  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  Solution

The correct option is C $$X$$ 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  $$N_X = N_o e^{-\lambda t}$$$$N_X = N_oe^{ -\frac{\ln 2 t}{T_{1/2}}} = N_o e^{ - \frac{\ln 2 t}{T_m}}$$     ($$\because T_{1/2} = T_m$$)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  $$N_Y = N_o e^{-\lambda t} = N_o e^{-\frac{t}{T_m}}$$$$\implies$$  $$N_X<N_Y$$  at any time $$t$$Thus, $$X$$ decays faster than $$Y$$.Physics

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