For the reaction,
`N_2O_5rarr2NO_2+1/2O_2`
Given : `-(d[N_2O_5])/(dt)=K_1[N_2O_5]`
`(d[NO_2])/(dt)=K_2[N_2O_5] and (d[O_2])/(dt)=K_3[N_2O_5]`
The relation between `K_1 , K_2 and K_3` is

Options

A. `2K_1=K_2=4K_3`
B. `K_1=K_2=K_3`
C. `2K_1=4K_2=K_3`
D. `K_1=2K_2=3K_3`

Answer

To find the relationship between the rate constants \( K_1 \), \( K_2 \), and \( K_3 \) for the reaction

\[ N_2O_5 \rightarrow 2NO_2 + \frac{1}{2}O_2, \]

we start by writing the rate expressions based on the stoichiometry of the reaction.

### Step 1: Write the rate expressions

From the stoichiometry of the reaction, we can express the rates of change of concentrations as follows:

1. The rate of disappearance of \( N_2O_5 \):
\[ -\frac{d[N_2O_5]}{dt} = K_1[N_2O_5] \]

2. The rate of formation of \( NO_2 \):
\[ \frac{d[NO_2]}{dt} = K_2[N_2O_5] \]

3. The rate of formation of \( O_2 \):
\[ \frac{d[O_2]}{dt} = K_3[N_2O_5] \]

### Step 2: Relate the rates using stoichiometry

From the balanced reaction, we can relate the rates of formation and disappearance based on their stoichiometric coefficients. The coefficients in the balanced equation are:

- For \( N_2O_5 \): 1
- For \( NO_2 \): 2
- For \( O_2 \): \( \frac{1}{2} \)

Thus, we can express the rates in terms of a common rate of reaction \( r \):

\[ r = -\frac{1}{1} \frac{d[N_2O_5]}{dt} = \frac{1}{2} \frac{d[NO_2]}{dt} = \frac{1}{\frac{1}{2}} \frac{d[O_2]}{dt} \]

### Step 3: Substitute the rate expressions

From the above relations, we can express \( K_2 \) and \( K_3 \) in terms of \( K_1 \):

1. From the rate of \( NO_2 \):
\[ r = K_1[N_2O_5] \implies \frac{d[NO_2]}{dt} = 2K_1[N_2O_5] \]
Therefore, we have:
\[ K_2 = 2K_1 \]

2. From the rate of \( O_2 \):
\[ r = K_1[N_2O_5] \implies \frac{d[O_2]}{dt} = \frac{1}{2}K_1[N_2O_5] \]
Therefore, we have:
\[ K_3 = \frac{1}{2}K_1 \]

### Step 4: Write the final relationship

Now we can summarize the relationships:

\[ K_2 = 2K_1 \quad \text{and} \quad K_3 = \frac{1}{2}K_1 \]

Thus, the final relationship among \( K_1 \), \( K_2 \), and \( K_3 \) can be expressed as:

\[ K_2 = 2K_1 \quad \text{and} \quad K_3 = \frac{1}{2}K_1 \]