In still other instances, apparently too high osmotic pressures, or too low molecular weights, have been found by the application of the Avogadro-van 't Hoff Hypothesis to solutions: for instance, the molecular weight of sodium, when dissolved in mercury, was found by Ramsay to vary from 21.6, in dilute, to 15.1 in concentrated solutions. But Cady found that the heat of dilution of sodium in mercury solution is considerable, and by taking this properly into account, Bancroft was able to show that the molecular weight, correctly calculated in a given experiment, is 22.7 (agreeing well with the theoretical weight 23), in place of 16.5, as calculated without making the required allowance for the heat of dilution.[26] These determinations are most instructive in showing that the sources of some of the most important deviations from the van 't Hoff-Avogadro principle, deviations which have been brought forward as arguments against its assumptions, are due, not to any untrustworthiness of the general principle, but to the error of neglecting to observe the limiting conditions of the formulation, or of neglecting to make corresponding corrections for the non-observance thereof.
«Summary.»—Van 't Hoff's theory of solution—that the osmotic pressure of substances in solution obeys the laws of gases, and that equal volumes of the most varied dilute solutions, having the same temperature and osmotic pressure, contain the same number of dissolved molecules, that number, namely, which would be found in the same volume of a gas at the same temperature and gas pressure,—accords thus, not only with the demands of thermodynamics,[27] but is also, within the limits demanded by the theory itself, in agreement with the best experimental measurements of osmotic pressures that have been made in recent years. The apparent exceptions, as in the cases just described and, as we shall find, in the case of electrolytic dissociation, are found to be no exceptions, when the conclusions, reached on the assumption that [p020] the theory is correct, are tested rigorously by independent methods of investigation.[28]
The fundamental laws of gases and the Avogadro Hypothesis may be condensed into the following general equation, expressing all of the laws, viz.: ‹P› ‹V› = ‹n› ‹R› ‹T›. This equation applies equally to the osmotic pressures of dilute solutions, the osmotic pressure being substituted for the gas pressure. In the equation, ‹T› is the absolute temperature of the gas or solution, ‹P› the gaseous or osmotic pressure, ‹V› the free space of the gas volume, ‹i.e.› the volume of the gas less the volume occupied by the gas molecules, or the volume of the pure solvent in the solution used, ‹i.e.› the volume of the solution less the volume of the solute. ‹R› is the so-called ‹gas-constant›, and represents ‹the work› done against the external pressure when one gram molecule, or mole, of the gas is heated one degree and allowed to expand, say at constant pressure ‹P›, against an external pressure ‹P›; ‹n› represents the number of gram molecules or moles of gas or solute used (the total weight of solute or gas, divided by the average weight of a mole in the gas or solute). If a given weight of a gas or solute is taken, and no dissociation or association occurs (such as would involve appreciable heats of dilution), then ‹n› is a given number; and, therefore, at a given temperature ‹T›, all the factors on the right side of the general equation being given numbers, ‹P› ‹V› ‹is a constant› (Boyle's law). For a given quantity of gas or solute (‹n› is a given number), kept at ‹constant volume› ‹V›, the pressure must vary as the absolute temperature (Gay-Lussac's law); ‹P› / ‹T› = ‹n› ‹R› / ‹V› = ‹a constant›. When the pressure, volume and temperature of two gases, or two dilute solutions, are equal, ‹n›, the number of gas or solute molecules present, must be the same (Avogadro-van 't Hoff Hypothesis); ‹n› = ‹P› ‹V› / (‹R› ‹T›), and all the factors of the right side are the same for the gases and solutions which we are comparing. Finally, if the pressure is expressed in atmospheres, the volume in litres, and the temperature in absolute degrees, the ‹gas-constant› ‹R› = ‹P› ‹V› / ‹T› = 1 × 22.4 / 273 = 0.082.