Rate Law

The rate law for radioactive decay states that the instantaneous rate of decay of a radioelement, at any given moment, is directly proportional to the number of undecayed nuclei present in the sample at that same moment. In symbols: $-\frac{dN}{dt} \propto N$, or, introducing a proportionality constant, $-\frac{dN}{dt} = \lambda N$.

The proportionality constant $\lambda$ introduced here is called the decay constant. Rearranging the rate law gives $\lambda = -\frac{dN}{dt} \times \frac{1}{N}$, which shows what $\lambda$ physically means: it is the FRACTION of the total nuclei present that decays in one unit of time (so a large $\lambda$ means a fast-decaying, short-lived radioelement, and a small $\lambda$ means a slow-decaying, long-lived one). Because the rate of decay depends only on $N$ itself, and on no other chemical or physical variable, this rate law is exactly the mathematical statement of the chemical-independence property of radioactivity described in section 13.4 -- and it is this rate law, taken toge …