For the hypothetical reaction `2A rarr3C` , the reaction rate r in terms of the rate of change of the concentration is given by

Options

A. `r=-(d[A])/(dt)`
B. `r=-1/2(d[A])/(dt)`
C. `r=1/3(d[A])/(dt)`
D. `r=(d[A])/(dt)`

Answer

To derive the expression for the reaction rate \( R \) in terms of the rate of change of concentration for the hypothetical reaction \( 2A \rightarrow 3C \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Reaction**:
The reaction given is \( 2A \rightarrow 3C \). Here, \( A \) is the reactant and \( C \) is the product.

2. **Define the Rate of Reaction**:
The rate of reaction \( R \) can be expressed in terms of the change in concentration of the reactants and products over time.

3. **Express the Rate in Terms of Reactants**:
For the reactant \( A \), since it is being consumed, the rate of disappearance can be expressed as:
\[ R = -\frac{1}{2} \frac{d[A]}{dt} \]
Here, the negative sign indicates that the concentration of \( A \) is decreasing, and the factor \( \frac{1}{2} \) comes from the stoichiometry of the reaction (2 moles of \( A \) are consumed).

4. **Express the Rate in Terms of Products**:
For the product \( C \), since it is being formed, the rate of appearance can be expressed as:
\[ R = \frac{1}{3} \frac{d[C]}{dt} \]
Here, the positive sign indicates that the concentration of \( C \) is increasing, and the factor \( \frac{1}{3} \) comes from the stoichiometry of the reaction (3 moles of \( C \) are produced).

5. **Final Expression for Rate**:
Thus, we can summarize the rate of reaction in terms of the concentrations of \( A \) and \( C \):
\[ R = -\frac{1}{2} \frac{d[A]}{dt} = \frac{1}{3} \frac{d[C]}{dt} \]

### Conclusion:
The rate of the reaction \( R \) can be expressed as:
\[ R = -\frac{1}{2} \frac{d[A]}{dt} \]
or
\[ R = \frac{1}{3} \frac{d[C]}{dt} \]