यदि किसी पदार्थ की अर्धायु \(20\) मिनट है, तो \(33 \%\) क्षय और \(67 \%\) क्षय के बीच का समय होगा। (मिनट मे)

\(\frac{ N _{1}}{ N _{0}}= e ^{-\lambda t _{1}}\)

\(0.67= e ^{-\lambda t _{1}}\)

\(\ln (0.67)=-\lambda t _{1}\)

\(N _{2}= N _{0} e ^{-\lambda t _{2}}\)

\(\frac{ N _{2}}{ N _{0}}= e ^{-\lambda t _{2}}\)

\(0.33= e ^{-\lambda t _{2}}\)

\(\ln (0.33)=-\lambda t _{2}\)

\(\ln (0.67)-\ln (0.33)=\lambda t _{1}-\lambda t _{2}\)

\(\lambda\left( t _{1}- t _{2}\right)=\ln \left(\frac{0.67}{0.33}\right)\)

\(\lambda\left( t _{1}- t _{2}\right) \cong \ln 2\)

\(t _{1}- t _{2} \simeq \frac{\ln 2}{\lambda}= t _{1 / 2}\)

Half life \(= t _{1 / 2}=20\,minutes.\)