A substance decays at a rate proportional to the amount of the substance itself. If half of the substance decays in one year, then what is the proportionality constant?

Concept:

It is given that the decay rate is proportional to the amount of substance. If the amount of the substance is N then, 

\(\frac{dN}{dt}\propto N \)

\(\frac{dN}{dt}=\lambda N\)

Here, \(\lambda\) is the decay constant.

The half life of any radioactive substance is given as: \(t_{1/2}=\frac{log_e2}{\lambda}\)

Explanation:

Given: The half life of the substance is : \(t_{1/2}=1years\).

Substituting in the half life formula : 

\(1=\frac{log_e2}{\lambda}\)

\(\lambda=log_e2\)