The rate equation for the second order reaction `2A+B rarr C` is found to be: `rate = k[A][B]`. The correct statement in relation of this reaction is that
(a) The value of `k` is independent of the initial concentration of `A` and `B`.
(b) `t_(1//2)` is a constant.
(c) The rate of formation of `C` is twice the rate of disappearance of `A`.
(d) The unit of `k` must be `s^(-1)`

Options

A. The value of `k` is indepedennt of the initial concentration of `A` and `B`.
B. `t_(1//2)` is a constant.
C. The rate of formation of `C` is twice the rate of disappearnce of `A`.
D. The unit of `k` must be `s^(-1)`

Explanation

To solve the question regarding the second order reaction \(2A + B \rightarrow C\) with the rate equation given as \( \text{rate} = k[A][B] \), we will analyze each of the provided statements one by one.

### Step-by-Step Solution:

1. **Understanding the Rate Constant \(k\)**:
- The rate constant \(k\) for a reaction is a proportionality factor that relates the rate of reaction to the concentrations of the reactants. It is dependent on temperature and is independent of the initial concentrations of the reactants.
- **Statement (a)**: "The value of \(k\) is independent of the initial concentration of \(A\) and \(B\)."
- This statement is **true** because \(k\) does not change with the concentration of reactants.

2. **Analyzing Half-Life \(t_{1/2}\)**:
- For a second-order reaction, the half-life is given by the formula:
\[ t_{1/2} = \frac{1}{k[A]_0} \]
- This indicates that the half-life \(t_{1/2}\) is inversely proportional to the initial concentration of \(A\). Therefore, it is not a constant.
- **Statement (b)**: "\(t_{1/2}\) is a constant."
- This statement is **false** because \(t_{1/2}\) varies with the initial concentration.

3. **Rate of Formation of Product \(C\)**:
- According to the stoichiometry of the reaction \(2A + B \rightarrow C\), the rate of formation of \(C\) can be expressed in terms of the rate of disappearance of \(A\):
\[ \text{Rate of formation of } C = \frac{1}{2} \left(-\frac{d[A]}{dt}\right) \]
- This means that the rate of formation of \(C\) is half the rate of disappearance of \(A\).
- **Statement (c)**: "The rate of formation of \(C\) is twice the rate of disappearance of \(A\)."
- This statement is **false** because it contradicts the stoichiometry.

4. **Units of the Rate Constant \(k\)**:
- For a second-order reaction, the units of the rate constant \(k\) can be derived from the rate equation:
\[ \text{Rate} = k[A][B] \]
- The rate has units of concentration per time (e.g., M/s), and the concentrations of \(A\) and \(B\) are in molarity (M). Thus:
\[ k = \frac{\text{Rate}}{[A][B]} = \frac{\text{M/s}}{\text{M} \cdot \text{M}} = \text{M}^{-1} \text{s}^{-1} \]
- **Statement (d)**: "The unit of \(k\) must be \(s^{-1}\)."
- This statement is **false** because the correct unit for \(k\) in a second-order reaction is \( \text{M}^{-1} \text{s}^{-1} \).

### Conclusion:
The correct statement in relation to this reaction is:
- **(a)** The value of \(k\) is independent of the initial concentration of \(A\) and \(B\).