# Alternative Form of Decay Equation in Terms of Half – Life Time

## Alternative Form of Decay Equation in Terms of Half – Life Time

Half – Life Time is defined as the time duration in which half of the total number of nuclei will decay or be left un-decayed. Say, for example, at any instant if we look into the quantity of a sample of radioactive element, it is observed that after every 3 hour some half – lives are only a millionth of a second for highly active elements and some less active elements have half – life in billions of years.

Let a radioactive element X decays to a daughter nucleus Y with a decay process nuclear reaction written as:

\[X\,\,\xrightarrow{\lambda }\,\,Y[/latex].

If initially N₀ nuclei of element X are present, then after time t number of nuclei present in the sample is given by decay equation given as:

\[N\,\,=\,\,{{N}_{0}}{{e}^{-\lambda t}}[/latex].

If we rearrange the equation, we have:

\[\ln \frac{N}{{{N}_{0}}}\,\,=\,\,-\lambda T[/latex].

We know that half – life of a substance is defined as:

\[T\,\,=\,\,\frac{\ln (2)}{\lambda }[/latex].

\[\Rightarrow \,\,\lambda \,\,=\,\,\frac{\ln (2)}{T}[/latex].

Therefore, we have:

\[\ln \left( \frac{N}{{{N}_{0}}} \right)\,\,=\,\,-\frac{\ln (2)}{T}t[/latex] (Or) \[\ln \left( \frac{N}{{{N}_{0}}} \right)\,\,=\,\,\ln {{(2)}^{-t/T}}[/latex].

Taking antilog on both the sides, we get:

\[\frac{N}{{{N}_{0}}}\,\,=\,\,{{(2)}^{-t/T}}[/latex] (Or) \[N\,\,=\,\,{{N}_{0}}{{(2)}^{-t/T}}[/latex].

This equation is an alternate form of decay equation useful for numerical applications.