QUANTUM THEORY 
51 
/ 
srred to 
present 
theory. 
(38-4), 
ntaneous 
atom at 
ro in the 
.(38-5). 
ed by the 
called by 
consisting 
ssion of a 
o increase 
> rays is so 
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absorbing 
t modified 
3 then not 
depend on 
which as- 
way affect 
istribution 
»nee for all 
Listribution 
formula is 
apart as to 
is in which 
an electron 
emission of 
radiation. If, as usual, we measure the energy of the system from a zero 
level with the electron just free of the atom and without kinetic energy, 
Xi will denote the energy of the electron in its orbit within the atom in 
the first state (a negative quantity) and y 2 will denote the positive kinetic 
energy of the free electron. If the free electron is in a region of zero 
potential and has velocity ( u , v, w) 
X 2 = ( u 2 + v 2 + w 2 ) 
(39-1). 
By Law I we may particularise the description of the states as minutely 
as we please. We shall take a system in state 2 to consist of an atom which 
has lost an electron, together with a free electron with velocity in the 
range u, v, w to u + du, v + dv, w + dw in an element of volume dxdydz 
at zero potential located in a specified manner with respect to the atom. 
Then the energy of a system in state 2 is given by (39-1). 
By (38-3) 
n 2 q ttr ml« + + + *>l*BT 
n. 
9i e 
-xi IRT 
If n' is the number of ionised atoms in the system, and dN is the average 
number* of free electrons in a range dxdydzdudvdw 
so that 
n 2 = n'dN, 
dN = (~, eX — — yq 2 e- m ( u2 + v * + w2 V 2RT (39-2). 
The factor in the bracket remains constant for different values of 
u, v, w, so that we obtain Maxwell’s law of velocities 
dN oc e~ m ^ u2 + v2 + w ' 2 d 2RT dudvdw dxdydz (39 - 3), 
except that we have not yet shown that the weight factor q 2 for the range 
of states considered is proportional to dudvdwdxdydz. It might well have 
been a function of u, v, w. 
Meanwhile the comparison of (39-2) with (39-3) shows that the con 
stant R hitherto unidentified is the same constant (Boltzmann’s constant) 
which occurs in the theory of gases. Assuming (39-3) the average value 
of u 2 is 
| u 2 e~ mu2 l 2RT du ~ I e~ mu2 l 2RT du 
-co J —00 
= RT/m. 
Hence the average kinetic energy \m ( u 2 + v 2 + w 2 ) is equal to 
f RT (39-4). 
Thus R is identified as | of the average kinetic energy of a molecule at 1° 
absolute. 
* The number dN is an infinitesimal fraction. When an infinitesimal range of a 
continuous distribution of states is considered, so that there is no longer a large 
number of systems in state 2, it becomes necessary to consider time-averages (or 
alternatively, probabilities) in order to smooth out the accidental fluctuations. 
4-2