[26] The velocity of the transmission of sound through gases and vapours closely bears on this. It = √(Kpg)/D(1 + at), where K is the ratio between the two specific heats (it is approximately 1·4 for gases containing two atoms in a molecule), p the pressure of the gas expressed by weight (that is, the pressure expressed by the height of a column of mercury multiplied by the density of a = 0·00367, and t the temperature. Hence, if K be known, and as D can he found from the composition of a gas, we can calculate the velocity of the transmission of sound in that gas. Or if this velocity be known, we can find K. The relative velocities of sound in two gases can he easily determined (Kundt).
If a horizontal glass tube (about 1 metre long and closed at both ends) be full of a gas, and be firmly fixed at its middle point, then it is easy to bring the tube and gas into a state of vibration, by rubbing it from centre to end with a damp cloth. The vibration of the gas is easily rendered visible, if the interior of the tube be dusted with lycopodium (the yellow powder-dust or spores of the lycopodium plant is often employed in medicine), before the gas is introduced and the tube fused up. The fine lycopodium powder arranges itself in patches, whose number depends on the velocity of sound in the gas. If there be 10 patches, then the velocity of sound in the gas is ten times slower than in glass. It is evident that this is an easy method of comparing the velocity of sound in gases. It has been demonstrated by experiment that the velocity of sound in oxygen is four times less than in hydrogen, and the square roots of the densities and molecular weights of hydrogen and oxygen stand in this ratio.
[27] If the conception of the molecular weights of substances does not give an exact law when applied to the latent heat of evaporation, at all events it brings to light a certain uniformity in figures, which otherwise only represent the simple result of observation. Molecular quantities of liquids appear to expend almost equal amounts of heat in their evaporation. It may be said that the latent heat of evaporation of molecular quantities is approximately constant, because the vis viva of the motion of the molecules is, as we saw above, a constant quantity. According to thermodynamics the latent heat of evaporation is equal to t + 273 / xA + yB (n′ - n) dp / dT × 13·59, where t is the boiling point, n′ the specific volume (i.e. the volume of a unit of weight) of the vapour, and n the specific volume of the liquid, dp/dT the variation of the tension with a rise of temperature per 1°, and 13·89 the density of the mercury according to which the pressure is measured. Thus the latent heat of evaporation increases not only with a decrease in the vapour density (i.e. the molecular weight), but also with an increase in the boiling point, and therefore depends on different factors.
[27 bis] The osmotic pressure, vapour tension of the solvent, and several other means applied like the cryoscopic method to dilute solutions for determining the molecular weight of a substance in solution, are more difficult to carry out in practice, and only the method of determining the rise of the boiling point of dilute solutions can from its facility be placed parallel with the cryoscopic method, to which it bears a strong resemblance, as in both the solvent changes its state and is partially separated. In the boiling point method it passes off in the form of a vapour, while in cryoscopic determinations it separates out in the form of a solid body.
Van't Hoff, starting from the second law of thermodynamics, showed that the dependence of the rise of pressure (dp) upon a rise of temperature (dT) is determined by the equation dp = (kmp/2T2)dT, where k is the latent heat of evaporation of the solvent, m its molecular weight, p the tension of the saturated vapour of the solvent at T, and T the absolute temperature (T = 273 + t), while Raoult found that the quantity (p - p′)/p (Chapter I., Note [50]) or the measure of the relative fall of tension (p the tension of the solvent or water, and p′ of the solution) is found by the ratio of the number of molecules, n of the substance dissolved, and N of the solvent, so that (p - p′)/p = Cn/(N + n) where C is a constant. With very dilute solutions p - p′ may be taken as equal to dp, and the fraction n/(N + n) as equal to n/N (because in that case the value of N is very much greater than n), and then, judging from experiment, C is nearly unity—hence: dp/p = n/N or dp = np/N, and on substituting this in the above equation we have (kmp/2T2)dT = np/N. Taking a weight of the solvent m/N = 100, and of the substance dissolved (per 100 of the solvent) q, where q evidently = nM, if M be the molecular weight of the substance dissolved, we find that n/N = qm/100M, and hence, according to the preceding equation, we have M = 0·02T2 / k · q / dT , that is, by taking a solution of q grms. of a substance in 100 grms. of a solvent, and determining by experiment the rise of the boiling point dT, we find the molecular weight M of the substance dissolved, because the fraction 0·02T2/k is (for a given pressure and solvent) a constant; for water at 100° (T = 373°) when k = 534 (Chapter I., Note [11]), it is nearly 5·2, for ether nearly 21, for bisulphide of carbon nearly 24, for alcohol nearly 11·5, &c. As an example, we will cite from the determinations made by Professor Sakurai, of Japan (1893), that when water was the solvent and the substance dissolved, corrosive sublimate, HgCl2, was taken in the quantity q = 8·978 and 4·253 grms., the rise in the boiling point dT was = O°·179 and 0°·084, whence M = 261 and 263, and when alcohol was the solvent, q = 10·873 and 8·765 and dT = 0°·471 and 0°·380, whence M = 266 and 265, whilst the actual molecular weight of corrosive sublimate = 271, which is very near to that given by this method. In the same manner for aqueous solutions of sugar (M = 342), when q varied from 14 to 2·4, and the rise of the boiling point from 0°·21 to 0°·035, M was found to vary between 339 and 364. For solutions of iodine I2 in ether, the molecular weight was found by this method to be between 255 and 262, and I2 = 254. Sakurai obtained similar results (between 247 and 262) for solutions of iodine in bisulphide of carbon.
We will here remark that in determining M (the molecular weight of the substance dissolved) at small but increasing concentrations (per 100 grms. of water), the results obtained by Julio Baroni (1893) show that the value of M found by the formula may either increase or decrease. An increase, for instance, takes place in aqueous solutions of HgCl2 (from 255 to 334 instead of 271), KNO3 (57–66 instead of 101), AgNO3 (104–107 instead of 170), K2SO4 (55–89 instead of 174), sugar (328–348 instead of 342), &c. On the contrary the calculated value of M decreases as the concentration increases, for solutions of KCl (40–39 instead of 74·5), NaCl (33–28 instead of 58·5), NaBr (60–49 instead of 103), &c. In this case (as also for LiCl, NaI, C2H3NaO2, &c.) the value of i (Chapter I., Note [49]), or the ratio between the actual molecular weight and that found by the rise of the boiling point, was found to increase with the concentration, i.e. to be greater than 1, and to differ more and more from unity as the strength of the solution becomes greater. For example, according to Schlamp (1894), for LiCl, with a variation of from 1·1 to 6·7 grm. LiCl per 100 of water, i varies from 1·63 to 1·89. But for substances of the first series (HgCl2, &c.), although in very dilute solutions i is greater than 1, it approximates to 1 as the concentration increases, and this is the normal phenomenon for solutions which do not conduct an electric current, as, for instance, of sugar. And with certain electrolytes, such as HgCl2, MgSO4, &c., i exhibits a similar variation; thus, for HgCl2 the value of M is found to vary between 255 and 334; that is, i (as the molecular weight = 271) varies between 1·06 and 0·81. Hence I do not believe that the difference between i and unity (for instance, for CaCl2, i is about 3, for KI about 2, and decreases with the concentration) can at present be placed at the basis of any general chemical conclusions, and it requires further experimental research. Among other methods by which the value of i is now determined for dilute solutions is the study of their electroconductivity, admitting that i = 1 + a(k - 1), where a = the ratio of the molecular conductivity to the limiting conductivity corresponding to an infinitely large dilution (see Physical Chemistry), and k is the number of ions into which the substance dissolved can split up. Without entering upon a criticism of this method of determining i, I will only remark that it frequently gives values of i very close to those found by the depression of the freezing point and rise of the boiling point; but that this accordance of results is sometimes very doubtful. Thus for a solution containing 5·67 grms. CaCl2 per 100 grms. of water, i, according to the vapour tension = 2·52, according to the boiling point = 2·71, according to the electroconductivity = 2·28, while for solutions in propyl alcohol (Schlamp 1894) i is near to 1·33. In a word, although these methods of determining the molecular weight of substances in solution show an undoubted progress in the general chemical principles of the molecular theory, there are still many points which require explanation.
We will add certain general relations which apply to these problems. Isotonic (Chapter I., Note [19]) solutions exhibit not only similar osmotic pressures, but also the same vapour tension, boiling point and freezing temperature. The osmotic pressure bears the same relation to the fall of the vapour tension as the specific gravity of a solution does to the specific gravity of the vapour of the solvent. The general formulæ underlying the whole doctrine of the influence of the molecular weight upon the properties of solutions considered above, are: 1. Raoult in 1886–1890 showed that
p - p′ / p · 100 / a · M / m = a constant C
where p and p′ are the vapour tensions of the solvent and substance dissolved, a the amount in grms. of the substance dissolved per 100 grms. of solvent, M and m the molecular weights of the substance dissolved and solvent. 2. Raoult and Recoura in 1890 showed that the constant above C = the ratio of the actual vapour density d′ of the solvent to the theoretical density d calculated according to the molecular weight. This deduction may now be considered proved, because both the fall of tension and the ratio of the vapour densities d′/d give, for water 1·03, for alcohol 1·02, for ether 1·04, for bisulphide of carbon 1·00, for benzene 1·02, for acetic acid 1·63. 3. By applying the principles of thermodynamics and calling L1 the latent heat of fusion and T1 the absolute (= t + 273) temperature of fusion of the solvent, and L2 and T2 the corresponding values for the boiling point, Van't Hoff in 1886–1890 deduced:—
Depression of freezing point / Rise of boiling point = L2 / L1 · T12 / T22