Question

For a given radioactive material of mean life \(\tau\) and half-life \(t_{1/2}\), the relationship between \(t_{1/2}\) and \(\tau\) is:

• A. \( t_{1/2} = \frac{\ln 2}{\tau} \)
• B. \( t_{1/2} = \tau \ln 2 \)
• C. \( t_{1/2} = \tau \)
• D. \( t_{1/2} = 2\tau \)
• E. \( t_{1/2} = \frac{\tau}{\ln 2} \)

Hint:

The half-life is related to the mean life by a simple logarithmic relationship.

Answer 1

The Correct Option is B

The relationship between the mean life \(\tau\) and the half-life \(t_{1/2}\) of a radioactive material is given by: \[ t_{1/2} = \tau \ln 2 \] This comes from the fact that the decay constant \(\lambda = \frac{1}{\tau}\) and the half-life is related to the decay constant by: \[ t_{1/2} = \frac{\ln 2}{\lambda} \] Substituting \(\lambda = \frac{1}{\tau}\), we get \(t_{1/2} = \tau \ln 2\).