Which of the following expression can be used to describe the instantaneous rate of the reaction ?
`2A+B rarr A_(2)B`
(a) `(1)/(2)(-d[A])/(d t)`
(b) `(-d[A])/(d t)`
(c) `(1)/(2) (d[A_(2)B])/(d t)`
(d) `(1)/(2) (-d[A])/(d t).(d[B])/(d t)`

Options

A. `(1)/(2)(-d[A])/(d t)`
B. `(-d[A])/(d t)`
C. `(1)/(2) (d[A_(2)B])/(d t)`
D. `(1)/(2) (-d[A])/(d t).(d[B])/(d t)`

Explanation

To determine the instantaneous rate of the reaction \(2A + B \rightarrow A_2B\), we need to consider the stoichiometry of the reaction and how the rate is expressed in terms of the change in concentration of the reactants and products over time.

### Step-by-Step Solution:

1. **Identify the Reaction and Stoichiometry**:
The reaction is given as \(2A + B \rightarrow A_2B\). Here, the coefficients indicate how many moles of each substance are involved in the reaction.

2. **Define the Rate of Reaction**:
The rate of a reaction can be expressed in terms of the change in concentration of the reactants and products. For the given reaction, the rate can be defined as:
\[ \text{Rate} = -\frac{1}{2} \frac{d[A]}{dt} = -\frac{1}{1} \frac{d[B]}{dt} = \frac{1}{1} \frac{d[A_2B]}{dt} \]
Here, the negative sign indicates that the concentration of reactants is decreasing, while the product's concentration is increasing.

3. **Choose the Correct Expression**:
From the definitions above, we can see that the instantaneous rate of the reaction can be expressed as:
\[ \text{Rate} = -\frac{1}{2} \frac{d[A]}{dt} \]
This matches option (a).

4. **Evaluate Other Options**:
- Option (b) \(-\frac{d[A]}{dt}\) does not account for the stoichiometric coefficient of 2.
- Option (c) \(\frac{1}{2} \frac{d[A_2B]}{dt}\) is incorrect because it does not have the correct stoichiometric relationship.
- Option (d) \(\frac{1}{2} (-\frac{d[A]}{dt})(\frac{d[B]}{dt})\) is not a valid expression for the rate of reaction.

### Conclusion:
The correct expression that can be used to describe the instantaneous rate of the reaction is:
\[ \text{Answer: (a) } \frac{1}{2} \left(-\frac{d[A]}{dt}\right) \]