If N₀ is the number of atoms at the beginning,
Thus
or the law of decay expresses the fact that the activity of a product at any time is proportional to the number of atoms which remain unchanged at that time.
This is the same as the law of monomolecular change in chemistry, and expresses the fact that there is only one changing system. If the change depended on the mutual action of two systems, the law of decay would be different, since the rate of decay in that case would depend on the relative concentration of the two reacting substances. This is not so, for not a single case has yet been observed in which the law of decay was affected by the amount of active matter present.
From the above equation (1)
or the number of systems changing in unit time is proportional to the number unchanged at that time.
In the case of recovery of activity, after an active product has been removed, the number of systems changing in unit time, when radio-active equilibrium is produced, is equal to λN₀. This must be equal to the number q₀ of new systems applied in unit time, or
λ has thus a distinct physical meaning, and may be defined as the proportion of the total number of systems present which change per second. It has different values for different types of active matter, but is invariable for any particular type of matter. For this reason, λ will be termed the “radio-active constant„ of the product.
We are now in a position to discuss with more physical definiteness the gradual growth of Th X in thorium, after the Th X has been completely removed from it. Let q₀ particles of Th X be produced per second by the thorium,