pressure are very great, yet, in the light of the exact verification of theory given by the experiments described above, it is evident that such failures to reach the limiting value in no wise invalidate the theory of osmotic equilibrium. They merely show that, in the conditions of the particular experiments, the thermodynamic equilibrium value of the osmotic pressure cannot be reached - the thermodynamic or theoretical osmotic pressure (which must be independent of the nature of the membrane provided it is truly semi-permeable) is a different thing from the equilibrium pressure actually reached in a given experiment, which measures the balance of ingress and egress of solvent through an imperfect semi-permeable membrane.

Dilute solutions of substances such as cane-sugar, as we have seen, give experimental values for the connected osmotic properties - pressure, freezing point and vapour pressure - in conformity with the theoretical values.

p y All these solutions are non-conductors of electricity. On the other hand, solution of mineral acids and salts conduct the current with chemical decomposition - they are called electrolytes. In order to explain the electrical properties of a solution, for instance of potassium chloride, we are driven to believe that each molecule of the salt is dissociated into two parts, potassium and chlorine, each associated with an electric charge equal in amount but opposite in sign. The movement in opposite directions of these charged ions constitutes the electric current in the solution. To explain the electrical properties of sulphuric acid in aqueous solution, the supposition of three ions, two of hydrogen and one of the chemical group S04, is necessary. Now measurements of osmotic properties of these solutions show that their osmotic pressures are abnormally great and that, at extreme dilution, the ratio of their osmotic pressures to that of equivalent solutions of non-electrolytes is equal to the number of ions indicated by the electrolytic properties. From the osmotic side also, then, electrolytic dissociation is indicated, and indeed, it was from this side that the idea was first suggested by S. Arrhenius in 1887. The subject is dealt with in Electrolysis and Electric conduction: § dealt with the relations between the properties of an ideally dilute solution, we now turn to the consideration of the general case where the simplifying assumption of great dilution is not made.

The height of the column of solution in fig. 13 required for osmotic equilibrium through a semi-permeable wall below is now very great, since the osmotic pressure of strong solutions may reach many hundred atmospheres. Hence we must not assume that the density of the vapour in the surrounding atmosphere is constant, or that the solution, when equilibrium is reached, is of uniform concentration throughout. The osmotic pressure (defined as the difference in the hydrostatic pressures of the solution and solvent when their vapour pressures are equal and they are consequently in equilibrium through a perfect semi-permeable membrane) may also depend on the absolute values of the hydrostatic pressures, as may the vapour pressure of the liquids.

To investigate the osmotic pressure of a' strong solution we may consider the hydrostatic pressure required to increase its vapour pressure to an equality with that of the solvent. The relation between hydrostatic pressure and the vapour pressure of a pure liquid may be obtained at once by considering the rise of liquid in a capillary tube. The difference in vapour pressure at the top and at the bottom of the column is p - p' = Pclp, as shown above for a column of solution. Writing v for I/a, the specific volume of the vapour at the pressure p, and V for I/p, the specific volume of the liquid at the pressure P, and restricting the result to small changes, we get vdp =VdP.

In considering the corresponding relation for a solution instead of a pure liquid, possible differences in concentration make the column method difficult of application, and it is better to attach the problem by means of an imaginary cycle of isothermal operation. The simplest way to do this is to imagine a vapour-sieve piston through which the vapour but not the liquid can pass. As we have explained above, such a vapour sieve may be constructed by boring a number of small enough holes through a solid not wetted by the liquid.

Let us imagine unit mass of solution of volume V confined in a cylinder ABC between a fixed vapour sieve B and a solid piston A A B C FIG. 14.

by which a pressure P is applied. The vapour at pressure p in equilibrium with the liquid is bounded by a solid piston C, which we can also move to change the pressure or volume.

With such an imaginary apparatus, H. L. Callendar has shown that the variation of vapour pressure of a solution with pressure is given by the expression V'dP = vdp, where V' is the change in volume of the solution when unit mass of solvent is mixed with it. The corresponding relation for a pure liquid can be regained by considering that at infinite dilution the liquid becomes pure solvent, and the change of volume becomes equal to the volume V.of solvent added.

The osmotic pressure Po is the difference of the hydrostatic pressures P' and P of the solution and the solvent when their vapour pressures are equal. Hence dPo=dP' - dP and dPo/dP=(V - V')/V' or dPo/dP' = (V - V')/V. If V =V' there is no change in osmotic pressure with hydrostatic pressure, and osmotic pressure depends on concentration and temperature only.

The relation between the equilibrium pressures P and P' for solution and solvent corresponding to the same value po of the vapour pressure is obtained by integrating the equation V'dP' = vdp between corresponding limits for solution and solvent. We get (P?' V'dP' = and (P VdP = C Po vdp, whence J f P' V'dP' - (P Vd .1 P = (P v?dp, I ?' J p J where p and p' are the vapour pressures of solvent and solution each under its own vapour pressure only.

If we measure the osmotic pressure Po when the solvent is under its own vapour pressure only, that is, when P = p = Po, the term involving V vanishes, and the limit of integration P' becomes Pod-p. If we assume that V', the volume change on dilution, varies regularly or not appreciably with pressure, we may write the first integral as V' (P o -{- p - p') where V' now denotes its mean value between the limits.

To evaluate the second integrals vdp we may subtract a constant b to represent the defect of the volume of the vapour from the ideal volume Rt/p. This gives V' (Po -{-p - p') = Rt log (p/p') - b(p - p'). For most experimental purposes the small terms involving the factor ( - p') may be neglected, and we have, approximately, PoV'=Rt log (ply). From this equation the osmotic pressure Po required to keep a solution in equilibrium as regards its vapour and through a semi-permeable membrane with its solvent, when that solvent is under its own vapour pressure, may be calculated from the results of observations on vapour pressure of solvent and solution at ordinary low hydrostatic pressures. The chief difficulty lies in the determination of the quantity V', the change in volume the solution under the pressure Po when unit mass of solvent is mixed with it. This determination involves a knowledge of the density and of the compressibility of the solution; the latter property is difficult to measure accurately.

In some solutions such as those of sugar the change in volume on dilution is nearly equal to the volume of solvent added; V' then becomes equal to V, the specific volume of the solvent. The osmotic pressures of strong sugar solutions were measured successfully by a direct method with semi-permeable membranes of copper ferrocyanide by Lord Berkeley and E. G. J. Hartley, who also determined the vapour pressures by passing a current of air successively through weighed vessels containing solution and water respectively.

Their table of comparison published in 1906 shows the following agreement: - It seems likely that measurements of vapour pressure and compressibility may eventually enable us to determine accurately osmotic pressures in cases where direct measurement is impossible.

The slope of the temperature vapour pressure curves in the neighbourhood of the freezing point of the solvent is given by the latest heat equation. The difference in the two slopes for water and ice is dp/dT - dp'/d T=L/Tv, Solutions. where L, the latent heat of fusion, is the difference between the heats of evaporation for ice and water, and v is the specific volume of the vapour.

The difference in the lowering of vapour pressures dp - dp' may be put equal to VdP/v, where P is the osmotic pressure, and V the specific volume of the solvent. We then get VdP=LdT/T.

In order to integrate this expression we need to know L and v as functions of the temperature and pressure. The latent heat L at any temperature is given by L=Lo - f 0 64 0 (s - s')dT, where Lo is value at To and s--s' is the difference in the specific heats of water and ice. The probable error in neglecting any variation of specific heat is small, and we may calculate L from the values of Lo - (s - s') (To - T), where s - s' is about 0.5 calories. The variation of L with pressure is probably small.

The volume of a gramme of water also depends on temperature and pressure. Approximately one degree lowering of freezing point corresponds with a change of 12 atmospheres in the osmotic pressure. From the known coefficients of compressibility and thermal expansion we find that V may be represented by the linear equation V=1.000+0.0008 A, where A is the lowering of the freezing point below o°.

Putting in these values and integrating we have, neglecting terms involving 0', P=12.06 0-0.021 O s where P is the osmotic pressure in atmospheres.

H. W. Morse and J. C. W. Frazer, who have made direct measurements of osmotic pressure of solution of cane-sugar, have also measured the freezing points of corresponding solutions. From these results the equation just given has been examined by G. N. Lewis.

Thus the theory of the connexion of osmotic pressure with freezing point (like that with vapour pressure) seems to give results which accord with experiments.

At the limit of dilution, when the concentration of a solution approaches zero, we have seen that thermodynamical theory, verified by experiment, shows that the osmotic pressure has the same value as the gas pressure of the same number of molecules in the same space. Gases at high pressures fail to conform to Boyle's law, and solu tions at moderate concentrations give osmotic pressures which increase faster than the concentration. The variation of gases from Boyle's law is represented in the equation of Van der Waals by subtracting a constant b from the total volume to represent the effect of the volume of the molecules themselves. The corresponding correction in solutions consists in counting only the volume of the solvent in which the solute is dissolved, instead of the whole volume of the solution.

140 120 100 80 60 40 2001-2345 FIG. 15.

In fig. 15 the curve I represents Boyle's law if the volume is taken to be that of the solution, and the curve II if the volume is that of the solvent. Even this correction is not sufficient in solution of sugar, where the theoretical curve II lies below the experimental observations. A further correction may be made by adding more empirical terms to the equation, but a more promising idea, due to J. H. Poynting and H. L. Callendar is to trace the effect of possible combination of molecules of solute with molecules of the solvent. These combined solvent molecules are thus removed from existence as solvent, the effective volume of which is reduced to that of the remaining free molecules of solvent. The greater the number of water molecules attached to one sugar molecule, the less the residual volume, and the greater the theoretical pressure. Callendar finds that five molecules of water in the case of cane-sugar or two molecules in the case of dextrose are required to bring the curves into conformity with the observations of Berkeley and Hartley, which in fig. 15 are indicated by crosses.

Solubility and Heat of Solution

The conceptions of osmotic pressure and ideal semi-permeable membranes enable us to deduce other thermodynamic relations between the different properties of solutions. As an example, let us take the following investigation: An engine cylinder may be imagined to possess a semi-permeable bottom and to work without friction. If it be filled with a solution and the bottom immersed in the pure solvent, pressure equal to the osmotic pressure must be exerted on the piston to maintain equilibrium. Such a system is in the thermo dynamic equilibrium. The slightest change in the load will cause motion in one direction or the other - the system is thermodynamically reversi ble. Such an arrangement may be put through a cycle of operations as in Carnot's engine (see Thermodynamics) and all the laws of reversible engines applied to it. If the solution in the cylinder be kept saturated by the presence FIG. 16.

of crystals of the solute, crystals will dissolve as solvent enters, and the solution remains saturated throughout. By an imaginary cycle of operations we may then justify the application to solutions of the latent heat equation which we have already assumed as applicable. In the equation dP/dT= X/T(v 2 - v 1), P is the osmotic pressure, T the absolute temperature and X the heat of solution of unit mass of the solute when dissolving to form a volume v2 - v1 of saturated solution in an osmotic cylinder. This process involves the performance of 6 7 8 9 an amount of osmotic work P(v - v). If the heat of solution be measured in a calorimeter, no work is done, so that, if we call this calorimetric heat of solution L, the two quantities are connected by the relation L = X+P(v - v). If L is zero or negligible, X = - P(v - v) and we have dP/dT = - P/T or dP/P = - dT/T, which on integration gives log P =log T+C, or P= kT, i.e. the osmotic pressure is proportional to the absolute temperature. This result must hold good for any solution, but if the solution be dilute when saturated, that is, if the solubility be small, the equation shows that if there be no heat effect when solid dissolves to form a saturated solution, the solubility is independent of temperature, for, in accordance with the gas laws, the osmotic pressure of a dilute solution of constant concentration is proportional to the absolute temperature. It follows that if the thermodynamic heat of solution be positive, that is, if heat be absorbed to keep the system at constant temperature, the solubility will increase with rising temperature, while if heat be evolved on dissolution, the solubility falls when the system is heated.

In all this investigation it should be noted that the heat of solution with which we are concerned is the heat effect when solid dissolves to form a saturated solution. It is not the heat effect when solid is dissolved in a large excess of solvent, and may differ so much from that effect as to have an opposite sign. Thus cupric chloride dissolves in much water with an evolution of heat, but when the solution is nearly saturated, it is cooled by taking up more of the solid.

In a very dilute solution no appreciable heat is evolved or absorbed when solvent is added, but such heat effects are generally found with more concentrated solutions. os The result is to change the relation between temperature and the osmotic pressure of a solution of constant concentration, a relation which, in very dilute solutions, is a direct proportionality.

The equation of available energy (see Energetics) A=U+ TdA/dT may be applied to this problem. The available energy A is the work which may be gained from the system by a small reversible isothermal operation with an osmotic cylinder, that is Pdv. If l is the heat of dilution per unit change of volume in a calorimeter where all the energy goes to heat, the change in internal energy U is measured by ldv. We then have Pdv =ldv+T (Pdv). Neglecting the volume change with temperature this gives P=I+TdP/aT for the relation required. In the case where l is negligible we have P/dP = T/dT, which on integration shows that the osmotic pressure, as in the special case of a dilute solution, is proportional to the absolute temperature.

Theories of Solution

The older observers, noticing the heat effects which often accompany dissolution, regarded solutions as chemical compounds of varying composition. The physical investigation of osmotic pressure, and its correlation by Van't Hoff with the pressure of a gas, brought forward a new aspect of the phenomena, and suggested an identity of physical modus operandi as well as of numerical value. On this view, the function of the solvent is to give space for the solute to diffuse, and the pressure on a semi-permeable membrane is due to the excess of solvent molecules entering over those leaving in consequence of the smaller number which impinge on the membrane from the side of the solution; the defect in the number must be proportional, roughly at any rate, to the number of solute molecules, present, that is, to the strength of the solution.

Whatever view, if any, be adopted as to the nature of a solution, the thermodynamic relations we have investigated equally hold good. It is the strength and weakness of thermodynamic methods that they are independent of theories of constitution. The results are true whatever theory be in vogue, but the results throw no light on the problem of which theory to choose. All the thermodynamic relations we have deduced hold on any theory of solution and favour no one theory rather than another. Whether osmotic pressure be due to physical impact or to chemical affinity it must necessarily have the gas value in a dilute solution, and be related to vapour pressure and freezing point in the way we have traced. But for any theory of solution to be tenable, it must at least be consistent with the known thermodynamic relations, verified as those relations are by experiment.

On certain assumptions required for the extension of the methods of the kinetic theory of gases to liquids, L. Boltzmann offered a demonstration of the law of osmotic pressure in dilute solutions, based on the idea that the mean energy of translation of a molecule should be the same in the liquid as in the gaseous state. But, whether or not the assumption underlying this demonstration be accepted, the similarity between solution and chemical action remains, and the osmotic law has been examined from this side by J. H. Poynting and by H. L. Callendar. The fundamental phenomenon they take to be the identity of vapour pressure, and consider the combination necessary to reduce the vapour pressure of a solution to the right value. If each molecule of the solute combines with a certain number of molecules of the solvent in such a way as to render them inactive for evaporation, we get a lowering of vapour pressure. Let us assume that the ratio p/p' of the vapour pressures of the solvent and solution is equal to the ratio of the number of free molecules of solvent to the whole number of molecules in the solution. Each molecular complex, formed by solution and solvent, is treated as a single molecule. If there are n molecules of solute to N of solvent originally, and each molecule of solute combines with a molecule of solvent, we get for the ratio of vapour pressures p/p'=(N - an)/(N - an+n), while the relative lowering of vapour pressure is (p - p')/p=n/(N - an). In the limit of dilution when n is very small compared with N this gives Raoult's experimental law that the relative lowering is n/N, which we deduced from the osmotic law, and conversely from which the osmotic law follows, while for more concentrated solutions agreement is obtained by assigning arbitrary values to a, which, as we have seen, is 5 in the case of cane-sugar.

Certain solvents, such as water, liquid ammonia or liquid hydrocyanic acid, possess the power of making some solutes, such as mineral salts and acids, when dissolved in them, conductors of electricity. The special properties of these solutions are dealt with under Electrolysis and Electric conduction, § In Liquids. Attempts have been made to co-ordinate this ionizing power of solvents with their dielectric constants, or with their chemical properties. On the lines of Poynting's theory of solution, each ion in electrolytes must combine with one or more molecules of solvent.

Diffusion in Solutions

The passage of dissolved substances through animal and vegetable membranes was the subject of many early experiments. It was found that substances like mineral salts, which crystallize well from solution, passed such membranes with comparative ease, while the jelly-like substances such as albumen passed with extreme slowness if at all. The first to make systematic experiments on the free diffusion of dissolved substances with no separating membrane was Thomas Graham (1804-1869), who immersed in a large volume of water a wide-mouthed bottle containing a solution, and after some time measured the quantity of substance which had diffused into the water. Again the two classes of substances mentioned above were found to be distinguished, and Graham called the slowly diffusible non-crystalline bodies colloids, in contrast to the quickly diffusible crystalloids. Graham showed that the diffusion was approximately proportional to the difference in concentration, and on these lines a theory of diffusion was founded on the lines of Fourier's treatment of the conduction of heat.

The quantity of substance which diffuses through unit area in one second may be taken as proportional to the difference in concentration between the fluids at that area and at another parallel area indefinitely near it. This difference in concentration is. proportional to the rate of variation - dc/dx of the concentration c with the distance x, so that the number of gramme-molecules of solute which, in a time dt, cross an area A of a long cylinder of constant cross section is dN = - DA(dc/dx)dt, where D is a constant known as the diffusion constant or the diffusivity.

The osmotic pressure of a solution depends on the concentration, and, if we regard the difference in that pressure as the effective force driving the dissolved substance through the solution, we are able to obtain the equation of diffusion in another form. When the solution is dilute enough for the osmotic pressure to possess. "the gas" value the equation becomes - dN = - 7 Adxdt, where R is the usual gas constant, T the absolute temperature, and F the force required to drive one gramme-molecule of the solute through the solution with unit velocity.

By comparison with the first equation we see that RT/F is equal to D, the diffusion constant. This constant can be measured experimentally, and for such a substance as sugar or water comes out about 0.3 at 20° C., the unit of time being the day. Hence the force required to drive one gramme-molecule of sugar through water with a velocity of one centimetre per second may be calculated as some thousands of millions of kilogrammes weight.

In the case of electrolytes we can go further, and calculate the diffusion constant itself from the theory of electrolytic dissociation (see Electric conduction, § In Liquids). On that theory the ions of a dilute solution migrate independently of each other. Since some ions are more mobile than others, a separation will ensue when water is placed in contact with a solution, the faster moving ion penetrating quicker into the water under the driving force of the osmotic pressure gradient. This separation causes a difference of potential, which can be calculated and is found to agree with the values obtained experimentally. The separation also sets up electrostatic forces, which increase until they are strong enough to drag the slower moving ions along faster, and to retard the naturally faster ions till they travel at the same rate. The resistance offered by the liquid, and therefore the force F, required to drive one grammemolecule through the liquid with unit velocity is the sum of the corresponding quantities for the individual ions. Now the velocities u and v of the opposite ions under unit potential gradient, and therefore U and V under unit force, are known from electrical data. Thus F, which is equal to I/U-1-1/V, is known. The osmotic pressure of an electrolyte consisting of two ions is double that of a non-electrolyte. Hence for a binary electrolyte the diffusion constant is measured by 2RT/F or 2UVRT/(U+V). This result gives a value of D for dilute hydrochloric acid equal to 2.49 to compare with the observed value of 2.30. Other substances give equally good agreements; thus sodium chloride has a calculated constant of 1.12 and an observed one of 1.11. Such concordance gives strong support to the theory of diffusion outlined above.

Colloidal Solutions

Besides a large number of animal and vegetable substances, many precipitates formed in the course of inorganic chemical reactions are non-crystalline and appear in the colloidal state, instances are the sulphides of antimony and arsenic and the hydroxides of iron and alumina. Some of these colloids dissolve in water or other liquids to form solutions called by Graham hydrosols; Graham named the solids formed by the setting or coagulation of these liquids hydrogels. Solutions of colloids in solvents such as water and alcohol seem to be divisible into two classes. Both mix with warm water in all proportions, and will solidify in certain conditions. One class, represented by gelatin, will redissolve on warming or diluting, while the other class, containing such substances as silica, albumen, and metallic, hydrosulphides, will solidify on heating or on the addition of electrolytes to form a solid "gel" which cannot be redissolved. Solidification of the first kind may be termed "setting," that of the second "coagulation." The power of coagulation of colloids shown by electrolytes depends in a curious manner on the chemical valency of the effective ion. The average of the coagulative powers of salts of univalent, divalent and trivalent metals have been found by experiment to be proportional to the numbers 1: 35: 1023. If we assume that a certain minimum electric charge must be brought into contact with a group of colloid particles to produce coagulation, twice as many univalent ions must collect to produce the same effect as a number of divalent ions, and three times as many as an effective number of trivalent ions. We can calculate, by the help of the kinetic theory and the theory of chances, the frequency with which the necessary conjunctions of ions will occur, and show that the general law will be that the coagulative powers should be in the ratios of 1: x: x 2. Putting x =32, we get r: 32: 1024 to compare with the experimental numbers. The ordinary surface energy of a two-phase system tends to diminish the area of contact, and thus to help the growth of the larger aggregates required for coagulation. A natural electric charge on the particles would oppose this tendency, and tend to increase the free surface and thus promote disintegration and solution. The function of the electrolyte may be to annul such a natural charge and thus allow the non-electric surface energy to produce coagulation. This explanation is supported by some experiments by W. B. Hardy, who found that certain colloids did possess electric charges, the sign of which depended on whether the surrounding liquid was slightly acid or slightly alkaline. At the neutral point, when the particles possessed no charge, their stability was destroyed, and they were precipitated. But recent experiments have shown that the simple theory of coagulation here outlined needs amplification in certain directions. The phenomena seem to be dependent on variables such as time, and are more complicated than seemed likely at first.

The size of the suspended particles in colloidal solutions varies greatly. In some solutions they are visible under a good microscope. In other cases, while too small to be directly visible, they are large enough to scatter and polarize a beam of light. In yet other solutions, the particles are smaller again, and seem to approach in size the larger molecules of crystalloid substances. It is not yet agreed whether colloid solution is the same in kind though different in degree from crystalloid solution or is a phenomenon of an entirely different order.

REFERENCES. - The properties and theory of solutions are Treated in all works on general physical chemistry; Ostwald's discussion in his Lehrbuch was translated into English in 1891 by M. M. P. Muir entitled Solution. Special works are W. C. D. Whetham, Theory of Solution (1902); W. Rothmund, Loslichkeit (1907). Solubility tables are given in Laridolt, Bornstein and Meyerhoffers, Tabellen (1905); A. M. Comey, Dictionary of Solubilities (Inorganic) (1896); A. Seidell, Dictionary of the Solubilities of Inorganic and Organic Substances (1907). (W. C. D. W.)