• 274 
IONISATION, DIFFUSION, ROTATION 
and the density of the electrons will be 
\s x Z x + s 2 Z 2 ) e (”*+**)l RT . 
By differentiation we find the density gradients at <f>, ifj = 0 to be 
(192 - 1} ’ 
and the gradient of the total charge density is 
eZ x s x ( . dcf> „ dxjj\ eZ 2 s 2 / . d(f> „ <hjj\ 
AtIT r 1 dz ~ eZl dz) + ~W r 2 dz ^ dz, 
e (Z x s x + Z 2 s 2 ) „ dift\ 
RT V dz + 6 dz) * 
Since the resultant charge is insignificant this must vanish; the condition 
gives dip __ Z 1 s 1 (A x — m) + Z 2 s 2 (A 2 — m) dj> 
dz Z x s x (Z x + 1) + Z 2 s 2 [Z 2 +1) dz 
\Z x s x (Z x + l) fx x + Z 2 s 2 (Z 2 + 1) | ^ noft. 9 .) 
1 Z& (Z x + 1) + Z 2 s 2 (Z t + 1) J dz ( 
where ¡x x , /x 2 are the average molecular weights for the two kinds of ion 
„ = A i + _ Z ' m „ - A * + Z * m - (192-25). 
^ Z x + 1 ’ ^ Z 2 + 1 V ' 
We can write (192-2) in the form 
( 192 ' 3) > 
where /x 0 is a mean between ¡x x and ¡x 2 , each ion being weighted proportion 
ately to Z (Z + 1). This is not quite the same weighting as in the average 
molecular weight of the material ¡x, the weighting being then proportional 
to (Z + 1). Evidently ¡x 0 > ¡x. 
Hence by (192-1) and (192-3) 
£-J^ (192-35), 
or dJ W A = m« z ' + l) ^- z ^ ) t (192 ' 4) 
by (192-25). 
If s is the number of free electrons, we find by setting A x = m, Z x = — 1 
in (192 ’ 35) d (l og a) = d± (ig2 . 6) 
cl/7s R r l ^ 6 ^/ 2 ' 
Hence by (192-4) and (192-5) 
i№)-]ErK + 1 ) 
It can easily be shown that this formula applies, however many sorts of 
ions may be present, ¡x 0 being the properly weighted mean for them all.