[19] In the light of recent work, especially by Morse and Frazer, the law would state, more exactly, that a substance in solution produces the osmotic pressure, at a given temperature, which it would exert, if it were contained as a gas, at the same temperature, ‹in the volume occupied by the pure solvent› of the solution. For sufficiently dilute solutions, the volume of the solution and the volume of the solvent may be considered identical; for more concentrated solutions, there is a decided difference, and the correct volume to use in calculation is the volume of the solvent alone, ‹i.e.› the volume of the solution reduced by the volume of the pure solute. This corresponds to the correction of the volume in the more accurate expression for the behavior of gases, developed by van der Waals; in place of ‹v›, the total gas volume, (‹v› − ‹b›), the total volume of the gas less the volume of the spheres of action of the gas particles, is used, especially for strongly compressed or concentrated gases. It may be added that van 't Hoff's thermodynamic proof involves the same correct definition of the volume that Morse and Frazer subsequently developed experimentally. ‹Cf.› Bancroft, ‹J. Phys. Chem.›, «10», 319 (1906). [20] One gram of cane sugar, C_{12}H_{22}O_{11} (the mol. wt. is 342) corresponds to 1 / 342 gram molecule or mole and, therefore, to 2.02 / 342 gram of hydrogen. The volume containing this quantity of hydrogen is 100.6 c.c.; a liter would contain 2.02 / 342 × 1000 / 100.6 gram of hydrogen. The pressure of a mole or 2.02 grams of hydrogen, contained in a liter at 0°, is 22.4 atmospheres, and the pressure of the quantity of hydrogen given above, in a liter, would be (2.02 × 1000) / (342 × 100.6) × (22.4 / 2.02) at 0°. At 36° C., for instance, the pressure would be 309 / 273 times as great, or ‹P›_{calculated} = (2.02 × 1000 × 22.4 × 309) / (342 × 100.6 × 2.02 × 273) = 0.735 atmosphere. [21] The exact relations are discussed in van 't Hoff's ‹Lectures on Physical Chemistry›, Part II, pp. 42–59, Nernst's ‹Theoretical Chemistry› (1904), pp. 142 and 148, and H. C. Jones's ‹The Elements of Physical Chemistry› (1909), pp. 252, 271. [22] ‹Vide› Raoult, ‹Scientific Memoir Series›, «4», 71, 127. [23] ‹I.e.› abnormally small depressions of freezing-points or elevations of boiling-points. [24] Nernst, ‹Theoretical Chemistry›, p. 486; Hendrixson, ‹Z. anorg. Chem.›, «13», 73 (1897). [25] ‹Cf.› Bancroft, ‹J. Phys. Chem.›, «10», 319 (1906). [26] For the discussion of other instances, ‹vide› Bancroft, ‹loc. cit.› [27] Footnote 3, p. 12. [28] For example, determinations of distribution coefficients (p. 18), heats of dilution (p. 19), conductivities and chemical activity (Chapters IV–VI). [p021] CHAPTER III «OSMOTIC PRESSURE AND THE THEORY OF SOLUTION II»