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The electron, its isolation and measurement and the determination of some of its properties

Chapter 32: APPENDIX A FROM MOBILITIES AND DIFFUSION COEFFICIENTS
By Robert Andrews Millikan
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About This Book

A systematic presentation of experimental and theoretical work that establishes the discrete, atomic character of electric charge and explains techniques used to isolate and measure the elementary charge. It traces historical ideas about electricity, extends electrolytic laws to gas conduction, analyzes ionization by penetrating radiation, examines Brownian motion in gases, and considers whether the electron is divisible. Later chapters discuss atomic structure and the behavior of radiant energy. Mathematical proofs, experimental data, and technical derivations are collected in appendices to keep the main text accessible to non-specialist readers.

APPENDIX A
FROM MOBILITIES AND DIFFUSION COEFFICIENTS

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 of Fig. 38 to be filled with gas through which ions are distributed and imagine that these ions are slowly diffusing upward. Let be the ionic concentration, i.e., the number of ions per cubic centimeter at any distance from the bottom of the vessel. Then the number of ions which pass per second through 1 sq. cm. taken perpendicular to at a distance from the bottom must be directly proportional to the concentration gradient and the factor of proportionality in a given gas is by definition the diffusion coefficient of the ions through this gas, i.e.,

But since is also equal to the product of the average velocity with which the ions are streaming upward at by the number of 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 to dynes, 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 take as 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 have Now if denote the velocity in unit field, a quantity which is technically called the “ionic mobility,” . Again since the partial pressure is proportional to , i.e., since , it follows that . Hence equation (43) reduces to or

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 have in which is the number of molecules per cubic centimeter in the air and is the pressure produced by them, i.e., is atmospheric pressure. We have then from equation (44)