For the reaction,
`N_(2)O_(5) rarr 2NO_(2)+1/2O_(2), Given`
`-(d[N_(2)O_(5)])/(dt)=K_(1)[NO_(2)O_(5)]`
`(d[NO_(2)])/(dt)=K_(2)[N_(2)O_(5)] and (d[O_(2)])/(dt)=K_(3)[N_(2)O_(5)]`
The relation in 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. None of these

Explanation

To solve the problem, we need to derive the relationship between the rate constants \( K_1 \), \( K_2 \), and \( K_3 \) based on the given reaction and the rate expressions.

### Step-by-Step Solution:

1. **Write the balanced chemical equation**:
The reaction is given as:
\[ N_2O_5 \rightarrow 2NO_2 + \frac{1}{2}O_2 \]

2. **Identify the stoichiometric coefficients**:
From the balanced equation, we can see that:
- The stoichiometric coefficient of \( N_2O_5 \) is 1.
- The stoichiometric coefficient of \( NO_2 \) is 2.
- The stoichiometric coefficient of \( O_2 \) is \( \frac{1}{2} \).

3. **Write the rate expressions**:
The rates of change of concentrations are given as:
- For \( N_2O_5 \):
\[ -\frac{d[N_2O_5]}{dt} = K_1 [N_2O_5] \]
- For \( NO_2 \):
\[ \frac{d[NO_2]}{dt} = K_2 [N_2O_5] \]
- For \( O_2 \):
\[ \frac{d[O_2]}{dt} = K_3 [N_2O_5] \]

4. **Relate the rates using stoichiometry**:
According to the stoichiometry of the reaction, we can express the rates in terms of the stoichiometric coefficients:
\[ -\frac{1}{1} \frac{d[N_2O_5]}{dt} = \frac{1}{2} \frac{d[NO_2]}{dt} = \frac{1}{4} \frac{d[O_2]}{dt} \]

5. **Substitute the rate expressions**:
Substitute the expressions for the rates into the stoichiometric relationships:
\[ -K_1 [N_2O_5] = \frac{1}{2} K_2 [N_2O_5] = \frac{1}{4} K_3 [N_2O_5] \]

6. **Cancel \( [N_2O_5] \)** (assuming it is not zero):
This gives us:
\[ -K_1 = \frac{1}{2} K_2 \]
and
\[ -K_1 = \frac{1}{4} K_3 \]

7. **Express \( K_2 \) and \( K_3 \) in terms of \( K_1 \)**:
From \( -K_1 = \frac{1}{2} K_2 \):
\[ K_2 = -2K_1 \]
From \( -K_1 = \frac{1}{4} K_3 \):
\[ K_3 = -4K_1 \]

8. **Final relationship**:
Therefore, we can summarize the relationships as:
\[ K_2 = 2K_1 \quad \text{and} \quad K_3 = \frac{1}{4} K_1 \]

### Conclusion:
The relationship between the rate constants is:
\[ 2K_1 = K_2 \quad \text{and} \quad K_3 = \frac{1}{4} K_1 \]