Radioactive Element Decay: Predicting Substance Amounts

[siddharth]

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?

 

[Gokul43201]

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 :

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

 

[Astronuc]

See also - for a discussion of radioactive decay

https://www.physicsforums.com/showthread.php?t=57309