The rate constant of a reaction , `A to `Product , with initial reactant concentration a mole /L ,is k mole /L/min. The `t_(1//2)` for the reaction is equal to

Options

A. `(0.6932)/k` min
B. `1/(ak)` min
C. `a/(2k)` min
D. `a/(4k)` min

Answer

To find the half-life (\(t_{1/2}\)) of a reaction with an initial concentration \(a\) (in moles/L) and a rate constant \(k\) (in moles/L/min), we first need to determine the order of the reaction based on the units of the rate constant \(k\).

### Step-by-step Solution:

1. **Identify the Units of Rate Constant \(k\)**:
The given unit of \(k\) is moles/L/min. This indicates the type of reaction order.
- For a zero-order reaction, the units of \(k\) are moles/(L·min).
- For a first-order reaction, the units of \(k\) are 1/min.
- For a second-order reaction, the units of \(k\) are L/(moles·min).

Since the unit of \(k\) is moles/L/min, we conclude that the reaction is **zero-order**.

**Hint**: Check the units of the rate constant to determine the order of the reaction.

2. **Use the Half-life Formula for Zero-order Reactions**:
The half-life for a zero-order reaction can be calculated using the formula:
\[ t_{1/2} = \frac{[A]_0}{2k} \]
where \([A]_0\) is the initial concentration of the reactant.

3. **Substitute the Given Values**:
Here, \([A]_0 = a\) (initial concentration) and \(k\) is the rate constant. Therefore, we can substitute these values into the half-life formula:
\[ t_{1/2} = \frac{a}{2k} \]

4. **Final Expression**:
Thus, the half-life of the reaction is given by:
\[ t_{1/2} = \frac{a}{2k} \]

### Final Answer:
The half-life (\(t_{1/2}\)) for the reaction is:
\[ t_{1/2} = \frac{a}{2k} \]