Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/483

Consider now the case in which there is conduction of heat without conduction of electricity. The flux of energy will in this case be given by the equation

$W=-\kappa {\frac {dT}{dx}}$,

where κ a denotes the thermal conductivity of the metal expressed in suitable units; or

$W=-\kappa .{\frac {3ma}{2q}}{\frac {da}{dx}}$.

If it be assumed that the conduction of heat in metals is effected by motion of the electrons, this expression may be compared with the preceding; thus we have

$\kappa {\tfrac {8}{9}}\pi ^{-{\frac {1}{2}}}\log N$;

and comparing this with the formula already found for the electric conductivity, we have

${\frac {\kappa }{\gamma }}={\tfrac {8}{9}}T\left({\frac {q}{e}}\right)^{2}$,

an equation which shows that the ratio of the thermal to the electric conductivity is of the form T × a constant which is the same for all metals. This result, accords with the law of Wiedemann and Franz,

Moreover, the value of q is known from the kinetic theory of gases; and the value of e has been determined by J.J. Thomson^[1] and his followers; substituting these values in the formula for κ/γ, a fair agreement is obtained with the values of κ/γ determined experimentally.

It was remarked by J. J. Thomson that if, as is postulated in the above theory, a metal contains a great number of free electrons in temperature equilibrium with the atoms, the specific heat of the metal must depend largely on the energy required in order to raise the temperature of the electrons. Thomson considered that the observed specific heats of metals are smaller than is compatible with the theory, and was thus

1. Cf. p. 407.