and the size of gaseous ions. 171 



velocity of translation of one ion relatively to the other ; in the 

 time t the ion will move through a distance Vt relatively to the 

 oppositely charged ion, and every point inside the cylinder, whose 

 volume is irr 2 Vt, will be at a distance not greater than r from the 

 ion at some point of its path. Thus, if there are n positive and 

 n negative ions per unit volume, the ion in the time t will come 

 within combining distance of mrr 2 Vt oppositely charged ions. 

 Thus the number of recombinations per ion per second is mrr 2 V, 

 and since there are n ions per unit volume, the number of re- 

 combinations per unit volume per second is n 2 7rr 2 V. But if a is 

 the coefficient of recombination, this number is by definition equal 

 to an 2 , thus oL = Trr 2 V, or substituting the value of r, 61 x 10~ 12 V. 

 If the ions in hydrogen had the same mass as a hydrogen mole- 

 cule, V at 0° C. would be ^2 x 1-8 x 10 5 and a Vo x lO" 6 . This 

 is the right order of magnitude for a, as at atmospheric pressure 

 a for hydrogen, air and carbonic acid is about 10 -6 . We should 

 expect the value of a determined by the equation a = 6'l x 10 _12 F 

 to be too small, for in determining it we have neglected the effect 

 of the surrounding gas on the motion of the ion ; this gas will 

 act like a resisting medium and will cause some ions to fall 

 together which would otherwise have escaped from each other's 

 action. We should expect that the effect of the surrounding gas 

 would be greater at high pressures than at low ones, so that a 

 would diminish with the pressure, a result shown very clearly in 

 Langevin's experiments. Since the equation a = 61xlO -12 F 

 must give too small a value for a, while the value got by this 

 equation on the supposition that the mass of the hydrogen ion is 

 equal to the mass of the hydrogen molecule is slightly too large, 

 we infer that V for the hydrogen ion must be less than V for the 

 hydrogen molecule, this requires the mass of the hydrogen ion 

 to be greater than that of the molecule ; the near agreement of 

 the theory and experiment indicates that the mass of the ion is 

 not a large multiple of that of the molecule. 



We have assumed that the kinetic energy of the ion was 

 determined by the temperature of the gas, this will be the case 

 when the electric field is weak, so that the velocity acquired by 

 an ion under the field is small compared with the average velocity 

 due to temperature. In electric fields strong enough to produce 

 discharge the velocity due to the field is much greater than that 

 due to temperature, and the kinetic energy is much greater than 

 that assumed in the preceding calculation. The value of a in 

 this case will be very much less than the value we have calcu- 

 lated, and we see that the value of a will diminish rapidly as the 

 strength of the field is increased. 



Let us now find the rate at which a charged molecule would 

 combine with an uncharged one to form a complex ion containing 



