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 \(N_2O_5\) into \(NO_2\) and \(O_2\), we can follow these steps:

### Step 1: Write the balanced chemical equation
The decomposition of \(N_2O_5\) can be represented as:
\[ N_2O_5 \rightarrow 2NO_2 + \frac{1}{2}O_2 \]

### Step 2: Define the rate of reaction
The rate of reaction can be expressed in terms of the change in concentration of the reactants and products:
\[ -\frac{d[N_2O_5]}{dt} = \frac{1}{2}\frac{d[NO_2]}{dt} = 2\frac{d[O_2]}{dt} = k_1[N_2O_5] \]

### Step 3: Express the rates in terms of \(k_1\)
From the rate expressions, we can derive the following relationships:
1. For \(N_2O_5\):
\[ -\frac{d[N_2O_5]}{dt} = k_1[N_2O_5] \]
2. For \(NO_2\):
\[ \frac{d[NO_2]}{dt} = 2k_1[N_2O_5] \quad \text{(let's denote this as } k_1' [N_2O_5] \text{)} \]
Thus, we have:
\[ k_1' = 2k_1 \]
3. For \(O_2\):
\[ \frac{d[O_2]}{dt} = \frac{1}{2}k_1[N_2O_5] \quad \text{(let's denote this as } k_1'' [N_2O_5] \text{)} \]
Thus, we have:
\[ k_1'' = \frac{1}{2}k_1 \]

### Step 4: Relate the rate constants
From the derived equations, we have:
- \(k_1' = 2k_1\)
- \(k_1'' = \frac{1}{2}k_1\)

### Step 5: Check the relationships
To find a relationship between \(k_1\), \(k_1'\), and \(k_1''\):
1. From \(k_1' = 2k_1\), we can express \(k_1\) in terms of \(k_1'\):
\[ k_1 = \frac{1}{2}k_1' \]
2. From \(k_1'' = \frac{1}{2}k_1\), we can express \(k_1\) in terms of \(k_1''\):
\[ k_1 = 2k_1'' \]

### Step 6: Final relationship
Combining these relationships, we can express:
\[ 2k_1 = k_1' \quad \text{and} \quad k_1 = 4k_1'' \]
Thus, we conclude:
\[ 2k_1 = k_1' = 4k_1'' \]

### Final Answer
The correct relationship is:
\[ 2k_1 = k_1' = 4k_1'' \]