﻿608 Prof. 0. W. Richardson : Some Applications 



§ 5. Ihe Electron Theory of Contact Electro motwe Force 

 and, Thermoelectricity . 



In the former paper the writer has shown how the relation 

 between the contact electromotive forces and the various 

 thermo-electric constants of substances, on the one hand, and 

 the number and potential energy of the internal free electrons, 

 on the other, may be determined. The development there 

 given involves the implicit assumption that the internal 

 electrons in any one substance may be treated as if they all 

 possessed the same potential energy. It also involved the 

 assumption that the heat liberated where electrons enter a 

 metal is equal to the difference in their potential energy 

 before and after condensation. The results of § 2 of 

 the present paper show that so much simplicity is not 

 warranted by the facts; though it may well be that the 

 supposedly characteristic constants n and iv of the previous 

 discussion may be replaced by suitably formed average 

 values. A more complete theory of these effects will now 

 be given. 



The state of the electrons at any point of any system is 

 defined by the average number v per unit volume at that 

 point at any instant, by the average potential energy co of 

 the electrons at that point, and by the temperature 6, which 

 is proportional to the average kinetic energy of the electrons. 

 We may imagine the interior of any conductor to be mapped 

 out by means of the level surfaces of co. The distribution of 

 these surfaces will be definite and characteristic for each 

 conductor. At points just outside any conductor co takes 

 the constant value co . In the state of equilibrium the 

 values of co are different and characteristic for different 

 conductors. 



This additional complexity will not affect the result which 

 we have imported from the dynamical theory of gases, 

 namely, that at any point in a system in equilibrium at 

 temperature 6, 



dn = vdT = Ke- b) i* 9 dT, (28) 



where dn is the number of electrons which participate in 

 thermal phenomena in the element of volume dr, and K is 

 constant throughout the system, being a function of 6 only. 



We shall now consider the formulae which determine the 

 equilibrium between the external free electrons and the 

 internal electrons which can become free. Confining our 

 attention to a single conductor, consider first the special case 

 in which there are a finite number p of finite internal regions 



