APPENDIX AFROM MOBILITIES AND DIFFUSION COEFFICIENTS
i038Fig. 38
Fig. 38
Fig. 38
If we assume that gaseous ions, which are merely charged molecules or clusters of molecules, act exactly like the uncharged molecules about them, they will tend to diffuse just as other molecules do and will exert a partial gas pressure of exactly the same amount as would an equal number of molecules of any gas. Imagine then the lower part of the vessel ofFig. 38to be filled with gas through which ions are distributed and imagine that these ions are slowly diffusing upward. Letbe the ionic concentration, i.e., the number of ions per cubic centimeter at any distancefrom the bottom of the vessel. Then the numberof ions which pass per second through 1 sq. cm. taken perpendicular toat a distancefrom the bottom must be directly proportional to the concentration gradientand the factor of proportionality in a given gas is by definition the diffusion coefficientof the ions through this gas, i.e.,
But sinceis also equal to the product of the average velocitywith which the ions are streaming upward atby the numberof ions per cubic centimeter at,i.e., since,we have from equation (42)The force which is acting on these-ions to cause this upward motion is the difference in the partial pressure of the ions at the top and bottom of a centimeter cube at the point.It is, therefore, equal todynes, and the ratio between the force acting and the velocity produced by it is
Now this ratio must be independent of the particular type of force which is causing the motion. Imagine then the same-ions set in motion, not by the process of diffusion, but by an electric field of strength.The total force acting on the-ions would then be,and if we takeas the velocity produced, then the ratio between the force acting and the velocity produced will now be.By virtue then of the fact that this ratio is constant, whatever kind of force it be which is causing the motion, we haveNow ifdenote the velocity in unit field, a quantity which is technically called the “ionic mobility,”.Again since the partial pressureis proportional to,i.e., since,it follows that.Hence equation (43) reduces toor
But if we assume that, so far as all pressure relations are concerned, the ions act like uncharged molecules (this was perhaps an uncertain assumption at the time, though it has since been shown to be correct), we havein whichis the number of molecules per cubic centimeter in the air andis the pressure produced by them, i.e.,is atmospheric pressure. We have then from equation (44)