Two radioactive elements A and B initially have same number of atoms. The half life of A is same as the average life of B . If λ_A and λ_B are decay constants of A and B respectively, then choose the correct relation

Correct answer: C. λ_ A =λ_ B 2 It is given that aligned & ( T_1 2 )_ A =(T_ av )_ B & T_ 1/2 = n 2 λ and T_ av =(1)/(λ) & n 2 λ_ A = 1 λ_ B λ_ A =λ_ B n 2 aligned

BITSAT 2023Paper 2

Exam: BITSAT
Year: 2023
Paper: Paper 2
Subject: Physics
Chapter: Nuclear Physics

BITSAT 2023 · Paper 2 · Physics · Nuclear Physics

Two radioactive elements A and B initially have same number of...

Two radioactive elements

A

and

B

initially have same number of atoms. The half life of

A

is same as the average life of

B

. If

\lambda_A

and

\lambda_B

are decay constants of

A

and

B

respectively, then choose the correct relation from the given options.

Options

A $\lambda_A=\lambda_B$
B $\lambda_A=2 \lambda_B$
C $\lambda_{\mathrm{A}}=\lambda_{\mathrm{B}} \ln 2$
D $\lambda_{\mathrm{A}} \ln 2=\lambda_{\mathrm{B}}$

Correct answer

C. $\lambda_{\mathrm{A}}=\lambda_{\mathrm{B}} \ln 2$

Step-by-step solution

It is given that

\begin{aligned} \\ & \left(\frac{\mathrm{T}_1}{2}\right)_{\mathrm{A}}=\left(\mathrm{T}_{\mathrm{av}}\right)_{\mathrm{B}} \\ \\ & \because \mathrm{T}_{1/2}=\frac{\ell \mathrm{n} 2}{\lambda} \text{\text{ and } } \mathrm{T}_{\mathrm{av}}=\frac{1}{\lambda} \\ \\ & \therefore \frac{\ell \mathrm{n} 2}{\lambda_{\mathrm{A}}}=\frac{1}{\lambda_{\mathrm{B}}} \Rightarrow \lambda_{\mathrm{A}}=\lambda_{\mathrm{B}} \ell n 2 \\ \end{aligned}

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