«Quantitative Evidence.»—Some of the most exact quantitative evidence bearing on these relations, such as the results of investigations, by Griffith and by Taylor, on the freezing-point depressions of solutions of electrolytes, may be briefly considered. The depression of the freezing-point of a given solvent by a solute is proportional to the concentration of the solute or proportional to its osmotic pressure. Further, according to the Van 't Hoff Hypothesis (p. 15), the osmotic pressure at a constant temperature is dependent only on the number of molecules present in unit volume, and not on the nature or composition of the molecules: the freezing-point of the solvent is depressed, likewise, proportionally to the total concentration of the solute, irrespective of the fact whether the solution contains only one, or more than one molecular species. ‹The ratio, observed depression / concentration›,[116] or ‹Δ› / ‹C›, ‹should be constant›,[117] therefore, in a given solvent, for dilute solutions of all kinds of solutes, simple or mixed. Griffith[118] found, for a solution of cane sugar, a non-electrolyte, in water, the ratio of the freezing-point depression to the concentration to be 1.858°. For instance, the freezing-point of a 0.01 molar solution of cane sugar (3.42 grams of cane sugar per liter; C_{12}H_{22}O_{11} = 342) is found to be −0.01858°, and 0.01858 / 0.01 = 1.858. This ratio should be the same, as stated above, according to van 't Hoff's theory of solutions, for dilute aqueous solutions of all solutes. But the ratio ‹Δ› / ‹C› for an aqueous solution of potassium chloride, an electrolyte, ‹was found to increase slowly› and ‹continuously› until in 0.0003 molar solution the ratio 3.72 was found, which is exactly twice the value obtained with cane sugar. The result indicates, therefore, a ‹gradual dissociation› of the potassium chloride with ‹increasing› dilution, and a ‹dissociation, ultimately, of each molecule› of the salt into ‹two› new molecules, in all respects exactly as demanded by the theory of Arrhenius. Loomis[119] found in a similar way a ratio of 3.61 for HCl, 3.71 [p069] for KOH, 3.60 for KCl, 3.67 for NaCl, 3.73 for HNO_{3}, etc., when 0.01 molar aqueous solutions were used. For similar solutions of calcium chloride CaCl_{2}, magnesium chloride MgCl_{2}, and sodium sulphate Na_{2}SO_{4}, the value 5.07 was found as the ratio between the depressions of the freezing-point and the concentration of the salts in extremely dilute solutions—a result showing, plainly, a dissociation of each salt into ‹three smaller molecules›. The limit 5.67 for such a dissociation is not quite reached in these cases, because salts of the types Me″X_{2} and Me_{2}′Y″ ionize less readily than do the electrolytes Me′X′, a fact also shown by their conductivities.