«Osmotic Pressure and the Laws of Gases.»—The work of van 't Hoff, which has proved of inestimable value to the development of chemistry, succeeded in demonstrating that, ‹for dilute solutions, the osmotic pressure›, as defined above, ‹obeys the common laws of gases›,[12]—‹that, in fact, a substance in a dilute solution has an osmotic pressure equal to the gas pressure which it would exert if it were a gas of the same volume and at the same temperature›.[13] Space does not permit the presentation of all the details of the evidence confirming this conclusion, but some of the most direct experimental proofs[14] will be considered. [p013] «Boyle's Law.»—Boyle's law for gases states that, at a constant temperature, the pressure of a gas changes inversely as its volume, or directly as its concentration. Mathematically we have ‹P› : ‹P′› = ‹V′› : ‹V› or ‹P› ‹V› = ‹P′› ‹V′› = a constant, and ‹P› : ‹P′› = ‹C› : ‹C′› or ‹P› : ‹C› = ‹P′› : ‹C′› = a constant. When van 't Hoff published his first paper on the subject, Pfeffer's results from the direct measurement of the osmotic pressures of cane-sugar solutions were available, and even these, although experimentally not as exact as more recent determinations, showed plainly that, at a given temperature, the osmotic pressure of a sugar solution varies directly as the concentration, or inversely as the volume containing a given weight of the sugar. At 13–16° we have: Concentration. Osmotic Pressure. Pressure/ mm. Mercury. Concentration. 1.00% 535 535 2.00% 1016 508 2.74% 1518 554 4.00% 2082 521 6.00% 3075 513 The ratio of pressure to concentration varies irregularly round a mean value of 526, and is approximately constant. The more recent, exceedingly careful measurements of Morse and Frazer confirm the conclusion, that Boyle's law holds for the osmotic [p014] pressures of dilute solutions; they find that the osmotic pressures of glucose and of cane-sugar solutions vary directly as the concentrations of the solutions, at a constant temperature.[15] «Gay-Lussac's Law.»—Gay-Lussac's law for gases states that, if the volume of gas is kept constant, its pressure increases by 1 / 273 of its value for every degree above 0° C., or ‹P›_{‹t›} = ‹P›_{0} (1 + ‹t› / 273). Expressing the temperature in absolute degrees, we have more simply: ‹P›_{‹t›} = ‹P›_{0} (‹T› / 273) or P_{t} / T = ‹P›_{0} / 273 = a constant.[16] That is, the pressure of a gas varies directly as its absolute temperature, if the volume is kept constant.