Question

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

• A. $X$ and $Y$ decay the same rate always
• B. $X$ will decay faster than $Y$
• C. $Y$ will decay faster than $X$
• D. $X$ and $Y$ have same decay rate initially

Answer 1

Answer: (B)

$X$ will decay faster than $Y$

Let the number of atoms of X and Y presented initially be $N_o$.

Number of atoms of X remaining after time $t$ is given by $N_X=N_o{e}^{-\lambda t}$

$N_X=N_o{e}^{-\frac{\ln 2t}{T_{1/2}}}=N_o{e}^{-\frac{\ln 2t}{T_m}}$ ($\because T_{1/2}=T_m$)

where and $T_{1/2}$ is the half-time of $X$ ans $T_m$ is the mean life time of $Y$.

Number of atoms of Y remaining after time $t$ is given by $N_Y=N_o{e}^{-\lambda t}=N_o{e}^{-\frac{t}{T_m}}$

$⟹$ $N_X<N_Y$ at any time $t$

Thus, $X$ decays faster than $Y$.