Principles of General Physiology

THE fact that the metal palladium allows hydrogen to pass freely through it, while refusing such passage to nitrogen, enabled Ramsay (1894, p. 206) to make an interesting experiment. A vessel of palladium was filled with nitrogen and connected to a mercury manometer. It was then immersed in an atmosphere of hydrogen and the mercury was seen to rise steadily in the manometer. Why did this happen? The reason is that hydrogen passes through the walls of the vessel until its concentration o? pressure becomes equal within and without ; but, as the nitrogen cannot escape to give room for the hydrogen which enters, the amount of gas inside the closed vessel must increase and the total pressure rise.

The fact can also be shown by the use of a membrane of water, or, rather, a parchmentpaper membrane soaked in water. Such a membrane is freely permeable to carbon dioxide, because the gas is soluble in water, but almost impermeable to oxygen and nitrogen. If, therefore, we take a bell-shaped vessel, and tie over the large end a wet parchment-paper membrane, connect the interior to a manometer, and then immerse the vessel in carbon dioxide, the pressure will rise rapidly inside for similar reasons as in the case of hydrogen and palladium.

Now we know, by what is usually known as Dalton's Law, that, in a mixture of gases at a certain pressure, this pressure is divided between the different gases in proportion to their relative volumes, or, in other words, the total pressure of a mixture of gases is equal to the sum of the pressures which each alone would exercise if it alone filled the vessel. Suppose that we have a mixture of nitrogen and carbon dioxide consisting of one-fifth nitrogen and four-fifths carbon dioxide at atmospheric pressure. The partial pressure of the nitrogen is one-fifth of 760 mm., that is, 152 mm. of mercury; this is also called its "tension." If such a mixture is put in a vessel as described above and pure carbon dioxide placed on the outer side of the membrane, the pressure will rise by carbon dioxide passing in until its tension is equal on both sides. But, since gases are compressible, the relative volume of the nitrogen will have been diminished by the process, so that it is better for the sake of description to imagine that, before immersion in the carbon dioxide atmosphere, we have raised the internal pressure by forcing in more of the gaseous mixture until the manometer reads 152 mm., that is, until the pressure is increased by the tension of the nitrogen while that of the carbon dioxide is that of the atmosphere. By this means we avoid the further inflow of carbon dioxide, and we find that the gauge remains stationary at 152 mm. of mercury, if the barometer stands at 760 mm. We have thus a measurement of the tension of nitrogen in the mixture. Of course, the pressure will not remain indefinitely at this point, since nitrogen is not absolutely insoluble in water, and it will therefore pass very slowly through the membrane, until the composition of the mixture is the same on both sides and no pressure will be shown on the manometer.

Let us now take an analogous experiment with a liquid system. We have seen in the preceding chapter that a membrane of copper ferrocyanide is freely permeable to water, while refusing passage to cane-sugar in solution in water. Pfeffer (1877) made a number of experiments in which the membrane was supported in the pores of a clay cell in order that it might be able to withstand the pressures developed. He found that these pressures, spoken of in the case of

solutions as "osmotic pressures," were directly proportional to the concentration of the solute and to the absolute temperature. The process by which water passes through a membrane from a solution on the one side to another solution on the opposite side had been known, since the time of Dutrochet (1827, p. 393), as "endosmosis" or "exosmosis," so that the pressure due to this passage of water was naturally called " osmotic" The experiments made by Pfeffer have served as the starting point of sub sequent work on osmotic pressure, especially as the basis of the important theory of solutions put forward by van't Hoff ; I have therefore given his portrait in Fig. 47.

The similarity of the process in gases and in solutions is obvious, but the relationship was not made clear until van't Hoff (1885, 1) was led by thermodynamic considerations (see Cohen's book, 1912, p. 225) to the view that the pressure developed by a substance in solution is identical with that which it would exert if converted into gas of the same volume and temperature ; in other words, the solute behaves as if it were in the dispersed molecular condition of a gas and the solvent were absent.

This statement does not necessarily imply that the state of the solute is actually that of a gas, although, as we shall see later, a kinetic theory, similar to that of gases, gives, on the whole, the most satisfactory explanation of the phenomena. It should be kept in mind that the facts of osmotic pressure, their connection with vapour pressure and so on, are independent of any theory of their origin. FIG. 47. PORTRAIT OF PFEFFER. On account of the Signature from Charter Book of the Royal Society.

1. " Loi de Boyle pour les Solutions. — La pression osmotique est proportionelle a la concentration, si la temperature reste invariable. 2. "Loi de Gay-Lussac pour les Solutions. — La pression osmotique est proportionelle a la temperature absolue, si la concentration reste invariable. Ce sont la les analogies qui ont ete demontrees et verifiees en detail dans le travail cite (the preceding paper, 1885, 1); elles ont rapport a la variation de la pression avec les circonstances. Je vais ajouter maintenant une troisieme

proposition, ayant rapport a la grandeur absolue de cette pression, et n'etant, en realite, autre chose qu'une extension de la loi d'Avogadro. 3. " Loi. d'Avogadro pour les Solutions. — La pression exercee par les gaz a une temperature determined, si un rneme nombre de molecules en occupe un volume donne, est egale a la pression osmotique qu'exerce dans les memes circonstances la grande majorite des corps, dissous dans les liquides quelconques." At normal temperature and pressure one gram-molecule of a gas occupies a volume of 22 '4 litres, so that if one gram-molecule of a solid be dissolved in 22'4 litres of water, its osmotic pressure should be one atmosphere, as may also be seen from the following consideration. To compress one gram-molecule of a gas to the volume of one litre, which is the volume occupied by any solute in what is known as molar concentration, requires, by Boyle's law, a pressure of 22 -4 atmospheres.

Let us take an example from one of Pfeffer's experiments. A 4 per cent, solution of canesugar gave at 15° an osmotic pressure of 208'2 cm. of mercury. By Gay-Lussac's law, molecule of the sugar weighs 342 g., so that the number of litres of a 4 per cent, solution required to contain 1 gram-molecule is '— = 8'55. Hence its osmotic pressure should be 76 x- = 199 cm. mercury, a very close agreement, considering the difficulty of the measuiv This example will serve to show the justification of van't HofFs point of view. The experiments of De Vries on isotonic solutions, referred to in the preceding chapter, gave further confirmation of its correctness.

Before proceeding further, we must insist on the fact that the theory was only intended to apply to dilute solutions. For the present purpose we may define dilute solutions as being those in which the number of molecules of the solute is so small in proportion to those of the solvent that any effects due to the mutual action of the molecules of the solute, to their actual volume, or to combination with the solvent, in the sense of hydration or solvation, may be neglected.

When we come to concentrated solutions, these factors have to be taken into account, as van't Hoff himself (see Cohen's book, 1912, p. 282) pointed out with reference to the treatment of the question from the kinetic point of view. In fact, the osmotic pressures of such solutions are found to be higher than the simple gas law would lead us to expect, the deviations becoming greater as the concentration rises. The most important work on concentrated solutions is that done by Morse and his collaborators in the United States (1901, etc, summary in 1914) and by Berkeley and Hartley in England (1906, 1). These experiments were made on solutions of cane sugar. A further series of measurements on calcium ferrocyanide was made in 1908 by Berkeley, Hartley, and Burton. As to the interesting methods employed by these observers, the reader is referred to the monograph by Morse (1914) and that by Findlay (1913). The preparation of the copper ferrocyanide membrane is of especial importance.

In the endeavour to find a formula which applies to concentrated solutions, as well as to dilute ones, it is obvious that, by the introduction of a sufficient number of empirical constants, this would not be difficult. On the other hand, if a physical meaning can be given to the constants introduced, although it may not, for the present, be possible to determine them by an independent method, such an expression is to be preferred. For this reason, in the following pages, I have adopted the point of view of van der Waals (1881) and, as regards details, that of Otto Stern (1913). This treatment consists in the application of the ra« der H'ttti/.*' i nuation of state to solutions, and it must not be supposed that no other point of view is possible. The point of view of the doctrine of energy, or thermodynamics, for example, as given by Findlay (1913), leads to a logarithmic formula and affords results which are, of course, cogent if based on correct foundations, but it does not seem to me to help us far in understanding the factors at work. Nernst (1911, p. 155) appears to be of the same opinion. It is pointed out by Arrhenius (1912, p. 6) in reference to the selection by van't Hoff of the thermodynamic method, that, at that time, the kinetic theory was not so manageable as the former. Boltzmann, however, brought the kinetic theory into favour again by reducing it to an application of the theory of probabilities. The application of the kinetic theory to liquids will be found discussed in Nernst's book (1911, pp. 212-219).

is not, in reality, of universal application even to gases, and fails especially under high compression. It gives more accurate results the higher the temperature, a fact which is significant in connection with the data obtained by Morse and his co-workers (1912, p. 29). The osmotic pressure of a molar solution (weight normal, see below) at 5° was found to be 1-115 times that calculated ; at 40° it was only 1 -085 times, and at 80" the values agreed.

The failure of the simple Boyle-Gay-Lussac law to express the behaviour of gases at any temperature and pressure led Van der Waals (1873, see Bibliography) to consider the causes of the failure, and to formulate a more general law, which is usually stated thus : — We notice that P is increased by a new factor, which is a function of V, while V itself is diminished by another factor, b. We will first consider this latter quantity, which has to do with the actual volume taken up by the molecules themselves. If molecules have a real concrete existence, and all recent work shows that they have, they must occupy space. The concordance between the values of Avogadro's constant, obtained by various methods as referred to in Chapter IV. above, is, in itself, sufficient proof of the actual existence of molecules. In gases at ordinary temperatures and pressures, the volume taken up by the molecules themselves is negligible in comparison with the space in which they are free to move. Larmor (1908) has pointed out that, if we imagine the molecule of a gas at atmospheric pressure to be magnified so that it has a diameter of 1 cm., there will only be one molecule in two litres ; or the space taken up by the actual molecules themselves is only about one four-thousandth part of the total volume of the gas. When the gas is compressed, the volume of the molecules is not diminished, so that the relative fraction of the volume taken up by them becomes more and more pronounced. V, therefore, in the simple gas equation, that is, the space free for the molecules to move in, is actually the volume as measured, diminished by the space occupied by the molecules. This space is not necessarily the size of the chemical molecules themselves, but the distance at which they begin to resist being pressed closer together, and is, according to van der Waals, four times the former quantity in the rarefied state. It diminishes to about half this value as the total volume of the gas decreases under pressure.

Turning to liquids, and remembering that van der Waals applies his formula to pure liquids, non-associated, that is, consisting of single molecules, we may, as a first approximation, expect that, if we reckon the concentration of our cane-sugar solution as being the number of grams dissolved in 100 c.c. of water, so that a 10 per cent, solution is made by adding 10 g. of sugar to 100 c.c. of water, instead of taking a solution containing 10 g. of sugar in 100 c.c. of solution, better correspondence of osmotic pressure measurements with the theoretical ones would be obtained. This is in fact the case, as the following numbers from the experiments of Morse and Fraser show : —

By taking, in this way, what are called weight-normal instead of volume-normal solutions, we are allowing for the volume of the molecules of the solute, or taking V-b instead of V in the simple equation. But this procedure, as the table shows, is not a complete solution of the question, and we must also take into consideration the remaining constant of van der Waals, viz., a, which refers to the mutual attraction of the molecules, and therefore acts in the opposite way to 6. This mutual attraction of the molecules has been already met with in Chapter III., in the case of liquids, as the internal pressure of Laplace, giving rise to the surface tension. These attracthr t'mvcs are naturally less the further the molecules are from one another. They are in fact inversely proportional to the square of the volume occupied by a given number of

molecules, i.e., — . We must then increase P, in the simple gas equation, by In the application of the van der Waals theory to solutions I propose to follow, in the main, the treatment of Otto Stern (1912), since it is, on the whole, capable of easier explanation than the similar one of Berkeley (1907). For a complete account, however, the original papers must be consulted. In the first place, we must not expect even dilute solutions to obey the simple gas law exactly, because the solvent itself is, as regards its molecular state, very concentrated when compared with a gas. In other words, its molecules are closely packed. According to van der Waals, at the boiling point, the volume of the molecules is about one-quarter of the entire space occupied by the liquid.

That there is space between the molecules of a liquid is shown by the fact, amongst others, that liquids are not altogether incompressible. Parsons and Cook (1911, p. 343) find that water at 4° can be compressed to 87 per cent, of its volume by a pressure of 4, 500 atmospheres, and ether at 35° to 80 per cent, of its volume by 4,000 atmospheres. Moreover, the molecules of the solvent affect those of the solute in both the attractive and the repulsive ways of the van der Waals equation ; so that it is, in point of fact, rather unexpected to find, even in the case of dilute solutions, that the osmotic pressure is so nearly equivalent to the gas pressure of the solute. The reason for this, according to Stern, is the presence of the semi-permeable membrane itself, which causes the effects due to both the attractive and repulsive forces to be compensated in dilute solutions in the following way : — As regards a, a molecule of the solute which hits against the membrane is surrounded on all sides by the solvent, since the membrane is permeable to these. The attractive forces are therefore equal on all sides, as if the membrane were not present, and play no part in the production of the osmotic pressure, which can only be affected by forces which are unequal on the two sides of the membrane. As regards 6, an increased osmotic pressure must undoubtedly be caused thereby, but a part of the total osmotic pressure, and, in fact, a part which is exactly equal to that due to the volume of the molecules, is taken up, not by the membrane, but by the molecules of the solvent in the act of passing through the membrane. A certain part of the membrane is occupied by molecules of the solvent, instead of membrane substance, so that a certain number of the molecules of the solute hit against these molecules of the solvent, instead of against the membrane, and are therefore inactive osmotically.

The above considerations apply only to dilute solutions, where the osmotic pressure is given by the simple gas law, and the number of molecules of the solute is so small in comparison with those of the solvent, its "molar fraction," that mutual action may be neglected. This mutual action cannot be neglected in more concentrated solutions, and Otto Stern has developed the following modification of the van der Waals formula : where TT is the osmotic pressure, al and bl are the van der Waals constants of the pure solute, a1>2 and brz are constants depending on the attraction and repulsion respectively between the molecules of solvent and solute, and x0 — x

is the difference between the concentration of the solvent outside the membrane and in the solution itself. We note that a of the van der "Waals equation is diminished by a factor expressing the attraction between the molecules of the solute and those of the solvent, which acts in the opposite direction as regards osmotic pressure to that between the molecules of the solute itself. The attraction between the molecules of the solvent and solute pulls the molecules of the solute away from each other, in opposition to their mutual attraction. The necessity for the introduction of *0 - x is that the concentration of the solvent inside the membrane is less than that outside by the space taken by the molecules of the solute. For similar reasons, the repulsive forces expressed by b are less than in the simpler case of a pure liquid.

The whole process of derivation of the formula is beyond the limits of this book, but there are one or two points to be noted in connection with it. Owing to the fact of its containing two additional constants, it is not to be wondered at that it can be made to satisfy experimental results. These new constants, unfortunately, cannot as yet be tested experimentally by an independent method, but, at the same time, it is a matter of some satisfaction to possess an equation, similar in form to that of van der Waals, containing only factors to which a physical meaning can be assigned.

If the solvent is an associated liquid, like water, the equation still applies, although, of course, the numerical values of the constants will not be the same. Consider further that the two van der Waals constants have opposed to them other constants by which their value is reduced, and it will be obvious that in solution a substance should obey the ideal gas law more closely than in the gaseous state. Suppose that we are dealing with two easily miscible substances whose critical points are not very far removed from one another, so that their molecular state may be considered to be similar, then a^^ and bV2 are of the same order as «! and bv Moreover, the difference between the concentrations of the pure solvent itself and that .which it has in the solution is nearly identical with the concentration of the solute, or xo~x. js very nearly equal to -, which is the concentration

of the solute ; x0- x is, therefore, practically unity. This being so, the factors representing a will nearly cancel out, as will also those representing b, and a gas in solution will obey the ideal gas law more closely than it does in the gaseous state. This remarkable result was tested by Otto Stern in the case of solutions of carbon dioxide in methyl and ethyl alcohols, acetone, and methyl and ethyl acetates, at low temperatures in order to avoid high pressures. The values actually measured were the absorption coefficients, and from these the osmotic pressures were calculated by a formula due to Nernst, taking account of the increase of the coefficient as the pressure increased.

The following numbers were obtained in the case of methyl alcohol at -78° C., and will serve as an illustration. The column headed "Theoretical osmotic pressure " gives the values calculated from the simple gas equation, and it will be noticed how closely the observed values correspond to these, deviating only at the higher pressures. The last column gives the corresponding pressures in the gaseous state, as calculated by the van der Waals formula.

The basis of the foregoing considerations has been that of the kinetic theory, according to which the osmotic pressure, developed by a solution constrained by a membrane permeable only to the solvent, is due to the impacts of the molecule <-t' the solute against the membrane through which they cannot pass (Nernst, 1911, p. 244). It is well to note that there are other views on the question, such as surface tension, attraction of solute for solvent, and so on, but it would exceed tinscope of the present book to discuss them. Although van:t Hoff made use of the therinodynamic method in the quantitative mathematical treatment of osin<»t itpressure, he interprets the phenomenon in terms of the kinetic theory as j,ri\m above (see p. 482 of his paper, 1887). For our purposes, the kinetic theory satisfies requirements best. Those who are interested in the question are referred to the monograph by Findlay (1913, pp. 65-76), and to the paper by Callendar (1908). Callendar remarks, " It is probable that all the theories possess some elements of truth, and that they may be to some extent merely different aspects of the same phenomenon."

There is one point that requires a few words. Many solutes are hydrated in solution in water. That is, each molecule is associated with a larger or smaller • number of water molecules. The result of this is that the number of molecules of water in a given volume is reduced, although that of the solute is not._ As far as dilute solutions are concerned, as Nernst (1911, pp. 271 and 469) points out, this fact will have no influence on the osmotic pressure, however measured. The number of molecules of water is so great in proportion to those of the solute that the fixation of a certain number of them will have no measurable effect. On the other hand, calculations of the osmotic pressure of concentrated solutions of cane-sugar, made on the hypothesis that each molecule is associated with five molecules of water, gives values more nearly approximating to those obtained experimentally (see Findlay, 1913, p. 42).

There is at present much difference of opinion as to the nature of this hydration. For example, it is stated by Callendar (1908, p. 498) that the conclusions of Jones and Bassett (1905) are "diametrically opposed" to his. The direct measurement of osmotic pressure, either by Pfeffer's method of measuring the pressure produced in the osmometer when one side of the membrane is immersed in water at atmospheric pressure, or by that of Berkeley by measuring the pressure necessary to be applied to the solution in order to prevent passage of solvent in either direction, is of considerable experimental difficulty, and only applicable in certain cases, owing to the fact that we know of so few appropriate semi-permeable membranes. In practice, the determination of other properties, which are related in a known way to the osmotic pressure, is usually resorted to. Fig. 48 shows the construction of some of the cells used by Morse.

Before passing to the indirect methods, a direct method due to Fouard (1911) will, perhaps, sometimes bo found useful. In speaking of the semi -permeable membranes prepared by Traube, that made by the action of tannin on gelatine was referred to. Fouard makes use of this, but instead of measuring the pressure in a manometer, he balances it by the use of solutions of cane-sugar of known osmotic pressure outside. It is clear that the approximate osmotic pressure of the solution inside should be known, in order to save a large number of preliminary trials. A small cylinder of silver gauze is taken, immersed in 6 per cent, collodion in order to form a film, washed with water, and then filled with 1 per cent, gelatine, which is then poured out. After soaking for five to six days in dilute tannin solution, it is ready for use. It should be kept in dilute solutions of the membrane formers, presumably gelatine inside and tannin outside, or vice versa. For use, it is connected with a capillary tube, bent liori/.oiitally BO as to be at the same level as the top of the outer solution. The solution whose osmotic pressure is to be measured is placed inside, so as to form a meniscus in the capillary tube. If the osmotic pressure of the cane-sugar solution is greater than that of the inner solution, water will pass out and the meniscus will move towards the cell and vice versa. By adding either water or sugar, as the case may be, a solution can be found which has the same

13. Solid glass stopper for use with substances which attack metals. b, Vent for solution, closed by valve at lower end of stopper 14. Glass manometer attachment for cells with straight necks. d, and e, Porcelain rings for compressing packing. f, Brass collar. y, h, i, and j, Brass pieces with which to close the cell, and also to adjust initial pressure. k, Vent for solution. 15. Glass manometer attachment for cells with straight necks. Like that of Fig.