For the reaction , `N_2O_4(g)underset(K_2)overset(K_1)hArr2NO_2(g)`, the rate of disappearance of `NO_2` will be

Options

A. `K_1[N_2O_4]-K_2[NO_2]^2`
B. `2K_1[N_2O_4]-2K_2[NO_2]^2`
C. `K_2[NO_2]^2-K_1[N_2O_4]`
D. `2K_2[NO_2]^2-2K_1[N_2O_4]`

Answer

To find the rate of disappearance of \( \text{NO}_2 \) for the reaction \( \text{N}_2\text{O}_4(g) \underset{K_2}{\overset{K_1}{\rightleftharpoons}} 2\text{NO}_2(g) \), we can follow these steps:

### Step 1: Write the Reaction and Identify the Rate Expressions
The reaction can be represented as:
\[ \text{N}_2\text{O}_4(g) \underset{K_2}{\overset{K_1}{\rightleftharpoons}} 2\text{NO}_2(g) \]
The forward reaction has a rate constant \( K_1 \) and the backward reaction has a rate constant \( K_2 \).

### Step 2: Define the Rate of Disappearance of \( \text{NO}_2 \)
The rate of disappearance of \( \text{NO}_2 \) is defined as:
\[ -\frac{d[\text{NO}_2]}{dt} \]
This negative sign indicates that the concentration of \( \text{NO}_2 \) is decreasing over time.

### Step 3: Write the Rate Expressions for Both Reactions
For the forward reaction:
\[ \text{Rate}_{\text{forward}} = K_1 [\text{N}_2\text{O}_4] \]
For the backward reaction:
\[ \text{Rate}_{\text{backward}} = K_2 [\text{NO}_2]^2 \]

### Step 4: Determine the Net Rate of Change of \( \text{NO}_2 \)
The net rate of change of \( \text{NO}_2 \) will be the difference between the rate of formation from the forward reaction and the rate of consumption from the backward reaction:
\[ \frac{d[\text{NO}_2]}{dt} = 2K_1 [\text{N}_2\text{O}_4] - 2K_2 [\text{NO}_2]^2 \]

### Step 5: Express the Rate of Disappearance of \( \text{NO}_2 \)
To find the rate of disappearance of \( \text{NO}_2 \), we take the negative of the net rate:
\[ -\frac{d[\text{NO}_2]}{dt} = -\left(2K_1 [\text{N}_2\text{O}_4] - 2K_2 [\text{NO}_2]^2\right) \]
This simplifies to:
\[ -\frac{d[\text{NO}_2]}{dt} = 2K_2 [\text{NO}_2]^2 - 2K_1 [\text{N}_2\text{O}_4] \]

### Final Answer
Thus, the rate of disappearance of \( \text{NO}_2 \) is:
\[ -\frac{d[\text{NO}_2]}{dt} = 2K_2 [\text{NO}_2]^2 - 2K_1 [\text{N}_2\text{O}_4] \]