JEE Mains · Physics · STD 12 - 13. Nuclei

Some nuclei of a radioactive material are undergoing radioactive decay. The time gap between the instances when a quarter of the nuclei have decayed and when half of the nuclei have decayed is given as: (where \(\lambda\) is the decay constant)

A \(\frac{2 \ln 2}{\lambda}\)
B \(\frac{1}{2} \frac{\ln 2}{\lambda}\)
C \(\frac{\ln \frac{3}{2}}{\lambda}\)
D \(\frac{\ln 2}{\lambda}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{\ln \frac{3}{2}}{\lambda}\)

Step-by-step Solution

Detailed explanation

\(\frac{3 N _{0}}{4}= N _{0} e ^{-\lambda t _{1}}\) \(\frac{ N _{0}}{2}= N _{0} e ^{-\lambda t _{2}}\) \(\ln (\frac 34)=-\lambda t _{1}\) \(\ln (\frac 12)=-\lambda t _{2}\) \(\ln (\frac 34)-\ln (\frac 12)=\lambda\left( t _{2}- t _{1}\right)\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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