CONTEMPORARY ADVANCES IN PHYSICS 351 



Ve = - MH cos e. (2) 



I propose now to show that Langevin's theory of magnetization is 

 obtained by applying to the potential-energy term Ve the same mode 

 of reasoning as is customarily and familiarly applied to the kinetic- 

 energy terms Ku • ■ • Kt. 



It is well known that the average kinetic energy of translation, 

 the average of the sum of the terms Ku and Kv and Kw, taken over 

 all the molecules of a gas of absolute temperature T, is proportional 

 to T; it is, in fact, given by the equation 



Ku+K, + K^ =\kT, (3) 



in which k stands for the ratio of the gas-constant R to the Loschmidt 

 number No (number of molecules per gramme-molecule).* The 

 average of each of these three terms separately is equal to ^kT; and 

 this result was generalized by Maxwell and by Boltzmann to the 

 three rotational terms in the expression for K, so that 



Ku = K^ = Kw = Kr = Ka = Kt = 2^T. (4) 



We go one step further in the analysis of the motion of the molecules. 

 Consider the distribution-function for any one of these six variables, 

 u for instance; it is given by Maxwell's formula: 



dN = NCu exp (- ^muykT)du, (5) 



in which dN stands for the number of molecules (among the N mole- 

 cules occupying unit volume) for which the velocity-components 

 along the x-axis lie between the values u and u + du. The constant 

 Cu is so adjusted that the integral of dN over the entire range of 

 values of u shall be equal to A'^; on being computed it turns out to be 

 ■ylm/lirkT. The quantity |wm^ is the one hitherto designated as Ku. 

 For the distribution-function with respect to u, which is the coefficient 

 of du in (5), and may be denoted by F{u), we therefore have: 



Fiu) = N^^lm/2^^kT^ exp. (- KJkT) (6) 



* The primitive way of deriving (3), reproduced in all elementary texts, is as 

 follows: Imagine a cubical vessel one cm. along each edge containing N molecules; 

 suppose that N/3 molecules are moving in lines parallel to each edge, with uniform 

 speed v; each face is then struck with Nv/6 impacts per second, and in each impact 

 an amount of momentum 2mv is communicated to the face, so that the average 

 pressure upon the surface is ^ = Nmv'/3. According to the well-known gas-law, 

 p = pRT/M (p standing for the density, M for the molecular weight of the gas); 

 hence Nmv~/3 = pRT/M, and recalling that p = Nm and that M/m = No and that 

 ^mv' is the kinetic energy X of a molecule, we have K = 3RT/2No = 3kT/2. The 

 same result is reached by more sophisticated methods of averaging. 



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