J.H. van `t Hoff
Zeitschrift fur physikalische Chemie
vol. 1, pp. 481-508 (1887)

In an investigation, whose essential aim was a knowledge of the laws of chemical equilibrium in solutions, it gradually became apparent that there is a deep-seated analogy-indeed, almost an identity-between solutions and gases, so far as their physical relations are concerned; provided that with solutions we deal with the so-called osmotic pressure, where with gases we are concerned with the ordinary elastic pressure. This analogy will be made as clear as possible in the following paper, the physical properties being considered first:


In considering the quantity, with which we shall chiefly have to deal in what follow, at first from the theoretical point of view, let us think of a vessel, A, completely filled for example, with an aqueous solution of sugar, the vessel being placed in water, B. If, now, the perfectly solid wall of the vessel is permeable to water, but impermeable to the dissolved sugar, the attraction of the water by the solution will, as is well known, cause the water to enter A, but this action will soon reach its limit due to the pressure produced by the water which enters (in minimal quantity). Equilibrium exists under these conditions, and the pressure exerted on the wall of the vessel we will designate in the following pages as osmotic pressure.

It is evident that this condition of equilibrium can be established in A also at the outset, that is, without previous entrance of water, by providing the vessel B with a piston which exerts a pressure equal to the osmotic pressure. We can then see that by increasing or diminishing the pressure on the piston it is possible to produce arbitrary changes in the concentration of the solution, through the walls of the vessel.

Let this osmotic pressure be described form an experimental stand-point by one of Pfeffer's experiments. An unglazed porcelain cell was used, which was provided with a membrane permeable to water, but not to sugar. This was obtained as follows: The cell, thoroughly moistened, so as to drive out the air, and filled with a solution of potassium ferrocyanide, was placed in a solution of copper sulphate. The potassium and copper salts came in contact, after a time, by diffusion, in the interior of the porous wall, and formed there a membrane having the desired property. Such a vessel was then filled with a one-per-cent. solution of sugar, and, after being closed by a cork with mano meter attached, was immersed in water. The osmotic pressure gradually makes its appearance through the entrance of some water, and after equilibrium is established it is read on the manometer. Thus, the one-per-cent. solution of sugar in question, which was diluted only an insignificant amount by the water which entered, showed at 6.8°, a pressure of 50.5 millimetres of mercury, therefore about 1/15 of an atmosphere.

The porous membrances here described will, under the name "semipermeable membranes," find extensive application in what follows, even though in some cases the practical application is, perhaps, still unrealized. They furnish a means of dealing with solutions, which bears the closest resemblance to that used with gases. This evidently arises from the fact that the elastic pressure, characteristic of the latter condition, is now introduced also for solutions as osmotic pressure. At the same time let stress be laid upon the fact that we are not dealing here with an artificially forced analogy, but with one which is deeply seated in the nature of the case. The mechanism by which, according to our present conceptions, the elastic pressure of gases is produced is essentially the same as that which gives rise to osmotic pressure in solutions. It depends, in the first case, upon the impact of the gas molecules against the wall of the vessel; in the latter, upon the impact of the molecules of the dissolved substance against the semipermeable membrance, since the molecules of the solvent, being present upon both sides of the membrane through which they pass, do not enter into consideration.

The great practical advantage for the study of solutions, which follows from the analogy upon which stress has been laid, and which leads at onece to quantitative results, is that the application of the second law of thermodynamics to solutions has now become extremely easy, since reversible processes, to which, as is well known, this law applies, can now be performed with the greatest simplicity. It has been already mentioned above that a cylinder, provided with semipermeable walls and piston, when immersed in the solvent, allows any desired change in concentration to be porduced in the solution beneath the piston by exerting a proper pressure upon the piston, just as a gas is compressed and can then expand; only that, in the first case the solvent, in these changes in volume, moves through the wall of the cylinder. Such processes can, in both cases, preserve the condition of reversibility with the same degree of ease, provided that the pressure of the piston is equal to the counter-pressure, i.e., with solutions, to the osmotic pressure.

We will now make use of this practical advantage, especially for the investigations of the laws which hold for "ideal solution," that is, for solutions which are diluted to such an extent that they are comparable with "ideal gases," and in which, therefore, the reciprocal action of the dissolved molecules can be neglected, as also the space occupied by these molecules, in comparison with the volume of the solution itself.


While up to the present, essentially only those changes have been dealt with which the osmotic pressure in solutions undergoes due to changes in concentration and temperature, and while the agreement with the corresponding laws which hold for gases manifested itself, we must now deal with the direct comparison of the two analogous quantities, elastic pressure and osmotic pressure of one and the same substance. It is evident that this applies to gases which have also been investigated in solution; and, as a matter of fact, it will be proved that, in case the law of Henry is satisfied, the osmotic pressure in solution is exactly equal to the elastic pressure as gas, at least at the same temperature and concentration.

For the purpose of demonstration, we will perfom a reversible cycle at constant temperature, by means of semipermeable walls, and then employ the second law of thermodynamics, which, in this case, as is known, leads to an extremely simple result, that no heat is transformed into work, or work into heat, and consequently the sum of all the work done must be equal to zero.

The reversible cycle is perfomed by two similarly arranged double cylinders, with pistons, like the one already described. One cylinder is partly filled with a gas (A), say oxygen, in contact with a solution of oxygen (B), saturated under the conditions of the experiment, for example, and aqueous solution. The wall bc allows only oxygen, but no water to pass through; the wall ab, on the contrary, allows water but not oxygen to pass, and in in contact on the outside with the liquid (E) in question. A reversible transformation can be made with such a cylinder; which amounts to this, that by raising the two pistons (1) and (2) oxygen is evolved form its aqueous solution as gas, while water is removed through ab. This transformation can take place so that the concentrations of gas and solution remain the same. The only difference between the two cylinders is in the concentrations which are present in them. These we will express in the following manner:

The unit of weight of the substance in question fills, in the left vessel, as gas and as solution, the volumes v and V respectively, in the right of v + dv and V+ dV; then in order that Henry's law be satisfied, the following relation must obtain:

v : V = (v + dv) : (V + dV)


v : V = dv : dV.

Let now the pressure and osmotic pressure of gas and solution, in case unit weight is present in unit volume, be respectively P and p (values which hereafter will be shown to be equal), the pressure in gas and solution is, then, from Boyle's law, respectively P/v and p/V.

If we now raise the pistons (1) and (2), and thus liberate a unit weight of gas from the solution, we increase, then, this gas volume v by dv, in order that it may have the concentration of the gas in the left vessel; if the gas just set free is forced into solution by lowering pistons (4) and (5), and thus the volume of the solution V+ dV diminished by dV in the cylinder with semipermeable walls, the cycle is then completed.

Six amounts of work are to be taken into consideration, whose sum, from what is stated above, must be equal to zero. We will designate these by numbers, whose meaning is self-evident. We have, then:

(1) + (2) + (3) + (4) + (5) + (6) = 0

But (2) and (4) are of equal value and opposite sign, since we are dealing with volume changes v and v + dv, in the opposite sense, which takes place at pressures which are inversely proportional to the volumes. For the same reasons the sum of (1) and (5) is zero; then , from the above relation:

(3) + (6) = 0.

The work done by the gas (3), in case it undergoes an increase in volume dv at a pressure P/v, is :

(3) = P/v dv,

while the work done by the solution (6), in case it undergoes a diminution in volume dV at an osmotic pressure p/V is:

(6) = p/V dV.

We obtain then :

P/v dv = p/V dV;

and since v : V = dv : dV, P and p must be equal, which was to be proved.

The conclusion here reached, which will be repeatedly confirmed in what follows, is, in turn, a new support to the law of Gay-Lussac applied to solutions. In case gaseous pressure and osmotic pressure are equal at the same temperature, changes in temperature must have also and equal influence on both. But, on the other hand, the relation found permits of an important extension of the law of Avogadro, which now finds application also to all solutions, if only osmotic pressure is considered instead of elastic pressure. At equal osmotic pressure and equal temperature, equal volumes of the most widely different solution's contain an equal number of molecules, and, indeed, the same number which, at the same pressure and temperature, is contained in an equal volume of a gas.


The well-known formula which expresses for gasses the two laws of Boyle and Gay-Lussac:

PV = RT,

is now, where the laws referred to are also applicable to liquids, valid also fro solutions, if we are dealing with the osmotic pressure. This holds even with the same limitaion which is also to be considered with gases, that the dilution shall be sufficiently great to allow one to disregard the reciprocal action of, and the space taken by, the dissolved particles.

If we wish to include in the above expression, also, the third, the law of Avogadro, this can be done in an exceedingly simple manner, following the suggestion of Horstmann, considering always kilogram-molecules of the substance in question; thus, 2k. hydrogen, 44 k. carbon dioxide, etc. Then R in the above equation has the same value for all gases, since at the same temperature and pressure the quantities mentioned occupy also the same volume. If this value is calculated, and the volume taken in Mr3, the pressure in K° per Mr2, and if, for example, hydrogen at 0° and atmospheric pressure is chosen:

P = 10333, V = 2/0.08956, T = 273, R = 845.05.

The combined expression of the laws of Boyle, Gay-Lussac, and Avogadro is, then:

PV = 845T

and in this form it refers not only to gases, but to all solutions, P being then always taken as osmotic presure.

In order that the formula last obtained may be hereafter easily applied, we give it finally a simpler form, by observing that the number of calories, which is equal to a kilogram-metre, therefore to the equivalent of work (A =1/423), stands in a very simple relation to R, indeed, AR = 2 (more exactly, about one-thousandth less).

Therefore, the following form can be chosen:

APV = 2T

which has the great practical advantage that the work done, of which we shall often speak hereafter, finds a very simple expression, in case it is calculated in calories.