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 \( NO_2 \) for the reaction

\[ N_2O_4(g) \underset{K_2}{\overset{K_1}{\rightleftharpoons}} 2NO_2(g) \]

we will analyze both the forward and backward reactions.

### Step 1: Write the rate expressions for both reactions

1. **Forward Reaction**: The forward reaction is the conversion of \( N_2O_4 \) to \( 2NO_2 \). The rate of this reaction can be expressed as:
\[ \text{Rate}_{\text{forward}} = K_1 [N_2O_4] \]

2. **Backward Reaction**: The backward reaction is the conversion of \( 2NO_2 \) back to \( N_2O_4 \). The rate of this reaction can be expressed as:
\[ \text{Rate}_{\text{backward}} = K_2 [NO_2]^2 \]

### Step 2: Determine the net rate of disappearance of \( NO_2 \)

The net rate of disappearance of \( NO_2 \) will be the rate of the forward reaction minus the rate of the backward reaction:
\[ \text{Rate}_{\text{disappearance}} = \text{Rate}_{\text{backward}} - \text{Rate}_{\text{forward}} \]

Substituting the expressions we derived:
\[ \text{Rate}_{\text{disappearance}} = K_2 [NO_2]^2 - K_1 [N_2O_4] \]

### Step 3: Write the final expression

Thus, the final expression for the rate of disappearance of \( NO_2 \) is:
\[ \text{Rate}_{\text{disappearance}} = 2K_2 [NO_2]^2 - K_1 [N_2O_4] \]

### Conclusion

The rate of disappearance of \( NO_2 \) is given by:
\[ \text{Rate}_{\text{disappearance}} = 2K_2 [NO_2]^2 - K_1 [N_2O_4] \]