Question

The activity of a radioactive sample is measured as \(N_0\) counts per minute at time \(t = 0\) and \(\frac{N_0}{e}\) counts per minute at time \(t = 3\) minute. The time (in minute) in which the activity reduces to half the value, is

• A. \(\frac{1}{3}log_e2\)
• B. \(3log_e2\)
• C. \(3log_{10}2\)
• D. \(\frac{3}{log_e2}\)

Hint:

The radius of the nth Bohr orbit is proportional to n squared.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The radius of the nth Bohr orbit is \(r_n = 0.529\,n^2\) angstrom, so \(r_n\propto n^2\).

Step 2: Ratio:
For \(n = 1, 2, 3, 4\): \(r_1 : r_2 : r_3 : r_4 = 1 : 4 : 9 : 16\). Options with \(\frac1n\) or \(\frac{1}{n^2}\) would describe velocity or energy, not radius.

Final Answer:
The radii are in the ratio \(1 : 4 : 9 : 16\), option (C). \[ \boxed{1 : 4 : 9 : 16} \]