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# A radioactive nucleus can decay by two different process . The half life for first process is $\;t_{1}\;$ and that for second process is $\;t_{2}\;$. Find the effective half life of nucleus

$(a)\;t_{1}+t_{2}\qquad(b)\;t_{1}\qquad(c)\;t_{2}\qquad(d)\;\large\frac{t_{1} t_{2}}{t_{1}+t_{2}}$

Answer : (d) $\;\large\frac{t_{1} t_{2}}{t_{1}+t_{2}}$
Explanation :
$-\large\frac{dN_{A}}{dt}=\large\frac{dN_{B}}{dt}+\large\frac{dN_{c}}{dt}$
$=\lambda_{1} N_{A}+\lambda_{2} N_{A}$
$=(\lambda_{1}+\lambda_{2}) N_{A}$
$\lambda_{H}=\lambda_{1}+\lambda_{2}\; \quad or \quad \large\frac{1}{T_{H}}=\large\frac{1}{t_{1}}+\large\frac{1}{t_{2}}$
$T_{H}=\large\frac{t_{1} t_{2}}{t_{1}+t_{2}}$
Where $\;T_{H}\;$ is effective half life of nucleus