एक रेडियो एक्टिव नाभिक का क्षय दो विभिन्न प्रक्रम में होता है। प्रथम प्रक्रम की अर्द्ध आयु \(5\) मिनट तथा दूसरे प्रक्रम की अर्द्धआयु \(30\) से. है। नाभिक की प्रभावी अर्द्धआयु की गणना \(\frac{\alpha}{11}\) से. की गयी है। \(\alpha\) का मान______________ है।

A \(301\)
B \(302\)
C \(300\)
D \(303\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(300\)

Step-by-step Solution

Detailed explanation

\(\frac{ dN _1}{ dt }=-\lambda_1 N \quad \frac{ dN _2}{ dt }=-\lambda_2 N\) \(\frac{ dN }{ dt }=-\left(\lambda_1+\lambda_2\right) N\) \(\Rightarrow \lambda_{\text {eq }}=\lambda_1+\lambda_2\) \(\Rightarrow \frac{1}{t_{1 / 2}}=\frac{1}{300}+\frac{1}{30}=\frac{11}{300}\)…

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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