MONATOMIC GAS: DIFFUSION, VISCOSITY, AND THERMAL CONDUCTION. 163 
By the equation of continuity (12'03), the last equation is equivalent to 
(12‘14) 
1 DT 1 Dy 0 _ 
T ThT ~~ 3 ^ E>T ~ 
or to 
(i2i5) = 0 = °- 
Since we are neglecting diffusion, m 0 (or A x m x + \ 2 m 2 ) is invariable, and (12'15) 
expresses the adiabatic law of expansion of a monatomic gas as a given element is 
followed throughout its motion, viz., 
(12-16) 
T = Up*. 
Thus (12T2) may be regarded as giving the correction to this law owing to 
diffusion, viscosity, and conduction. The left-hand side represents the net rate of 
increase of energy of molecular agitation, being the increase corresponding to the 
rising temperature after allowing for the change of heat energy produced by adiabatic 
expansion or compression. The right-hand side indicates that this is due to the 
following causes :— 
(a) We are considering the fluid contained in a volume element which moves with 
the mean motion of the fluid ; but in addition to this motion there is one of inter¬ 
diffusion within the element, there being a stream of ± (v 0 u' 0 , v 0 v' 0 , v 0 v/ 0 ) molecules 
per unit time per unit volume in opposite directions. The ^-components of the 
effective forces on each of these molecules are X, -m, 
D t 
and X 2 — m 2 respectively, 
the second term depending 
on 
Dw 0 
Dt 
being present, since we are throughout concerned 
with the energy relative to the mean motion (u 0 , v 0 , w 0 ). The work done on the 
fluid by these forces is expressed, in thermal units, by the first right-hand term of 
( 12 - 12 ). 
(^) Owing to the motion of diffusion there will be an increase in the number of 
molecules m x and an equal decrease in the number of molecules m 2 , at the rate of 
dv' 0 div' 0 
dy dz 
motion is the same for molecules m 1 and m 2 , so that this change affects only the 
energy of mean motion, which is \m x c 2 for a molecule m x , and %m 2 c 0 2 for a molecule 
m 2 . The increase of energy due to this cause is represented by the second right-hand 
term of (12T2). 
(y) The third term gives the heat generated by the viscous forces acting in the 
gas, being, in fact, twice the “ dissipation function ’’ of the viscous motion.* 
'j per unit time per unit volume. The mean energy of peculiar 
( du'o 
\ dx 
+ 
VOL. CCXVII.—A. 
* Cf. Lamb, ‘ Hydrodynamics,’ p. 518. 
2 A 
