Pfeffer's measurements, with solutions of 1 g. of sugar in 100 c.c. of water (the volume of the solution is 100.6 c.c.), were shown to prove, that the observed osmotic pressures agreed excellently with the gas pressures, calculated for the equimolar weight of hydrogen, in the same volume and at the same temperature:
| Temperature. | Osmotic Pressure | |
|---|---|---|
| Found. Atmosphere. | Calculated.[20] Atmosphere. | |
| 6.8 | 0.664 | 0.665 |
| 13.7 | 0.691 | 0.681 |
| 14.2 | 0.671 | 0.682 |
| 15.5 | 0.684 | 0.686 |
| 22 | 0.721 | 0.701 |
| 32 | 0.716 | 0.725 |
| 36 | 0.746 | 0.735 |
Morse's more recent and more exact results show, that the osmotic pressure of solutions of cane sugar and of glucose (corrected for the volume occupied by the sugar, see footnote, p. [15]) agrees within 6% with the values demanded by van 't Hoff's theory, being about 6% larger for concentrations ranging from 0.1 to 1.0 molar. The difference of 6% is noteworthy and is probably due to secondary causes, but suggests extended investigation of its source.
Indirect Determinations of Osmotic Pressure.
Apparent Exceptions.
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.
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.
Chapter II Footnotes
[3] Even after a solution of uniform concentration of the solute is formed, the tendency toward diffusion, and the diffusion itself, and the resulting pressure must still persist. But a state of dynamic (or flowing) equilibrium must be considered now to exist, the loss caused by the moving away of the solute, from a given part of the solution, being balanced by the diffusion (into that part) of the solute from the neighboring parts. Whether one ascribes the diffusion to inherent molecular velocities of the solute, or to an attraction between solvent and solute, the discrete particles of the solute in a solution of uniform concentration will continue to have such inherent velocities (Chap. III), and will also continue to be surrounded by pure solvent, exactly as in solutions of unequal concentrations, where the diffusion may be observed, because the net result, in such a case, is a one-sided action.