TS EAMCET · Physics · Nuclear Physics

A radioactive element which can decay by two processes, has half-life \(t_1\) for first process and half-life \(t_2\) for second process. Let \(\langle t\rangle\) be the effective average-life of this element. Which of the following is correct?

A \(\langle t\rangle < \frac{t_1 t_2}{t_1+t_2}\)
B \(\langle t\rangle=\frac{t_1 t_2}{t_1+t_2}\)
C \(\langle t\rangle>\frac{t_1 t_2}{t_1+t_2}\)
D \(\langle t\rangle=\ln 2\left(\frac{t_1+t_2}{t_1 t_2}\right)\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\langle t\rangle=\frac{t_1 t_2}{t_1+t_2}\)

Step-by-step Solution

Detailed explanation

Radioactive decay rate is directly proportional to number of nucleus present at any instant, \(\frac{d N}{d t}=-\lambda N\) \(\ldots(\mathrm{i})\) Here, \(\lambda\) is a constant related to half-life \(\left(\lambda=\frac{\ln 2}{T_{1 / 2}}\right)\), where, \(T_{1 / 2}\) is…

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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