57* CONDUCTION OF HEAT 465 



57*. Conductivity 



The integration, so far as it can be carried out, gives that the 

 energy which crosses unit area in unit time in the direction of 

 increasing x has the value 



u 



1 (2-2107 k~ l - 



The first of these terms, which is constant, may be put into the 

 more intelligible form 



In the opposite direction, that of decreasing x, there passes 

 a quantity of kinetic energy, the expression for which differs 

 from that just given only by the sign of the second term. The 

 flow of heat in the positive direction resulting from both 

 transfers, which, in accordance with Fourier's theory, is put 

 proportional to the conductivity f and to the differential coefficient 

 of the temperature $ with negative sign, is the difference between 

 the two magnitudes, or 



_ F = - fyrmN^MFdu 6 B -^2-2107 fc-'-wwV*"" 1 . 



Now, as we found before in 19*, p. 388, 



km = 47T- 1 Q- 2 = 47T- 1 Q " VC 1 + $) 



where fi is the mean molecular speed at the temperature 9 = 0* C. 

 and a is the coefficient of expansion ; consequently 



dk yi -10 -2 n i a\-2^ akm d 



m-=- = 47r 1 Q0 2 a(l + a3) 2 -=- = j -=-. 



dx dx 1 -f a$dx 



We therefore obtain for the conductivity 

 f = $K-*N(km)*a(l + a-9)- 1 f^ 6 B-^ww 8 - 2-2107 &->- *""". 



To reduce this expression to thermal units for energy is ex- 

 pressed in it in mechanical units we note that the kinetic energy 

 of unit mass is 



while on the other hand the heat at temperature 5 C. (or absolute 

 temperature $ + a" 1 ), which is equivalent to it, is 



c(* + a' 1 ), 



H H 



