A radioactive sample disintegrates via two independent decay processes having half lives T121 and T122 respectively- The effective half - life- T12 of the nuclei is-

We know that: ( dNdt )_1=Nλ _1, ( dNdt )_2=Nλ _2 Now, dNdt= ( dNdt )_1+ ( dNdt )_2 Nλ _eff=Nλ _1+Nλ _2 1T_12=1T_12^(1)+1T_12^(2) T_12=T_12^(1)T_12^(2) ...

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Standard XII

Physics

Nucleus: Common Terms

A radioactive...

Question

A radioactive sample disintegrates via two independent decay processes having half lives $T_{\frac{1}{2}}^{(1)}$ and $T_{\frac{1}{2}}^{(2)}$ respectively. The effective half - life, $T_{\frac{1}{2}}$ of the nuclei is:

None of these

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T_{\frac{1}{2}} = T_{\frac{1}{2}}^{(1)} + T_{\frac{1}{2}}^{(2)}

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T_{\frac{1}{2}} = \frac{T_{\frac{1}{2}}^{(1)} + T_{\frac{1}{2}}^{(2)}}{T_{\frac{1}{2}}^{(1)} - T_{\frac{1}{2}}^{(2)}}

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T_{\frac{1}{2}} = \frac{T_{\frac{1}{2}}^{(1)} T_{\frac{1}{2}}^{(2)}}{T_{\frac{1}{2}}^{(1)} + T_{\frac{1}{2}}^{(2)}}

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Solution

The correct option is D $T_{\frac{1}{2}} = \frac{T_{\frac{1}{2}}^{(1)} T_{\frac{1}{2}}^{(2)}}{T_{\frac{1}{2}}^{(1)} + T_{\frac{1}{2}}^{(2)}}$

We know that: ${(- \frac{d N}{d t})}_{1} = N λ_{1}, {(- \frac{d N}{d t})}_{2} = N λ_{2}$ Now, $\frac{d N}{d t} = {(\frac{d N}{d t})}_{1} + {(\frac{d N}{d t})}_{2}$

$N λ_{e f f} = N λ_{1} + N λ_{2}$

$\frac{1}{T_{\frac{1}{2}}} = \frac{1}{T_{\frac{1}{2}}^{(1)}} + \frac{1}{T_{\frac{1}{2}}^{(2)}}$

$T_{\frac{1}{2}} = \frac{T_{\frac{1}{2}}^{(1)} T_{\frac{1}{2}}^{(2)}}{T_{\frac{1}{2}}^{(1)} + T_{\frac{1}{2}}^{(2)}}$

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A radioactive sample disintegrates via two independent decay processes having half lives T(1)12 and T(2)12 respectively. The effective half - life, T12 of the nuclei is:

The correct option is D T12=T(1)12T(2)12T(1)12+T(2)12 We know that: (−dNdt)1=Nλ1,(−dNdt)2=Nλ2Now, dNdt=(dNdt)1+(dNdt)2 Nλeff=Nλ1+Nλ2 1T12=1T(1)12+1T(2)12 T12=T(1)12T(2)12T(1)12+T(2)12

A radioactive sample disintegrates via two independent decay processes having half lives $T_{\frac{1}{2}}^{(1)}$ and $T_{\frac{1}{2}}^{(2)}$ respectively. The effective half - life, $T_{\frac{1}{2}}$ of the nuclei is: