THE ELECTRIC AND LUMINIFEROUS MEDIUM. 
279 
solution across a layer in which the concentration varies, w^hen the steady state is 
attained both ions must diftuse together at the same rate notwithstanding their 
different mobilities u and v, measured by Kohlrausch as the values of their mean 
velocities due to unit electric force. Now the mean steady velocity of migration of a 
single ion is equal to this mobility divided by its electric charge e and multiplied by 
the force which causes its motion : this force consists of an electric part — edYjdx^ 
where V is the electric potential set up during the transition to the steady state of 
diffusion, and of an osmotic part, do determine the latter, observe that when a 
solution is separated from the pure solvent by a permeable osmotic partition, the 
solvent is restrained from passing across only by an osmotic pressure acting against 
it : this means that to maintain the steady state without diffusion the osmotic 
partition must exert more pressure by the amount iJ on the solution than on the 
puie solvent. If we consider a layer of the actual solution, of cross-section unity and 
thickness Sa^, there would thus have to be a bodily force dpjdx. Sa; exerted on it if 
the diffusion of its ions were prevented : therefore — dpjdx. Bx is the aggregate of 
the forces acting on the contained ions and producing diffusion, that arise from the 
gradient of concentration. If n be the number of ions per unit volume, the mean 
force per ion is thus — n~^dpldx : this is not a mere hypothesis founded on a vague 
analogy of osmotic pressure with ordinary hydrostatic pressure, but gives a precise 
measure of an actual force on a constituent of the medium. The number d^jdt of 
single ions of either kind that is driven across unit area of a geometrical interface in 
a solution of varying concentration by these forces is thus given, after Nernst, by 
dN 
dt 
dY 
— nu 
cLc 
11 u di) 
c n dx ’ 
also 
dY 
nv , 
dx 
11 V 
c n 
dp ^ 
dx 
If D denote the coefficient of diffusion of the solution, d'^jdt = — D dnidx ; and 
by the gaseous law which applies to osmotic pressure of very dilute solutions 
p ~ neliT. Hence immediately 
X^T ^ ^ 
"10 V ’ dx V -{• u nc dd: ’ 
so that the integrated potential difference across the diffusion layer is 
UT {y — u)l{v -f u) . log Pz/'Pi- It follows that when steady diffusion is allowed to go 
on, the back electromotive force at the junction of the fluids is thereby reduced in 
the ratio of the difference to the sum of the ionic mobilities. The agreement with 
experiment of these expressions for Vo — V^, and for the ordinary diffusion coefficient 
D of a solution as thus determined electrically, constitutes two distinct tests of the 
general validity of this diftusion scheme, and of the hypothesis of independent 
mobility of the ions of which it is a corollary. 
In the case of a solution only partially dissociated, like that of acetic acid referred to 
in § 58, piovided the time of association of two paired ions is on the average large 
compared with the time of relaxation of the system, these expressions for D and 
