Question

If the mean life of a radioactive substance is 12 minutes, then the time (in minutes) at which the fraction of atoms disintegrated becomes $\frac{e^2 - 1}{e^2}$ is

• A. 36
• B. 6
• C. 24
• D. 12

Hint:

Remember that the decay constant ($\lambda$) is the reciprocal of the mean life ($\tau$). Be careful to distinguish between the fraction of atoms \textit{remaining} ($N/N_0$) and the fraction of atoms \textit{disintegrated} ($1 - N/N_0$). The decay law $N = N_0 e^{-\lambda t}$ directly gives the fraction remaining.

Answer 1

The Correct Option is C

Step 1: Identify Given Information and Relevant Formulas
The mean life of a radioactive substance is given: $\tau = 12$ minutes. The fraction of atoms disintegrated is given: $f_d = \frac{e^2 - 1}{e^2}$. We need to find the time ($t$) in minutes. The relationship between mean life ($\tau$) and decay constant ($\lambda$) is: \[ \lambda = \frac{1}{\tau} \] The radioactive decay law states that the number of undecayed nuclei ($N$) at time ($t$) is related to the initial number of nuclei ($N_0$) by: \[ N = N_0 e^{-\lambda t} \quad \implies \quad \frac{N}{N_0} = e^{-\lambda t} \] The fraction of atoms disintegrated ($f_d$) is given by: \[ f_d = \frac{N_0 - N}{N_0} = 1 - \frac{N}{N_0} \] Step 2: Calculate the Decay Constant ($\lambda$)
Using the given mean life: \[ \lambda = \frac{1}{12} \text{ min}^{-1} \] Step 3: Determine the Fraction of Atoms Remaining ($\frac{N}{N_0}$)
We are given the fraction of atoms disintegrated ($f_d$). We can use this to find the fraction of atoms remaining: \[ \frac{e^2 - 1}{e^2} = 1 - \frac{N}{N_0} \] Rearrange the equation to solve for $\frac{N}{N_0}$: \[ \frac{N}{N_0} = 1 - \frac{e^2 - 1}{e^2} \] \[ \frac{N}{N_0} = \frac{e^2 - (e^2 - 1)}{e^2} \] \[ \frac{N}{N_0} = \frac{1}{e^2} \] Step 4: Equate and Solve for Time ($t$)
Now, use the decay law equation $\frac{N}{N_0} = e^{-\lambda t}$ and the calculated fraction remaining: \[ e^{-\lambda t} = \frac{1}{e^2} \] We know that $\frac{1}{e^2}$ can be written as $e^{-2}$. So: \[ e^{-\lambda t} = e^{-2} \] Comparing the exponents: \[ -\lambda t = -2 \] \[ \lambda t = 2 \] Solve for $t$: \[ t = \frac{2}{\lambda} \] Substitute the value of $\lambda$ from Step 2: \[ t = \frac{2}{1/12} \] \[ t = 2 \times 12 \] \[ t = 24 \text{ minutes} \] Step 5: Analyze the Options
\begin{itemize} \item Option (1): 36. Incorrect. \item Option (2): 6. Incorrect. \item Option (3): 24. Correct, as it matches our calculated time. \item Option (4): 12. Incorrect. \end{itemize}