How many atoms decay in one mean life time of a radioactive sample-

Options

A. `37%`
B. `63%`
C. `50%`
D. `100%`

Answer

To solve the question of how many atoms decay in one mean lifetime of a radioactive sample, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Mean Lifetime**:
The mean lifetime (τ) of a radioactive sample is defined as the average time it takes for a single atom to decay. Mathematically, it is given by:
\[ \tau = \frac{1}{\lambda} \]
where \( \lambda \) is the decay constant.

2. **Radioactive Decay Equation**:
The rate of decay of a radioactive substance is described by the equation:
\[ \frac{dN}{dt} = -\lambda N \]
where \( N \) is the number of undecayed atoms at time \( t \).

3. **Rearranging the Equation**:
We can rearrange this equation to facilitate integration:
\[ \frac{dN}{N} = -\lambda dt \]

4. **Integrating Both Sides**:
We integrate both sides. The left side will be integrated from \( N_0 \) (initial number of atoms) to \( N \) (number of atoms at time \( t \)), and the right side from \( 0 \) to \( t \):
\[ \int_{N_0}^{N} \frac{dN}{N} = -\lambda \int_{0}^{t} dt \]
This gives us:
\[ \ln\left(\frac{N}{N_0}\right) = -\lambda t \]

5. **Substituting Mean Lifetime**:
We substitute \( t = \tau = \frac{1}{\lambda} \):
\[ \ln\left(\frac{N}{N_0}\right) = -\lambda \left(\frac{1}{\lambda}\right) = -1 \]

6. **Exponentiating**:
To solve for \( N \), we exponentiate both sides:
\[ \frac{N}{N_0} = e^{-1} \]
Hence,
\[ N = N_0 e^{-1} \approx 0.3679 N_0 \]

7. **Calculating Decayed Atoms**:
The number of atoms that have decayed in one mean lifetime is given by the difference between the initial number of atoms and the number remaining:
\[ \text{Decayed atoms} = N_0 - N = N_0 - 0.3679 N_0 = 0.6321 N_0 \]

8. **Converting to Percentage**:
To express this as a percentage:
\[ \text{Percentage decayed} = 0.6321 \times 100\% \approx 63.21\% \]

### Final Answer:
Approximately 63% of the atoms decay in one mean lifetime of a radioactive sample.

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