and let N be the number of particles of Th X present at any time t after the original Th X was removed. The number of particles of Th X which change every second is λN, where λ is the radio-active constant of Th X. Now, at any time during the process of recovery, the rate of increase of the number of particles of Th X = the rate of production – the rate of change; that is
The solution of this equation is of the form
where a and b are constants.
Now when t is very great, the number of particles of Th X present reach a maximum value N₀.
and the equation becomes
This is equivalent to the equation already obtained in section 130, since the intensity of the radiation is always proportional to the number of particles present.
134. Influence of conditions on the rate of decay. Since the activity of any product, at any time, may be taken as a measure of the rate at which chemical change takes place, it may be used as a means of determining the effect of conditions on the changes occurring in radio-active matter. If the rate of change should be accelerated or retarded, it is to be expected that the value of the radio-active constant λ will be increased or decreased, i.e. that the decay curve will be different under different conditions.
No such effect, however, has yet been observed in any case of radio-active change, where none of the active products produced are allowed to escape from the system. The rate of decay is unaltered by any chemical or physical agency, and in this respect the changes in radio-active matter are sharply distinguished from ordinary chemical changes. For example, the rate of decay of activity from any product takes place at the same rate when the substance is exposed to light as when it is kept in the dark, and at the same rate in a vacuum as in air or any other gas at atmospheric pressure. Its rate of decay is unaltered by surrounding the active matter by a thick layer of lead under conditions where no ordinary radiation from outside can affect it. The activity of the matter is unaffected by ignition or chemical treatment. The material giving rise to the activity can be dissolved in acid and re-obtained by evaporation of the solution without altering the activity. The rate of decay is the same whether the active matter is retained in the solid state or kept in solution. When a product has lost its activity, resolution or heat does not regenerate it, and as we shall see later, the rate of decay of the active products, so far examined, is the same at a red heat as at the temperature of liquid air. In fact, no variation of physical or chemical conditions has led to any observable difference in the decay of activity of any of the numerous types of active matter which have been examined.
135. Effect of conditions on the rate of recovery of activity. The recovery of the activity of a radio-element with time, when an active product is separated from it, is governed by the rate of production of fresh