Callendar is to trace the effect of possible combination of molecules of solute with molecules of the solvent.
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.
There must, then, be a relation between the rate of change of the concentration and the osmotic pressure gradient, and thus we may consider the osmotic pressure gradient as a force driving the solute through a viscous medium.
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.
The vapour pressure of the solution of a non-volatile solute is less than the vapour pressure of the pure solvent.
The quantity of substance, or solute, which a given quantity of liquid or solvent will dissolve in presence of excess of the solute measures the solubility of the solute in the given solvent in the conditions of temperature and pressure.
The theoretical value for the depression of the freezing point of a dilute solution per gramme-equivalent of solute per litre is 1857° C. Completely ionized solutions of salts with two ions should give double this number or 3.714°, while electrolytes with three ions should have a value of 5.57°.
Of crystals of the solute, crystals will dissolve as solvent enters, and the solution remains saturated throughout.
Membranes which allow a solvent to pass freely but are impervious to a solute when dissolved in that solvent.
The solubilities of solids may be expressed in terms of the mass of solute which will dissolve in loo grammes of water.
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.
It is assumed that each molecule of solute combines with a molecules of solvent according to the ordinary law of chemical combination, and that the number a, representing the degree of hydration, remains constant within wide limits of temperature and concentration.
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.
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.
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 case of solutions, if the absorption of the solvent is negligible, the effect of increasing the concentration of the absorbing solute is the same as that of increasing the thickness in the same ratio.
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.
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
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.