Molecules of a Gas in a Field of Force. 639 



§ 2. The Theory of Thermoelectric Phenomena. 

 1. The Peltier Effect. 

 To find the heat liberated when N electrons cross the 

 junction between two metals, consider two planes parallel 

 to the surface of separation, one on each side of it. The 

 planes are to be taken so far from the separating surface 

 that in each case the values of p, V, U, and w are those 

 characteristic, at the given temperature^ of the metal in 

 which each surface lies. Then the total energy of the 

 N electrons crossing one of the surfaces is the value of 

 pV + U + Nw at that surface, and the energy liberated at 

 the junction by the passage of the N electrons will be the 

 difference of the value of the sum above at the two surfaces. 

 Since N electrons transport Ne units of electricity, the 

 Peltier coefficient P 12 in ergs per unit quantity of electricity 

 will be given by 



NeP 12 = Pl V 1 -p 2 Y 2 + Ui - U 2 + Nfa - w 2 ) 



= |(U 1 -U 2 )-JV^ (18) 





~ 12UT li?J, /T?J, 0=1 i ,og i 



If Xi and x 2 are both small these reduce to 



-p RT , p 2 RT . n 2 

 P12 = — log — = — log -, ... (20) 

 * Pi € ? 'i 



the results given by the classical dynamics*. When x\ 

 and # 2 are both large, which will be the case with metals 

 at very low temperatures, 



P H -IT a H(Vi»-VflT* (21) 



This equation has also been deduced by Keesom j-. 



2. The Thomson Effect. 

 Similar considerations to those used in dealing with the 

 Peltier effect can be employed in calculating the Thomson 

 effect. The difference lies in the fact that the differences in 

 p, U, V, and w are now due to a difference of temperature 

 in a single material, instead of being conditioned by different 

 materials at the same temperature. Under the new con- 

 ditions one may suppose that the equilibrium of the electrons 

 is established when the gradient of the potential energy 



* O. W. Richardson, Phil. Mag. vol. xxiii. p. 269 (1912). 

 f Luc. cit. 



