14 THE DIFFUSION OF GASES THROUGH 



Let m a and m h be the masses of air and of hydrogen transpiring per second, 

 into and out of the imprisoned volume, v. Hence, if a is the area of the 

 water level at v, 



-m=- (m h - m a ) = a (k h dpjdl - k a dpjdl) (9) 



is the variation of mass per second, if k h and k a are the coefficients for hydro- 

 gen and air, respectively. But 



dp a B-T:-p a p h -h" Pw g dp h 



(10) 



dl h" + 2h'" ti' + 2h'" dl h" + 2h'" 



where p u is the density of water and g the acceleration of gravity. Hence 



- in = a{k h - k a ) h , IJr h 2]r + ok a 1+ P 2 l>» /h » ( JI ) 



Therefore, when p h vanishes in the lapse of time, the diffusion of air alone 

 is in question and m will be constant. The datum actually measured, 

 however, is m' = mp/p , where p is the density of the imprisoned mixed gas 

 and p the density of the initial gas, hydrogen, all at the supposedly fixed 

 mean temperature and pressure assumed. Thus it will be necessary to 

 refer equation (11) to diffusion by volume and write, k being the coefficient, 



-v = a(K h -K a ) h „ +2h „, +™a h „+ 2h „, (12) 



In equation (12) if p h were to vanish or become negligible, the transpira- 

 tion of air would alone be in question and v would be constant, supposing 

 that K h and K a are really constant, or that the phenomenon is homogeneous. 

 For a diffusion and a transpiration phenomenon may occur side by side, 

 subject to different laws ; the first depending upon the degree of mixture of 

 the gases and rapidly vanishing as the mixture is more nearly complete, 

 the second depending upon the head h" . It does not follow, however, in 

 view of §15, that p n will vanish first; for when p n — h"p w g, p a = B — ir, and 

 the influx of air must cease, because the air gradient has vanished. Hence 

 thereafter hydrogen and air will both diffuse out of the swimmer; for any 

 further diminution of p h means an increase of p a which is now greater than 

 B — 7T. A mixture of gases thus diffuses which grows continually richer in 

 air and poorer in hydrogen, until it is nearly pure air. 



Equations (1 1) and (12) are not integrable, since mis independent of m and 

 v independent of v. 



17. Continued. Flotation. — Admitting equation (12), the endeavor 

 must now be made to express p h or its equivalent in terms of quantities 

 belonging to the mixture, or to express m in terms of p, the density of the 

 imprisoned gases. 



Let P be the artificial barometric pressure on flotation, and let p' a and 

 p h be the corresponding pressures of the dry air and hydrogen imprisoned. 



