`2N_(2)O_(5)(g)to4NO_(2)+O_(2)(g)`
What is the ratio of the rate of decomposition of `N_(2)O_(5)`
to rate of formation of `NO_(2)`

Options

A. `1:2`
B. `2:1`
C. `1:4`
D. `4:1`

Explanation

To find the ratio of the rate of decomposition of \( N_2O_5 \) to the rate of formation of \( NO_2 \) in the reaction:

\[ 2N_{2}O_{5}(g) \rightarrow 4NO_{2}(g) + O_{2}(g) \]

we can follow these steps:

### Step 1: Write the Rate Expressions
For a general reaction of the form:

\[ aA \rightarrow bB + cC \]

the rate of reaction can be expressed in terms of the change in concentration of the reactants and products.

For our reaction, we can express the rates as follows:

- Rate of decomposition of \( N_2O_5 \):
\[ \text{Rate}_{decomposition} = -\frac{1}{2} \frac{d[N_2O_5]}{dt} \]
(The negative sign indicates that the concentration of \( N_2O_5 \) is decreasing.)

- Rate of formation of \( NO_2 \):
\[ \text{Rate}_{formation} = \frac{1}{4} \frac{d[NO_2]}{dt} \]
(The positive sign indicates that the concentration of \( NO_2 \) is increasing.)

### Step 2: Set Up the Ratio
To find the ratio of the rate of decomposition of \( N_2O_5 \) to the rate of formation of \( NO_2 \), we can write:

\[ \text{Ratio} = \frac{\text{Rate}_{decomposition}}{\text{Rate}_{formation}} = \frac{-\frac{1}{2} \frac{d[N_2O_5]}{dt}}{\frac{1}{4} \frac{d[NO_2]}{dt}} \]

### Step 3: Simplify the Ratio
Now we can simplify the ratio:

\[ \text{Ratio} = \frac{-\frac{1}{2}}{\frac{1}{4}} \cdot \frac{d[N_2O_5]}{d[NO_2]} \]

Calculating the numerical part:

\[ \text{Ratio} = -\frac{1}{2} \cdot 4 = -2 \]

Since we are interested in the absolute value of the ratio, we take the positive value:

\[ \text{Ratio} = 2 \]

### Conclusion
Thus, the ratio of the rate of decomposition of \( N_2O_5 \) to the rate of formation of \( NO_2 \) is:

\[ \boxed{2} \]