In a mean life of a radioactive sample

Options

A. about `2//3` of substance disintefrates
B. about `2//3` of substanced disintegrates
C. about `90%` of the substance disintegrates
D. almost all the substance disintegrates

Explanation

To solve the problem regarding the mean life of a radioactive sample, we can follow these steps:

### Step 1: Understand the decay formula
The decay of a radioactive substance can be described by the equation:
\[ n = n_0 e^{-\lambda t} \]
where:
- \( n \) is the number of undecayed nuclei at time \( t \),
- \( n_0 \) is the initial number of nuclei,
- \( \lambda \) is the decay constant,
- \( e \) is the base of the natural logarithm.

### Step 2: Relate mean life to decay constant
The mean life (or average lifetime) \( \tau \) of a radioactive sample is given by:
\[ \tau = \frac{1}{\lambda} \]

### Step 3: Substitute mean life into the decay formula
Using the mean life, we can express the decay formula in terms of \( \tau \):
\[ n = n_0 e^{-\frac{t}{\tau}} \]

### Step 4: Determine the amount of substance remaining after one mean life
After one mean life (\( t = \tau \)):
\[ n = n_0 e^{-1} \]
Since \( e^{-1} \) is approximately \( \frac{1}{e} \), we can say:
\[ n \approx \frac{n_0}{e} \]

### Step 5: Calculate the amount of substance disintegrated
The amount of substance that has disintegrated can be calculated as:
\[ \text{Disintegrated} = n_0 - n \]
Substituting the value of \( n \):
\[ \text{Disintegrated} = n_0 - \frac{n_0}{e} = n_0 \left(1 - \frac{1}{e}\right) \]

### Step 6: Calculate the fraction of substance disintegrated
The fraction of the substance that has disintegrated is:
\[ \frac{\text{Disintegrated}}{n_0} = 1 - \frac{1}{e} \]
Using the approximation \( e \approx 2.718 \), we find:
\[ 1 - \frac{1}{e} \approx 1 - 0.3679 \approx 0.6321 \]
This means approximately 63.21% of the substance has disintegrated.

### Step 7: Determine the fraction in terms of \( n_0 \)
To express this as a fraction of \( n_0 \):
\[ \text{Disintegrated} \approx 0.6321 n_0 \]
This can be approximated as:
\[ \text{Disintegrated} \approx \frac{2}{3} n_0 \]

### Conclusion
Thus, approximately \( \frac{2}{3} \) of the substance has disintegrated after one mean life.

### Final Answer
The correct option is B: about \( \frac{2}{3} \) of the substance has disintegrated.

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