The half - life of `._6C^(14) `, if its `lamda` is `2.31 xx10^(-4) " year"^(-1)` is

Options

A. `3.5 xx10^4 ` years
B. `3xx10^3` years
C. `2xx10^2` years
D. `4xx10^3` years

Answer

To find the half-life of carbon-14 given its decay constant (λ), we can use the formula for half-life (t₁/₂) in terms of the decay constant:

\[ t_{1/2} = \frac{0.693}{\lambda} \]

### Step-by-Step Solution:

1. **Identify the given values**:
- The decay constant (λ) is given as \( 2.31 \times 10^{-4} \, \text{year}^{-1} \).

2. **Substitute the value of λ into the half-life formula**:
\[ t_{1/2} = \frac{0.693}{2.31 \times 10^{-4}} \]

3. **Calculate the denominator**:
- First, calculate \( 2.31 \times 10^{-4} \):
\[ 2.31 \times 10^{-4} = 0.000231 \]

4. **Perform the division**:
\[ t_{1/2} = \frac{0.693}{0.000231} \]

5. **Calculate the result**:
- Performing the division:
\[ t_{1/2} \approx 3000 \, \text{years} \]

6. **Final answer**:
- Therefore, the half-life of carbon-14 is approximately \( 3000 \, \text{years} \) or \( 3.0 \times 10^{3} \, \text{years} \).