Question

A radioactive nucleus can decay by two different processes. The half-life for the first process is \(t_1\), and that for the second process is \(t_2\). The effective half-life \(\tau\) of the nucleus is given by:

• A. \(\dfrac{1}{\tau} = \dfrac{1}{t_1} + \dfrac{1}{t_2}\)
• B. \(\tau = t_1 + t_2\)
• C. \(\tau = t_1 - t_2\)
• D. \(\tau = t_1 t_2\)

Hint:

Independent decay channels add their decay constants; since \(t_{1/2}\propto 1/\lambda\), the reciprocals of the half-lives add.

Answer 1

The Correct Option is A

Step 1: When a nucleus can decay through two independent channels, the total decay probability per unit time is the sum of the individual decay constants:
\[ \lambda = \lambda_1 + \lambda_2. \]
Step 2: The half-life is related to the decay constant by \(t_{1/2} = \dfrac{\ln 2}{\lambda}\), so \(\lambda = \dfrac{\ln 2}{t_{1/2}}\).
Step 3: Substitute for each channel:
\[ \frac{\ln 2}{\tau} = \frac{\ln 2}{t_1} + \frac{\ln 2}{t_2}. \]
Step 4: Cancel the common factor \(\ln 2\):
\[ \frac{1}{\tau} = \frac{1}{t_1} + \frac{1}{t_2}. \]
\[ \boxed{\dfrac{1}{\tau} = \dfrac{1}{t_1} + \dfrac{1}{t_2}} \]