After how many days will 1/20th of the radioactive element remain, if the half life of the element is 6.931 days?

Options

A. 23.03 days
B. 25.12days
C. 28.38 days
D. 29.96 days

Answer

To solve the problem of how many days it will take for 1/20th of a radioactive element to remain, given that its half-life is 6.931 days, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Half-Life**:
The half-life (t_half) of a radioactive element is the time required for half of the radioactive atoms to decay. In this case, t_half = 6.931 days.

2. **Determine the Remaining Fraction**:
We want to find out after how many days (t) only 1/20th of the original amount (N0) remains. This means:
\[ N = \frac{N_0}{20} \]

3. **Use the Radioactive Decay Formula**:
The formula for radioactive decay is given by:
\[ N = N_0 e^{-\lambda t} \]
where \( \lambda \) is the decay constant.

4. **Set Up the Equation**:
Substitute \( N \) with \( \frac{N_0}{20} \):
\[ \frac{N_0}{20} = N_0 e^{-\lambda t} \]
Dividing both sides by \( N_0 \) (assuming \( N_0 \neq 0 \)):
\[ \frac{1}{20} = e^{-\lambda t} \]

5. **Take the Natural Logarithm**:
To solve for \( t \), take the natural logarithm of both sides:
\[ \ln\left(\frac{1}{20}\right) = -\lambda t \]

6. **Express \( \lambda \) in Terms of Half-Life**:
The decay constant \( \lambda \) is related to the half-life by the formula:
\[ \lambda = \frac{0.6931}{t_{half}} \]
Substituting \( t_{half} = 6.931 \) days:
\[ \lambda = \frac{0.6931}{6.931} \]

7. **Substitute \( \lambda \) Back into the Equation**:
Now, substitute \( \lambda \) into the logarithmic equation:
\[ \ln\left(\frac{1}{20}\right) = -\left(\frac{0.6931}{6.931}\right) t \]

8. **Calculate \( t \)**:
Rearranging gives:
\[ t = -\frac{6.931 \cdot \ln\left(\frac{1}{20}\right)}{0.6931} \]
Calculate \( \ln\left(\frac{1}{20}\right) \):
\[ \ln\left(\frac{1}{20}\right) = -\ln(20) \approx -2.996 \]
Thus:
\[ t = -\frac{6.931 \cdot (-2.996)}{0.6931} \approx 29.96 \text{ days} \]

9. **Final Answer**:
Therefore, it will take approximately **29.96 days** for 1/20th of the radioactive element to remain.