the number of atoms n_{t} which change per second at the time t is given by
n_{t}/n_{0} = e^{-λt},
where n_{0} is the initial number which change per second. On this view, then, the law of decay expresses the result that the number of atoms changing in unit time, diminishes according to an exponential law with the time. The number of atoms N_{t} which remain unchanged after an interval t is given by
N_{t} = [integral]_{t}^[infinity] n_{t} dt
= (n_{0}/λ)e^{-λt}.
If N_{0} is the number of atoms at the beginning,
N_{0} = n_{0}/λ.
Thus N_{t}/N_{0} = e^{-λt} (1),
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 mono-molecular 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)
dN_{t}/dt = -λN_{t},
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
