The rate of reaction.
`2N_(2)O_(5) to 4NO_(2) + O_(2)`
can be written in three ways.
`(-d[N_(2)O_(5)])/(dt) = k[N_(2)O_(5)]`
`(d[N_(2)O_(5)])/(dt) =( k^(')[N_(2)O_(5)])`
`(d[O_(2)])/(dt) = (k^(')[N_(2)O_(5)])`
The relation between k and `k^(')` are:

Options

A. `k^(') = 2k, k^(')=k`
B. `k^(') = 2k, k^('')=k/2`
C. `k^(') = 2k, k^('')`
D. `k^(')=k,k^('')=k`

Explanation

To find the relationship between the rate constants \( k \) and \( k' \) for the reaction

\[ 2N_{2}O_{5} \rightarrow 4NO_{2} + O_{2}, \]

we need to analyze the rate expressions given in the question.

### Step-by-Step Solution:

1. **Identify the Rate Expressions**:
The rate of reaction can be expressed in terms of the change in concentration of reactants and products. The three expressions provided are:
- \(-\frac{d[N_{2}O_{5}]}{dt} = k[N_{2}O_{5}]\)
- \(\frac{d[N_{2}O_{5}]}{dt} = k'[N_{2}O_{5}]\)
- \(\frac{d[O_{2}]}{dt} = k'[N_{2}O_{5}]\)

2. **Understanding the Stoichiometry**:
From the balanced equation, we see that:
- 2 moles of \( N_{2}O_{5} \) decompose to produce 4 moles of \( NO_{2} \) and 1 mole of \( O_{2} \).
- This gives us the stoichiometric coefficients: for every 2 moles of \( N_{2}O_{5} \) that react, 4 moles of \( NO_{2} \) and 1 mole of \( O_{2} \) are produced.

3. **Relating the Rates**:
Using the stoichiometry of the reaction, we can relate the rates of change of concentrations:
\[ -\frac{1}{2} \frac{d[N_{2}O_{5}]}{dt} = \frac{1}{4} \frac{d[NO_{2}]}{dt} = \frac{1}{2} \frac{d[O_{2}]}{dt} \]

4. **Expressing the Rates**:
From the first expression, we can express the rate of disappearance of \( N_{2}O_{5} \):
\[ -\frac{d[N_{2}O_{5}]}{dt} = 2k[N_{2}O_{5}] \]
Therefore, we can write:
\[ \frac{d[NO_{2}]}{dt} = 4k[N_{2}O_{5}] \]
and
\[ \frac{d[O_{2}]}{dt} = 2k[N_{2}O_{5}] \]

5. **Relating \( k \) and \( k' \)**:
From the expressions for \( \frac{d[NO_{2}]}{dt} \) and \( \frac{d[O_{2}]}{dt} \), we can identify:
\[ k' = 2k \quad \text{(for } NO_{2}\text{)} \]
and
\[ k' = k \quad \text{(for } O_{2}\text{)} \]

6. **Final Relationship**:
Thus, we conclude that:
\[ k' = 2k \quad \text{and} \quad k'' = \frac{1}{2}k \]

### Summary of Relationships:
- For \( NO_{2} \): \( k' = 2k \)
- For \( O_{2} \): \( k'' = \frac{1}{2}k \)