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13.Nuclei
medium
A radioactive nucleus can decay by two different processes. Half-life for the first process is $3.0\, hours$ while it is $4.5\, hours$ for the second process. The effective half- life of the nucleus will be $.........\,hours.$
A
$3.75$
B
$0.56$
C
$0.26$
D
$1.80$
(JEE MAIN-2022)
Solution
$\lambda_{\text {eq }}=\lambda_{1}+\lambda_{2}$
$\frac{\ln 2}{\left( t _{1 / 2}\right)_{ eq }}=\frac{\ln 2}{\left( t _{1 / 2}\right)_{1}}+\frac{\ln 2}{\left( t _{1 / 2}\right)_{2}}$
$\left( t _{1 / 2}\right)_{ eq }=\frac{\left( t _{1 / 2}\right)_{1} \times\left( t _{1 / 2}\right)_{2}}{\left( t _{1 / 2}\right)_{1}+\left( t _{1 / 2}\right)_{2}}$
$=\frac{3 \times 4.5}{3+4.5}=\frac{3 \times 4.5}{7.5}=\frac{3 \times 3}{5}=1.8 \; hr$
Standard 12
Physics
