Half life of `C^19` is 5700 years. Find its decay constant

Options

A. `12.16 x 10^-5 year^-1`
B. `12.16 x 10^-3 year^-1`
C. `8.52 x 10^-5 year^-1`
D. `8.53 x 10^-3 year^-1`

Answer

To find the decay constant of \( C^{19} \) given its half-life, we can use the relationship between half-life (\( t_{1/2} \)) and decay constant (\( \lambda \)). The formula is:

\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]

### Step-by-Step Solution:

1. **Identify the half-life**:
The half-life of \( C^{19} \) is given as \( t_{1/2} = 5700 \) years.

2. **Use the decay constant formula**:
Substitute the half-life into the decay constant formula:
\[ \lambda = \frac{\ln(2)}{5700} \]

3. **Calculate \( \ln(2) \)**:
The natural logarithm of 2 is approximately:
\[ \ln(2) \approx 0.693 \]

4. **Substitute \( \ln(2) \) into the equation**:
\[ \lambda = \frac{0.693}{5700} \]

5. **Perform the division**:
To simplify the calculation, convert \( 5700 \) to scientific notation:
\[ 5700 = 57 \times 10^2 \]
Now, substitute this back into the equation:
\[ \lambda = \frac{0.693}{57 \times 10^2} \]

6. **Calculate \( \frac{0.693}{57} \)**:
Performing the division gives:
\[ \frac{0.693}{57} \approx 0.01215 \]

7. **Express in scientific notation**:
Now, we can express this as:
\[ \lambda \approx 12.15 \times 10^{-5} \text{ year}^{-1} \]

8. **Final result**:
Therefore, the decay constant \( \lambda \) is approximately:
\[ \lambda \approx 12.16 \times 10^{-5} \text{ year}^{-1} \]

### Final Answer:
The decay constant of \( C^{19} \) is \( 12.16 \times 10^{-5} \text{ year}^{-1} \).