[14] As a result of numerous vain endeavors, as well as of much direct evidence of a positive character, the scientific world has, for many years, held the opinion that any sort of "perpetual motion machine" is impossible. Every one now admits that a machine which would be able to work continuously, without consuming energy, is an impossibility—that is, that a "perpetuum mobile of the first class," as it is called, is impossible (law of the conservation of energy or first law of thermodynamics). From this law it does not of necessity follow, however, that it would be impossible to make a machine or device that would convert continuously into available energy or work, say, the enormous amounts of heat energy of the earth or of large bodies of water ("dissipated energy") which would thereby be cooled below the temperatures of their surroundings. Such a hypothetical process has been termed a "perpetuum mobile of the second class"; it has never been realized and is universally conceded to be an impossibility; the so-called "second law of thermodynamics" gives expression to this fact.
Now van 't Hoff [Z. phys. Chem., 1, 481 (1887)] showed, first, that a gas like oxygen, nitrogen, hydrogen, etc., which is soluble in proportion to its gas pressure (Henry's law), must exert, in solution, an osmotic pressure equal to the gas pressure, which it would have, if present in the same quantity as a gas in the same volume at the same temperature; for, if such were not the case, the solution and gas could be used to produce a perpetuum mobile of the second class, which, according to the above law, is an impossibility. Similar proofs were given by Rayleigh [Nature, 55, 253 (1897)] and by Larmor [Phil. Trans., 190, 266 (1897), Nature, 55, 545 (1897)] that the principle applies to solutions of other solutes.
Provided, then, that we have (1) perfect semipermeable membranes, (2) sufficiently dilute solutions, and (3) none but negligible heats of dilution (p. [12]), van 't Hoff's generalization, concerning the relation of osmotic pressure and the laws of gases, must hold, if the perpetuum mobile of the second class is impossible, as is demanded by the second law of thermodynamics.
[15] See p. [15] in regard to the relation for concentrated solutions.
[16] The pressure P0 of a given quantity (weight) of a gas at 0° C., in a given constant volume, is also a given number and consequently P0/273 is a constant under these conditions.
[17] The slight differences in the ionization of copper sulphate solutions of 14% and 17% and at 20° and 80° are not included in the calculation, ionization being unknown, when van 't Hoff made his calculations.
[18] Am. Chem. J., 41, 258 (1909).
[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, C12H22O11 (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 Pcalculated = (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.