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 the half-life of a first-order reaction:

### Step-by-Step Solution:

1. **Identify the decay constant (λ)**:
The decay constant for Carbon-14 is given as:
\[ \lambda = 2.31 \times 10^{-4} \text{ year}^{-1} \]

2. **Use the half-life formula**:
The formula for the half-life (t₁/₂) of a first-order reaction is:
\[ t_{1/2} = \frac{0.693}{\lambda} \]

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

4. **Calculate the half-life**:
Performing the division:
\[ t_{1/2} = \frac{0.693}{2.31 \times 10^{-4}} \approx 3000 \text{ years} \]

5. **Express the result in scientific notation**:
The result can be expressed in scientific notation as:
\[ t_{1/2} \approx 3.0 \times 10^{3} \text{ years} \]

### Final Answer:
The half-life of Carbon-14 is approximately \( 3.0 \times 10^{3} \) years.

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