# The half­-life period of a radio­-active element X  is same as the mean life time of another radio­-active element Y . Initially they have the same number of atoms. Then Option 1) X and Y decay at same rate always Option 2) X will decay faster than Y Option 3) Y will decay faster than X Option 4) X and Y have same decay rate initially

As we learnt in

Number of nuclei in terms of half life -

$N=\frac{N_{0}}{2^{t/t_{1/2}}}$

- wherein

Very useful to determine number of nuclei in terms of half life

$T_{1/2},\: hal\! f\: life\: o\! f X= \tau _{\gamma }mean\; life\; of\; Y$

$\frac{ln2}{\lambda _{X}}= \frac{1}{\lambda _{Y}}\Rightarrow \lambda _{X}= \lambda _{Y}ln2$

$\lambda _{X}> \lambda _{Y}$

$\therefore A_{X}=A_{0}e^{-\lambda X^{t}}; A_{Y}=A_{0}e^{-\lambda Y^{t}};$

X will decay faster than Y

Correct option is 2

Option 1)

X and Y decay at same rate always

This is an incorrect option.

Option 2)

X will decay faster than Y

This is the correct option.

Option 3)

Y will decay faster than X

This is an incorrect option.

Option 4)

X and Y have same decay rate initially

This is an incorrect option.

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