The Diffusion of Ions into Gases. 



193 



In order to obtain the coefficient of diffusion when the reduction in 

 the conductivity has been found experimentally, the following problem 

 presents itself : — 



If a small quantity of a gas, A, is mixed with another gas, B, and 

 the mixture passed along a tube, the sides of which completely absorb 

 A, to find what quantity of A emerges from the tube with B. 



It will be immediately seen that if the gases diffuse rapidly into 

 each other, a large proportion of the molecules of the gas A will come 

 into contact with the surface of the tube and will there be absorbed. 

 If on the other hand the rate of interdiffusion is very small, the 

 molecules of A will travel down the tube in straight lines parallel to 

 the axis of the tube, and practically none of them will come into con- 

 tact with the surface. 



The complete solution of the above problem may be obtained from 

 the following equations : — 



1 dp „ 



• = ~Tx +nXe ' 



1 dp TT 



K r dy 



1 dp „ 1 ,, r 



-pw = - -^ + n/je + -j)\\ , 



and the equation of continuity (pu) + ^- (pv) + ~ (pw) =0; where 



n is the number of ions per cubic centimetre ; p their partial pressure ; 

 e the charge on each ion ; X, Y, and Z the electric forces at any point ; 

 u, v, and w the velocities of the ions ; W the velocity of the gas B 

 through the tube ; k the coefficient of diffusion of the ions into the gas B. 



The partial differential coefficient with respect to the time, is omitted 

 from the equation of continuity, as we need only consider the steady 

 state. 



The term dp/dz may be omitted from the third equation, as it is small 

 compared with the other terms. 

 2Y 



W = — 2 (a 2 - r 2 ), where V is the mean velocity of the gas B, 



defined by the condition 7ra 2 V/ = total volume of gas crossing any section 

 in a time t, a the radius of the tube, and r the perpendicular distance 

 of any point from the axis. 



The forces X, Y, Z vanish since the electrification is too small to con- 

 tribute appreciably to the motion of the ions. 



The boundary conditions are : — 



p = when r = a, 



p = constant when z — 0. 



