Depression of freezing point = AT12a / L1M1

Rise of boiling point = AT22a / L2M1

where A = 0·01988 (or nearly 0·02 as we took it above), a is the weight in grms. of the substance dissolved per 100 grms. of the solvent, M1 the molecular weight of the dissolved substance (in the solution), and M the molecular weight of this substance according to its composition and vapour density, then i = M/M1. The experimental data and theoretical considerations upon which these formulæ are based will be found in text-books of physical and theoretical chemistry.

[28] A similar conclusion respecting the molecular weight of liquid water (i.e. that its molecule in a liquid state is more complex than in a gaseous state, or polymerized into H8O4, H6O3 or in general into nH2O) is frequently met in chemico-physical literature, but as yet there is no basis for its being fully admitted, although it is possible that a polymerization or aggregation of several molecules into one takes place in the passage of water into a liquid or solid state, and that there is a converse depolymerization in the act of evaporation. Recently, particular attention has been drawn to this subject owing to the researches of Eötvös (1886) and Ramsay and Shields (1893) on the variation of the surface tension N with the temperature (N = the capillary constant a2 multiplied by the specific gravity and divided by 2, for example, for water at 0° and 100° the value of a2 = 15·41 and 12·58 sq. mm., and the surface tension 7·92 and 6·04). Starting from the absolute boiling point (Chapter II., Note [29]) and adding 6°, as was necessary from all the data obtained, and calling this temperature T, it is found that AS = kT, where S is the surface of a gram-molecule of the liquid (if M is its weight in grams, s its sp. gr., then its sp. volume = M/s, and the surface S = ∛(M/s)2), A the surface tension (determined by experiment at T), and k a constant which is independent of the composition of the molecule. The equation AS = kT is in complete agreement with the well-known equation for gases vp = RT (p. [140]) which serves for deducing the molecular weight from the vapour density. Ramsay's researches led him to the conclusion that the liquid molecules of CS2, ether, benzene, and of many other substances, have the same value as in a state of vapour, whilst with other liquids this is not the case, and that to obtain an accordance, that is, that k shall be a constant, it is necessary to assume the molecular weight in the liquid state to be n times as great. For the fatty alcohols and acids n varies from 1½ to 3½, for water from 2¼ to 4, according to the temperature (at which the depolymerization takes place). Hence, although this subject offers a great theoretical interest, it cannot be regarded as firmly established, the more so since the fundamental observations are difficult to make and not sufficiently numerous; should, however, further experiments confirm the conclusions arrived at by Professor Ramsay, this will give another method of determining molecular weights.

[28 bis] Their variance is expressed in the same manner as was done by Van't Hoff (Chapter I., Notes [19] and [49]) by the quantity i, taking it as = 1 when k = 1·05, in that case for KI, i is nearly 2, for borax about 4, &c.

[29] We will cite one more example, showing the direct dependence of the properties of a substance on the molecular weight. If one molecular part by weight of the various chlorides—for instance, of sodium, calcium, barium, &c.—be dissolved in 200 molecular parts by weight of water (for instance, in 3,600 grams) then it is found that the greater the molecular weight of the salt dissolved, the greater is the specific gravity of the resultant solution.

Molecular
weight		Sp. gr. at 15° Molecular
weight		Sp. gr. at 15°
HCl 36·51·0041CaCl21111·0236
NaCl 58·51·0106NiCl21301·0328
KCl 74·51·0121ZnCl21361·0331
BeCl2 801·0138BaCl22081·0489
MgCl2 951·0203

[29 bis] With respect to the optical refractive power of substances, it must first be observed that the coefficient of refraction is determined by two methods: (a) either all the data are referred to one definite ray—for instance, to the Fraunhofer (sodium) line D of the solar spectrum—that is, to a ray of definite wave length, and often to that red ray (of the hydrogen spectrum) whose wave length is 656 millionths of a millimetre; (b) or Cauchy's formula is used, showing the relation between the coefficient of refraction and dispersion to the wave length n = A + B / λ , where A and B are two constants varying for every substance but constant for all rays of the spectrum, and λ is the wave length of that ray whose coefficient of refraction is n. In the latter method the investigation usually concerns the magnitudes of A, which are independent of dispersion. We shall afterwards cite the data, investigated by the first method, by which Gladstone, Landolt, and others established the conception of the refraction equivalent.

It has long been known that the coefficient of refraction n for a given substance decreases with the density of a substance D, so that the magnitude (n - 1) ÷ D = C is almost constant for a given ray (having a definite wave length) and for a given substance. This constant is called the refractive energy, and its product with the atomic or molecular weight of a substance the refraction equivalent. The coefficient of refraction of oxygen is 1·00021, of hydrogen 1·00014, their densities (referred to water) are 0·00143 and 0·00009, and their atomic weights, O = 16, H = 1; hence their refraction equivalents are 3 and 1·5. Water contains H2O, consequently the sum of the equivalents of refraction is (2 × 1·5) + 3 = 6. But as the coefficient of refraction of water = 1·331, its refraction equivalent = 5·958, or nearly 6. Comparison shows that, approximately, the sum of the refraction equivalents of the atoms forming compounds (or mixtures) is equal to the refraction equivalent of the compound. According to the researches of Gladstone, Landolt, Hagen, Brühl and others, the refraction equivalents of the elements are—H = 1·3, Li = 3·8, B = 4·0, C = 5·0, N = 4·1 (in its highest state of oxidation, 5·3), O = 2·9, F = 1·4, Na = 4·8, Mg = 7·0, Al = 8·4, Si = 6·8, P = 18·3, S = 16·0, Cl = 9·9, K = 8·1, Ca = 10·4, Mn = 12·2, Fe = 12·0 (in the salts of its higher oxides, 20·1), Co = 10·8, Cu = 11·6, Zn = 10·2, As = 15·4, Bi = 15·3, Ag = 15·7, Cd = 13·6, I = 24·5, Pt = 26·0, Hg = 20·2, Pb = 24·8, &c. The refraction equivalents of many elements could only be calculated from the solutions of their compounds. The composition of a solution being known it is possible to calculate the refraction equivalent of one of its component parts, those for all its other components being known. The results are founded on the acceptance of a law which cannot be strictly applied. Nevertheless the representation of the refraction equivalents gives an easy means for directly, although only approximately, obtaining the coefficient of refraction from the chemical composition of a substance. For instance, the composition of carbon bisulphide is CS2 = 76, and from its density, 1·27, we find its coefficient of refraction to be 1·618 (because the refraction equivalent = 5 + 2 × 16 = 37), which is very near the actual figure. It is evident that in the above representation compounds are looked on as simple mixtures of atoms, and the physical properties of a compound as the sum of the properties present in the elementary atoms forming it. If this representation of the presence of simple atoms in compounds had not existed, the idea of combining by a few figures a whole mass of data relating to the coefficient of refraction of different substances could hardly have arisen. For further details on this subject, see works on Physical Chemistry.