# Decay of an element

Homework Helper
Gold Member
My school textbook says that "The decay of a radioactive element is a random process and does not depend on external factors such as temperature". But if the decay is a random process, how can we accuratley predict the amount of substance after t seconds using the rate law?? Did I miss something?

## Answers and Replies

Related Atomic and Condensed Matter News on Phys.org
Gokul43201
Staff Emeritus
Science Advisor
Gold Member
Actually, it's because radioactive decay is random that we can derive the rate law. The route is through probability.

Given a radioactive nucleus, you can never tell when the next decay event is going to happen, because it is equally likely to happen any time. So, if you have a large enough radioactive sample, then in any small time interval $\Delta t$, the number of decay events expected will be proportional to the number of nucleii in the sample.

Or, $$~~ lim_{\Delta t \rightarrow 0} (\frac {\Delta N}{\Delta t}) = \frac {dN}{dt}~~ \alpha ~~N$$

This is exactly what gives you the first-order rate law :

$$N(t) = N(0)~e^{-kt}$$