The half-life period of a radioactive element | vedclass

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(C) Given that the half-life of $X$ equals the mean life of $Y$: $(T_{1/2})_X = (	au)_Y$.We know that $(T_{1/2})_X = \frac{0.693}{\lambda_X}$ and $(\

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$Y$ will decay at a faster rate than $X$.

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(C) Given that the half-life of $X$ equals the mean life of $Y$: $(T_{1/2})_X = (	au)_Y$.We know that $(T_{1/2})_X = \frac{0.693}{\lambda_X}$ and $(	au)_Y = \frac{1}{\lambda_Y}$.Equating these, we get $\frac{0.693}{\lambda_X} = \frac{1}{\lambda_Y}$, which implies $\lambda_X = 0.693 \lambda_Y$.Since $0.693 The rate of decay is given by $R = \lambda N$.Initially, the number of atoms $N$ is the same for both elements. Since $\lambda_Y > \lambda_X$, the decay rate $R_Y = \lambda_Y N$ will be greater than $R_X = \lambda_X N$.Therefore, $Y$ will decay at a faster rate than $X$.

Column-$I$	Column-$II$
$A$. $500$ million years ago	$1$. Sea weeds and few plants
$B$. $350$ million years ago	$2$. Jawless fish
$C$. $320$ million years ago	$3$. Invertebrates

The half-life period of a radioactive element $X$ is the same as the mean lifetime of another radioactive element $Y$. Initially, both of them have the same number of atoms. Then:

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