[4] The specific heats here given refer to different limits of temperature, but in the majority of cases between 0° and 100°; only in the case of bromine the specific heat is taken (for the solid state) at a temperature below -7°, according to Regnault's determination. The variation of the specific heat with a change of temperature is a very complex phenomenon, the consideration of which I think would here be out of place. I will only cite a few figures as an example. According to Bystrom, the specific heat of iron at 0° = 0·1116, at 100° = 0·1114, at 200° = 0·1188, at 300° = 0·1267, and at 1,400° = 0·4031. Between these last limits of temperature a change takes place in iron (a spontaneous heating, recalescence), as we shall see in Chapter XXII. For quartz SiO2 Pionchon gives Q = 0·1737 + 394t10-6 - 27t210-9 up to 400°, for metallic aluminium (Richards, 1892) at 0° 0·222, at 20° 0·224, at 100° 0·232; consequently, as a rule, the specific heat varies slightly with the temperature. Still more remarkable are H. E. Weber's observations on the great variation of the specific heat of charcoal, the diamond and boron:

0°100°200°600°900°
Wood charcoal0·150·230·290·440·46
Diamond0·100·190·220·440·45
Boron0·220·290·35——

These determinations, which have been verified by Dewar, Le Chatelier (Chapter VIII., Note [13]), Moissan, and Gauthier, the latter finding for boron AQ = 6 at 400°, are of especial importance as confirming the universality of Dulong and Petit's law, because the elements mentioned above form exceptions to the general rule when the mean specific heat is taken for temperatures between 0° and 100°. Thus in the case of the diamond the product of A × Q at 0° = 1·2, and for boron = 2·4. But if we take the specific heat towards which there is evidently a tendency with a rise of temperature, we obtain a product approaching to 6 as with other elements. Thus with the diamond and charcoal, it is evident that the specific heat tends towards 0·47, which multiplied by 12 gives 5·6, the same as for magnesium and aluminium. I may here direct the reader's attention to the fact that for solid elements having a small atomic weight, the specific heat varies considerably if we take the average figures for temperatures 0° to 100°:

Li = 7Be = 9B = 11C = 12
Q =0·940·420·240·20
AQ =6·63·82·62·4

It is therefore clear that the specific heat of beryllium determined at a low temperature cannot serve for establishing its atomicity. On the other hand, the low atomic heat of charcoal, graphite, and the diamond, boron, &c., may perhaps depend on the complexity of the molecules of these elements. The necessity for acknowledging a great complexity of the molecules of carbon was explained in Chapter [VIII]. In the case of sulphur the molecule contains at least S6 and its atomic heat = 32 × 0·163 = 5·22, which is distinctly below the normal. If a large number of atoms of carbon are contained in the molecule of charcoal, this would to a certain extent account for its comparatively small atomic heat. With respect to the specific heat of compounds, it will not be out of place to mention here the conclusion arrived at by Kopp, that the molecular heat (that is, the product of MQ) may be looked on as the sum of the atomic heats of its component elements; but as this rule is not a general one, and can only be applied to give an approximate estimate of the specific heats of substances, I do not think it necessary to go into the details of the conclusions described in Liebig's ‘Annalen Supplement-Band,’ 1864, which includes a number of determinations made by Kopp.

[5] It must be remarked that in the case of oxygen (and also hydrogen and carbon) compounds the quotient of MQ/n, where n is the number of atoms in the molecule, is always less than 6 for solids; for example, for MgO = 5·0, CuO = 5·1, MnO2 = 4·6, ice (Q = 0·504) = 3, SiO2 = 3·5, &c. At present it is impossible to say whether this depends on the smaller specific heat of the atom of oxygen in its solid compounds (Kopp, Note [4]) or on some other cause; but, nevertheless, taking into account this decrease depending on the presence of oxygen, a reflection of the atomicity of the elements may to a certain extent be seen in the specific heat of the oxides. Thus for alumina, Al2O3 (Q = 0·217), MQ = 22·3, and therefore the quotient MQ/n = 4·5, which is nearly that given by magnesium oxide, MgO. But if we ascribe the same composition to alumina, as to magnesia—that is, if aluminium were counted as divalent—we should obtain the figure 3·7, which is much less. In general, in compounds of identical atomic composition and of analogous chemical properties the molecular heats MQ are nearly equal, as many investigators have long remarked. For example, ZnS = 11·7 and HgS = 11·8; MgSO4 = 27·0 and ZnSO4 = 28·0, &c.

[6] If W be the amount of heat contained in a mass m of a substance at a temperature t, and dW the amount expended in heating it from t to t + dt, then the specific heat Q = dW(m × dt). The specific heat not only varies with the composition and complexity of the molecules of a substance, but also with the temperature, pressure, and physical state of a substance. Even for gases the variation of Q with t is to be observed. Thus it is seen from the experiments of Regnault and Wiedemann that the specific heat of carbonic anhydride at 0° = 0·19, at 100° = 0·22, and at 200° = 0·24. But the variation of the specific heat of permanent gases with the temperature is, as far as we know, very inconsiderable. According to Mallard and Le Chatelier it is = 0·0006 / M per 1°, where M is the molecular weight (for instance, for O2, M = 32). Therefore the specific heat of those permanent gases which contain two atoms in the molecule (H2, O2, N2, CO, and NO) may be, as is shown by experiment, taken as not varying with the temperature. The constancy of the specific heat of perfect gases forms one of the fundamental propositions of the whole theory of heat and on it depends the determination of temperatures by means of gas-thermometers containing hydrogen, nitrogen, or air. Le Chatelier (1887), on the basis of existing determinations, concludes that the molecular heat—that is, the product MQ—of all gases varies in proportion to the temperature, and tends to become equal (= 6·8) at the temperature of absolute zero (that is, at -273°); and therefore MQ = 6·8 + a(273 + t), where a is a constant quantity which increases with the complexity of the gaseous molecule and Q is the specific heat of the gas under a constant pressure. For permanent gases a almost = 0, and therefore MQ = 6·8—that is, the atomic heat (if the molecule contains two atoms) = 3·4, as it is in fact (Chapter IX., Note [17 bis]. As regards liquids (as well as the vapours formed by them), the specific heat always rises with the temperature. Thus for benzene it equals 0·38 + 0·0014t. R. Schiff (1887) showed that the variation of the specific heat of many organic liquids is proportional to the change of temperature (as in the case of gases, according to Le Chatelier), and reduced these variations into dependence with their composition and absolute boiling point. It is very probable that the theory of liquids will make use of these simple relations which recall the simplicity of the variation of the specific gravity (Chapter II., Note [34]), cohesion, and other properties of liquids with the temperature. They are all expressed by the linear function of the temperature, a + bt, with the same degree of proximity as the property of gases is expressed by the equation pv = Rt.

As regards the relation between the specific heats of liquids (or of solids) and of their vapours, the specific heat of the vapour (and also of the solid) is always less than that of the liquid. For example, benzene vapour 0·22, liquid 0·38; chloroform vapour 0·13, liquid 0·23; steam 0·475, liquid water 1·0. But the complexity of the relations existing in specific heat is seen from the fact that the specific heat of ice = 0·502 is less than that of liquid water. According to Regnault, in the case of bromine the specific heat of the vapour = 0·055 at (150°), of the liquid = 0·107 (at 30°), and of solid bromine = 0·084 (at -15°). The specific heat of solid benzoic acid (according to experiment and calculation, Hess, 1888) between 0° and 100° is 0·31, and of liquid benzoic acid 0·50. One of the problems of the present day is the explanation of those complex relations which exist between the composition and such properties as specific heat, latent heat, expansion by heat, compression, internal friction, cohesion, and so forth. They can only be connected by a complete theory of liquids, which may now soon be expected, more especially as many sides of the subject have already been partially explained.

[7] According to the above reasons the quantity of heat, Q, required to raise the temperature of one part by weight of a substance by one degree may be expressed by the sum Q = K + B + D, where K is the heat actually expended in heating the substance, or what is termed the absolute specific heat, B the amount of heat expended in the internal work accomplished with the rise of temperature, and D the amount of heat expended in external work. In the case of gases the last quantity may be easily determined, knowing their coefficient of expansion, which is approximately = 0·00368. By applying to this case the same argument given at the end of Note [11], Chapter I., we find that one cubic metre of a gas heated 1° produces an external work of 10333 × 0·00368, or 38·02 kilogrammetres, on which 38·02/424 or 0·0897 heat units are expended. This is the heat expended for the external work produced by one cubic metre of a gas, but the specific heat refers to units of weight, and therefore it is necessary in order to know D to reduce the above quantity to a unit of weight. One cubic metre of hydrogen at 0° and 760 mm. pressure weighs 0·0896 kilo, a gas of molecular weight M has a density M/2, consequently a cubic metre weighs (at 0° and 760 mm.) 0·0448M kilo, and therefore 1 kilogram of the gas occupies a volume 1/0·0448M cubic metres, and hence the external work D in the heating of 1 kilo of the given gas through 1° = 0·0896/0·0448M, or D = 2/M.

Taking the magnitude of the internal work B for gases as negligible if permanent gases are taken, and therefore supposing B = 0, we find the specific heat of gases at a constant pressure Q = K + 2 M, where K is the specific heat at a constant volume, or the true specific heat, and M the molecular weight. Hence K = Q - 2/M. The magnitude of the specific heat Q is given by direct experiment. According to Regnault's experiments, for oxygen it = 0·2175, for hydrogen 3·405, for nitrogen 0·2438; the molecular weights of these gases are 32, 2, and 28, and therefore for oxygen K = 0·2175 - 0·0625 = 0·1550, for hydrogen K = 3·4050 - 1·000 = 2·4050, and for nitrogen K = 0·2438 - 0·0714 = 0·1724. These true specific heats of elements are in inverse proportion to their atomic weights—that is, their product by the atomic weight is a constant quantity. In fact, for oxygen this product = 0·155 × 16 = 2·48, for hydrogen 2·40, for nitrogen 0·7724 × 14 = 2·414, and therefore if A stand for the atomic weight we obtain the expression K × A = a constant, which may be taken as 2·45. This is the true expression of Dulong and Petit's law, because K is the true specific heat and A the weight of the atom. It should be remarked, moreover, that the product of the observed specific heat Q into A is also a constant quantity (for oxygen = 3·48, for hydrogen = 3·40), because the external work D is also inversely proportional to the atomic weight.