Note 7 — Chapter XIV · THE VALENCY AND SPECIFIC HEAT OF THE METALS. MAGNESIUM. CALCIUM, STRONTIUM, BARIUM, AND BERYLLIUM

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.

In the case of gases we distinguish the specific heat at a constant pressure c′ (we designated this quantity above by Q), and at a constant volume c. It is evident that the relation between the two specific heats, k, judging from the above, is the ratio of Q to K, or equal to the ratio of 2·45n + 2 to 2·45n. When n = 1 this ratio k = 1·8; when n = 2, k = 1·4, when n = 3, k = 1·3, and with an exceedingly large number n, of atoms in the molecule, k = 1. That is, the ratio between the specific heats decreases from 1·8 to 1·0 as the number of atoms, n, contained in the molecule increases. This deduction is verified to a certain extent by direct experiment. For such gases as hydrogen, oxygen, nitrogen, carbonic oxide, air, and others in which n = 2, the magnitude of k is determined by methods described in works on physics (for example, by the change of temperature with an alteration of pressure, by the velocity of sound, &c.) and is found in reality to be nearly 1·4, and for such gases as carbonic anhydride, nitric dioxide, and others it is nearly 1·3. Kundt and Warburg (1875), by means of the approximate method mentioned in Note 29, Chapter VII., determined k for mercury vapour when n = 1, and found it to be = 1·67—that is, a larger quantity than for air, as would be expected from the above.

It may be admitted that the true atomic heat of gases = 2·43, only under the condition that they are distant from a liquid state, and do not undergo a chemical change when heated—that is, when no internal work is produced in them (B = 0). Therefore this work may to a certain extent be judged by the observed specific heat. Thus, for instance, for chlorine (Q = 0·12, Regnault; k = 1·33, according to Straker and Martin, and therefore K = 0·09, MK = 6·4), the atomic heat (3·2) is much greater than for other gases containing two atoms in a molecule, and it must be assumed, therefore, that when it is heated some great internal work is accomplished.

In order to generalise the facts concerning the specific heat of gases and solids, it appears to me possible to accept the following general proposition: the atomic heat (that is, AQ or QM/n, where M is the molecular weight and n the number of molecules) is smaller (in solids it attains its highest value 6·8 and in gases 3·4), the more complex the molecule (i.e. the greater the number (n) of atoms forming it) and so much smaller, up to a certain point (in similar physical states) the smaller the mean atomic weight M/n.