Define the term 'Half-life' of a radioactive substance. Two different radioactive substances have half-lives $T_1$ and $T_2$ and number of undecayed atoms at an instant $N_1$ and $N_2$, respectively. Find the ratio of their activities at that instant.
CBSE 55-1-3 PAPER SET 2020
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Half-life: It is the time interval after which the activity of a radioactive sample reduces to half of initial value.
$\because\text{R}=\lambda\text{N}$
$\Rightarrow\frac{\text{R}_1}{\text{R}_2}=\frac{\lambda_1}{\lambda_2}\times\frac{\text{N}_1}{\text{N}_2}$
$\because\lambda=\frac{0.693}{\text{T}_\frac{1}{2}}$
$\Rightarrow\frac{\lambda_1}{\lambda_2}=\frac{\text{T}_2}{\text{T}_1}$
$\therefore\frac{\text{R}_1}{\text{R}_2}=\frac{\text{T}_2}{\text{T}_1}\times\frac{\text{N}_1}{\text{N}_2}$
$\because\text{R}=\lambda\text{N}$
$\Rightarrow\frac{\text{R}_1}{\text{R}_2}=\frac{\lambda_1}{\lambda_2}\times\frac{\text{N}_1}{\text{N}_2}$
$\because\lambda=\frac{0.693}{\text{T}_\frac{1}{2}}$
$\Rightarrow\frac{\lambda_1}{\lambda_2}=\frac{\text{T}_2}{\text{T}_1}$
$\therefore\frac{\text{R}_1}{\text{R}_2}=\frac{\text{T}_2}{\text{T}_1}\times\frac{\text{N}_1}{\text{N}_2}$
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