The half-life period of a radioactive element | vedclass

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(B) The half-life of $X$ is $T_{1/2, X} = \frac{0.693}{\lambda_X}$.The mean life of $Y$ is $	au_Y = \frac{1}{\lambda_Y}$.Given $T_{1/2, X} = 	au_Y$,

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$Y$ will decay faster than $X$

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(B) The half-life of $X$ is $T_{1/2, X} = \frac{0.693}{\lambda_X}$.The mean life of $Y$ is $	au_Y = \frac{1}{\lambda_Y}$.Given $T_{1/2, X} = 	au_Y$,we have $\frac{0.693}{\lambda_X} = \frac{1}{\lambda_Y}$,which implies $\frac{\lambda_X}{\lambda_Y} = 0.693$.The decay rate $R$ is given by $R = \lambda N$.Given initial number of atoms $N_X = N_Y = N_0$,the initial decay rates are $R_X = \lambda_X N_0$ and $R_Y = \lambda_Y N_0$.Taking the ratio,$\frac{R_X}{R_Y} = \frac{\lambda_X}{\lambda_Y} = 0.693$.Since $0.693 < 1$,it follows that $R_X < R_Y$,meaning $Y$ decays faster than $X$ initially.

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

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