Question

If a radioactive element disintegrates for its mean life, the fraction of original amount remaining is ________.

• A. $e$
• B. $1-e$
• C. $1-\frac{1}{e}$
• D. $\frac{1}{e}$
• E. $2e$

Hint:

After one mean life, approximately 36.8% of the sample remains.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
This question relates two important concepts in radioactive decay: the law of radioactive decay and the mean life of a radioactive element. We need to find the fraction of nuclei that have not decayed after a time interval equal to the mean life.
Step 2: Key Formula or Approach
1. Law of Radioactive Decay: The number of undisintegrated nuclei (\(N\)) remaining at time \(t\) from an initial sample of \(N_0\) nuclei is given by: \[ N(t) = N_0 e^{-\lambda t} \] where \(\lambda\) is the decay constant.
2. Mean Life (\(\tau\)): The mean life (or average lifetime) of a radioactive nucleus is the reciprocal of the decay constant. \[ \tau = \frac{1}{\lambda} \] We need to calculate the fraction \(\frac{N(t)}{N_0}\) at the specific time \(t = \tau\).
Step 3: Detailed Explanation
1. Set the time equal to the mean life.
We are given that the time elapsed is equal to the mean life, so \(t = \tau\).
2. Substitute this time into the decay equation.
\[ N(\tau) = N_0 e^{-\lambda \tau} \] 3. Substitute the relationship between mean life and decay constant.
We know that \(\tau = \frac{1}{\lambda}\), which means \(\lambda\tau = 1\).
Substitute this into the exponent:
\[ N(\tau) = N_0 e^{-1} \] 4. Find the fraction remaining.
The fraction of the original amount remaining is \(\frac{N(\tau)}{N_0}\).
\[ \frac{N(\tau)}{N_0} = \frac{N_0 e^{-1}}{N_0} = e^{-1} \] This can be written as:
\[ \frac{1}{e} \] Numerically, \(\frac{1}{e} \approx \frac{1}{2.718} \approx 0.368\). This means that after one mean life, approximately 36.8% of the original radioactive nuclei remain.
Step 4: Final Answer
The fraction of the original amount remaining is \(\frac{1}{e}\).