Consider the decomposition of `N_2O_5` as `N_2O_5rarr2NO_2+1/2O_2` The rate of reaction is given by `(-d[N_2O_5])/(dt)=1/2(d[NO_2])/(dt)=2(d[O_2])/(dt)=k_1[N_2O_5]" Therefore,"(-d[N_2O_5])/(dt)=k_1[N_2O_5] , (+d[NO_2])/(dt)=2k_1[N_2O_5]=k_1^(')[N_2O_5],(d[O_2])/(dt)=1/2k_1[N_2O_5]=k_1^('')[N_2O_5]` Choose the correct option.

Options

A. `4k_1=2k_1^(')=k_(1)^('')`
B. `4k_1=k_1^(')=2k_(1)^('')`
C. `k_1=2k_1^(')=k_(1)^('')`
D. `2k_1=k_1^(')=4k_(1)^('')`

Answer

To solve the problem regarding the decomposition of \( N_2O_5 \) given by the reaction:

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

we start by analyzing the rate of the reaction. The rate of reaction can be expressed in terms of the change in concentration of the reactants and products over time.

### Step-by-Step Solution:

1. **Write the Rate Expressions**:
The rate of the reaction can be expressed as:
\[ -\frac{d[N_2O_5]}{dt} = k_1[N_2O_5] \]
This indicates that the rate of decrease of \( N_2O_5 \) is proportional to its concentration.

2. **Relate the Rates of Products**:
From the stoichiometry of the reaction, we can relate the rates of formation of the products:
\[ \frac{d[NO_2]}{dt} = 2 \left(-\frac{d[N_2O_5]}{dt}\right) = 2k_1[N_2O_5] \]
\[ \frac{d[O_2]}{dt} = \frac{1}{2} \left(-\frac{d[N_2O_5]}{dt}\right) = \frac{1}{2}k_1[N_2O_5] \]

3. **Introduce New Rate Constants**:
We can define new rate constants for the products:
\[ \frac{d[NO_2]}{dt} = k_1' [N_2O_5] \quad \text{where } k_1' = 2k_1 \]
\[ \frac{d[O_2]}{dt} = k_1'' [N_2O_5] \quad \text{where } k_1'' = \frac{1}{2}k_1 \]

4. **Establish Relationships Between Rate Constants**:
From the definitions of \( k_1' \) and \( k_1'' \):
\[ k_1' = 2k_1 \]
\[ k_1'' = \frac{1}{2}k_1 \]

5. **Compare Rate Constants**:
We can compare these constants:
\[ k_1' = 2k_1 \quad \text{and} \quad k_1'' = \frac{1}{2}k_1 \]

6. **Final Relationships**:
From the above relationships, we can summarize:
\[ k_1' = 2k_1, \quad k_1'' = \frac{1}{2}k_1 \]

### Conclusion:
The relationships we derived show how the rate constants for the products relate to the rate constant of the reactant. The correct option based on the relationships derived is:

**Option 4**: \( k_1' = 2k_1 \) and \( k_1'' = \frac{1}{2}k_1 \).