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The half-life period of element $\mathrm{X}$ is same as the mean life time of element $Y$. Assume initially $\mathrm{X}$ and $\mathrm{Y}$ have same number of atoms. Then
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The correct answer is:
$\mathrm{Y}$ decays faster than $\mathrm{X}$
For element X,
$\left(\mathrm{t}_{1 / 2}\right)_{\mathrm{x}}=\frac{0.693}{\lambda_{\mathrm{x}}}$
For element $Y$,
$\begin{aligned}
& \left(\mathrm{t}_{\text {moan }}\right)_Y=\frac{1}{\lambda_{\mathrm{y}}} \\
& \frac{0.693}{\lambda_{\mathrm{x}}}=\frac{1}{\lambda_{\mathrm{y}}} \Rightarrow \lambda_{\mathrm{x}}=0.693 \lambda_{\mathrm{y}} \\
& \lambda_{\mathrm{x}} < \lambda_{\mathrm{y}}
\end{aligned}$
Hence, $Y$ will decay faster rate than $X$.
$\left(\mathrm{t}_{1 / 2}\right)_{\mathrm{x}}=\frac{0.693}{\lambda_{\mathrm{x}}}$
For element $Y$,
$\begin{aligned}
& \left(\mathrm{t}_{\text {moan }}\right)_Y=\frac{1}{\lambda_{\mathrm{y}}} \\
& \frac{0.693}{\lambda_{\mathrm{x}}}=\frac{1}{\lambda_{\mathrm{y}}} \Rightarrow \lambda_{\mathrm{x}}=0.693 \lambda_{\mathrm{y}} \\
& \lambda_{\mathrm{x}} < \lambda_{\mathrm{y}}
\end{aligned}$
Hence, $Y$ will decay faster rate than $X$.
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