When the ions are not removed from the gas, they will increase until the number of ions of one sign which combine with ions of the opposite sign in any time is equal to the number produced by the ionizing agent in that time. We can easily calculate the number of free ions at any time after the ionizing agent has commenced to act.
Let q be the number of ions (positive or negative) produced in one cubic centimetre of the gas per second by the ionizing agent, n1, n2, the number of free positive and negative ions respectively per cubic centimetre of the gas. The number of collisions between positive and negative ions per second in one cubic centimetre of the gas is proportional to n1n2. If a certain fraction of the collisions between the positive and negative ions result in the formation of an electrically neutral system, the number of ions which disappear per second on a cubic centimetre will be equal to αn1 n2, where α is a quantity which is independent of n1, n2; hence if t is the time since the ionizing agent was applied to the gas, we have
Thus n1 − n2 is constant, so if the gas is uncharged to begin with, n1 will always equal n2. Putting n1 = n2 = n we have
| dn/dt = q − αn2 | (1), |
the solution of which is, since n = 0 when t = 0,
| n = k(ε2kαt − 1)ε2kαt + 1 | (2), |
if k2 = q/α. Now the number of ions when the gas has reached a steady state is got by putting t equal to infinity in the preceding equation, and is therefore given by the equation
We see from equation (1) that the gas will not approximate to its steady state until 2kαt is large, that is until t is large compared with 1/2kα or with 1/2√ (qα). We may thus take ½√ (qα) as a measure of the time taken by the gas to reach a steady state when exposed to an ionizing agent; as this time varies inversely as √q we see that when the ionization is feeble it may take a very considerable time for the gas to reach a steady state. Thus in the case of our atmosphere where the production of ions is only at the rate of about 30 per cubic centimetre per second, and where, as we shall see, α is about 10−6, it would take some minutes for the ionization in the air to get into a steady state if the ionizing agent were suddenly applied.
We may use equation (1) to determine the rate at which the ions disappear when the ionizing agent is removed. Putting q=0 in that equation we get dn/αt = −αn2.
Hence
| n = n0/(1 + n0 αt) | (3), |
where n0 is the number of ions when t = 0. Thus the number of ions falls to one-half its initial value in the time 1/n0α. The quantity α is called the coefficient of recombination, and its value for different gases has been determined by Rutherford (Phil. Mag. 1897 [5], 44, p. 422), Townsend (Phil. Trans., 1900, 193, p. 129), McClung (Phil. Mag., 1902 [6], 3, p. 283), Langevin (Ann. chim. phys. [7], 28, p. 289), Retschinsky (Ann. d. Phys., 1905, 17, p. 518), Hendred (Phys. Rev., 1905, 21, p. 314). The values of α/e, e being the charge on an ion in electrostatic measure as determined by these observers for different gases, is given in the following table:—
| Townsend. | McClung. | Langevin. | Retschinsky. | Hendred. | |
| Air | 3420 | 3380 | 3200 | 4140 | 3500 |
| O2 | 3380 | ||||
| CO2 | 3500 | 3490 | 3400 | ||
| H2 | 3020 | 2940 |
The gases in these experiments were carefully dried and free from dust; the apparent value of α is much increased when dust or small drops of water are present in the gas, for then the ions get caught by the dust particles, the mass of a particle is so great compared with that of an ion that they are practically immovable under the action of the electric field, and so the ions clinging to them escape detection when electrical methods are used. Taking e as 3.5×10−10, we see that α is about 1.2×10−6, so that the number of recombinations in unit time between n positive and n negative ions in unit volume is 1.2×10−6n2. The kinetic theory of gases shows that if we have n molecules of air per cubic centimetre, the number of collisions per second is 1.2×10−10n2 at a temperature of 0° C. Thus we see that the number of recombinations between oppositely charged ions is enormously greater than the number of collisions between the same number of neutral molecules. We shall see that the difference in size between the ion and the molecule is not nearly sufficient to account for the difference between the collisions in the two cases; the difference is due to the force between the oppositely charged ions, which drags ions into collisions which but for this force would have missed each other.
Several methods have been used to measure α. In one method air, exposed to some ionizing agent at one end of a long tube, is slowly sucked through the tube and the saturation current measured at different points along the tube. These currents are proportional to the values of n at the place of observation: if we know the distance of this place from the end of the tube when the gas was ionized and the velocity of the stream of gas, we can find t in equation (3), and knowing the value of n we can deduce the value of α from the equation
where n1, n2 are the values of n at the times t1, t2 respectively. In this method the tubes ought to be so wide that the loss of ions by diffusion to the sides of the tube is negligible. There are other methods which involve the knowledge of the speed with which the ions move under the action of known electric forces; we shall defer the consideration of these methods until we have discussed the question of these speeds.
In measuring the value of α it should be remembered that the theory of the methods supposes that the ionization is uniform throughout the gas. If the total ionization throughout a gas remains constant, but instead of being uniformly distributed is concentrated in patches, it is evident that the ions will recombine more quickly in the second case than in the first, and that the value of α will be different in the two cases. This probably explains the large values of α obtained by Retschinsky, who ionized the gas by the α rays from radium, a method which produces very patchy ionization.
Variation of α with the Pressure of the Gas.—All observers agree that there is little variation in α with the pressures for pressures of between 5 and 1 atmospheres; at lower pressures, however, the value of α seems to diminish with the pressure: thus Langevin (Ann. chim. phys., 1903, 28, p. 287) found that at a pressure of 15 of an atmosphere the value of α was about 15 of its value at atmospheric pressure.
Variation of α with the Temperature.—Erikson (Phil. Mag., Aug. 1909) has shown that the value of α for air increases as the temperature diminishes, and that at the temperature of liquid air −180° C., it is more than twice as great as at +12° C.
Since, as we have seen, the recombination is due to the coming together of the positive and negative ions under the influence of the electrical attraction between them, it follows that a large electric force sufficient to overcome this attraction would keep the ions apart and hence diminish the coefficient of recombination. Simple considerations, however, will show that it would require exceedingly strong electric fields to produce an appreciable effect. The value of α indicates that for two oppositely charged ions to unite they must come within a distance of about 1.5×10−6 centimetres; at this distance the attraction between them is e2×1012/2.25, and if X is the external electric force, the force tending to pull them apart cannot be greater than Xe; if this is to be comparable with the attraction, X must be comparable with e×1012/2.25, or putting e = 4×10−10, with 1.8×102; this is 54,000 volts per centimetre, a force which could not be applied to gas at atmospheric pressure without producing a spark.
Diffusion of the Ions.—The ionized gas acts like a mixture of gases, the ions corresponding to two different gases, the non-ionized gas to a third. If the concentration of the ions is not uniform, they will diffuse through the non-ionized gas in such a way as to produce a more uniform distribution. A very valuable series of determinations of the coefficient of diffusion of ions through various gases has been made by Townsend (Phil. Trans., 1900, A, 193, p. 129). The method used was to suck the ionized gas through narrow tubes; by measuring the loss of both the positive and negative ions after the gases had passed through a known length of tube, and allowing for the loss by recombination, the loss by diffusion and hence the coefficient of diffusion could be determined. The following tables give the values of the coefficients of diffusion D on the C.G.S. system of units as determined by Townsend:—
| Gas. | D for +ions. | D for −ions. | Mean Value of D. | Ratio of D for − to D for +ions. |
| Air | .028 | .043 | .0347 | 1.54 |
| O2 | .025 | .0396 | .0323 | 1.58 |
| CO2 | .023 | .026 | .0245 | 1.13 |
| H2 | .123 | .190 | .156 | 1.54 |
| Gas. | D for +ions. | D for −ions. | Mean Value of D. | Ratio of D for − to D for +ions. |
| Air | .032 | .037 | .0335 | 1.09 |
| O2 | .0288 | .0358 | .0323 | 1.24 |
| CO2 | .0245 | .0255 | .025 | 1.04 |
| H2 | .128 | .142 | .135 | 1.11 |
It is interesting to compare with these coefficients the values of D when various gases diffuse through each other. D for hydrogen through air is .634, for oxygen through air .177, for the vapour of
