the Quantum Theory

Law of Dulong and Petit.

It is well known that the product of specific heat and atomic weight is very nearly the same, about 6 in the usual units, for many solid elements, e.g., copper, iron, zinc, etc. This is the law of Dulong and Petit. There are many exceptions to the law, notably among elements of low atomic weight, such as boron and carbon (graphite, diamond). (See HEAT.) It should be pointed out that the specific heat of a solid is usually measured under the condition of constant pressure and that we are concerned here with the specific heat measured under the condition of constant volume. The latter is nearly always less than the former, and never greater. Let us examine the law from the point of view of the classical theory. We should expect the heat energy, expressed in mechanical units (ergs), in a gram atom of a solid element to be where N is the number of atoms in a gram atom (or the number of molecules in a gram molecule), since the energy of a single atom in a solid element may be regarded as associated with three independent vibrations in mutually perpendicular directions and must therefore be equal to 3k T according to the classical theory. The atomic heat at constant volume, i.e., the amount of heat required to raise the temperature of a gram atom one degree will be therefore To express it in the usual units, we divide by the mechanical equivalent of heat, i.e., by 4.2X to' ergs per calorie, and so we get where R =8.3isX i ergs per degree is the gas constant for one gram molecule of a gas. It is easily verified that the quantity (Is) is very nearly equal to 6. The elements which deviate from the law of Dulong and Petit at ordinary temperatures conform to it more and more closely as the temperature is raised, and con versely, those which agree with the law at ordinary temperatures deviate from it more and more the lower the temperature at which the measurements are taken. We have here an illustration of the competence of the classical theory to deal with phenomena on a sufficiently macroscopic scale.

Planck's theory was first applied to the problem of atomic heats or specific heats by Einstein, and later and more completely by Debye. Debye assumed that the heat energy of a solid ele ment is the energy of elastic vibrations of all frequencies from very low values to a certain upper limit which was fixed by the con sideration that, as we have already seen, the theory must asymp totically approach the classical theory when the energy involved is very great (or the temperature very high). This upper limit is low enough (or the corresponding lengths of the elastic waves big enough) to justify us in regarding the material as uniform and to ignore in the calculation the fact that it is made of atoms and therefore granular in structure. The problem is closely analogous to that of black body radiation. The difference lies mainly in the fact that here we are dealing with the vibrations of a material and not with aether vibrations only. The number of vibrations in the frequency range from P to v--Fdy will now be per unit volume, where vt is the velocity of transverse waves in the solid and vi that of longitudinal waves. This formula is

based on the fact that we may have both transverse and longi tudinal waves in a solid material and it should be compared with formula (8) above. Each of these vibrations has the average energy expressed by formula (to), so that we get for the total energy in the unit volume the expression the upper limit being fixed in accordance with the principle re ferred to above. Strictly speaking we ought to take into account the aether vibrations as well as those of the material; but the additional energy is negligible by comparison with that given by (i7). The reason for this is that the velocities vt and vi are very small compared with C the velocity of aether waves. Since formula (17) must reduce to the corresponding classical one at high temperatures, we must fix v,, so that the total number of vibrations is equal to 3N where N is the number of molecules in the unit volume, or When we substitute this result in (i7) we get for the energy in the unit volume of the material, If we agree that N in formula (19) shall represent the number of atoms in a gram atom of the element, instead of the number in the unit volume, then (19) will give the heat energy in a gram atom at the temperature T.

Since can be calculated from the velocities of transverse and longitudinal waves in the solid element (equation 18), it follows that the heat energy in a gram molecule and therefore also the atomic heat (the heat required to raise the temperature of a gram atom by unity) can be found from a knowledge of the elastic moduli, Young's modulus and the modulus of rigidity of the material.

If we use the letter E for the energy of a gram atom, and ab breviate by representing hv AT by 0, which amounts to replacing by formula (ig) will take the shape ture. We may say then finally, that the atomic heat of a solid element (measured in terms of the centigrade degree, or in terms of the same unit for all elements) is the same function of the temperature (now measured in terms of the characteristic tem perature of the particular element as a unit) for all solid elements.

Reference to fig. 3 will show how well this theoretical result is supported by the facts. The atomic heats are given in terms of 3N K, i.e., to get the actual atomic heats the ordinates must be multiplied by 5.96. Debye's theory has been improved and ex tended by Born and Karman, who have taken the crystalline structure of the solid into account.

The Constants of Planck and Boltzmann.

The universal constants h and k which have appeared in the formulae of the theories of full radiation and the specific heats of solid elements are appropriately named after Planck and Boltzmann. The data provided by investigators of full radiation enable us to assign to them the values, These values do not differ materially from those originally com puted by Planck.