In the decompoistion of `N_(2)O_(5)(g)`,
`2N_(2)O_(5)(g)overset(k_(obs))rarr4NO_(2)(g) + O_(2)(g)`,
the observed rate law is given by
`(d[O_(2)])/(dt) = k_(obs)[N_(2)O_(5)]`.
Which of the following proposed mechanics is conisstent with the rate law?
a. `N_(2)O_(5)(g)overset(k)rarrN_(2)O_(3)(g) + O_(2)(g)`
b. `N_(2)(g) + NO_(3)(g)overset(k_(1))rarrNO(g) + NO_(2)(g)+O_(2)(g)`

Explanation

To solve the problem, we need to analyze the proposed mechanisms and see which one is consistent with the given rate law for the decomposition of \( N_2O_5 \).

### Step-by-Step Solution:

1. **Understand the Given Reaction and Rate Law**:
The decomposition reaction is:
\[ 2N_2O_5(g) \overset{k_{obs}}{\rightarrow} 4NO_2(g) + O_2(g) \]
The observed rate law is:
\[ \frac{d[O_2]}{dt} = k_{obs}[N_2O_5] \]
This indicates that the rate of formation of \( O_2 \) is directly proportional to the concentration of \( N_2O_5 \).

2. **Analyze Proposed Mechanism (a)**:
The first proposed mechanism is:
\[ N_2O_5(g) \overset{k}{\rightarrow} N_2O_3(g) + O_2(g) \]
- For this mechanism, the rate law can be expressed as:
\[ \text{Rate} = k[N_2O_5] \]
- This matches the form of the observed rate law, where the rate of formation of \( O_2 \) is proportional to the concentration of \( N_2O_5 \). Thus, this mechanism is consistent with the given rate law.

3. **Analyze Proposed Mechanism (b)**:
The second proposed mechanism is:
\[ N_2(g) + NO_3(g) \overset{k_1}{\rightarrow} NO(g) + NO_2(g) + O_2(g) \]
- For this mechanism, the rate law can be expressed as:
\[ \text{Rate} = k_1[N_2][NO_3] \]
- Here, the rate depends on the concentrations of \( N_2 \) and \( NO_3 \), which does not match the observed rate law that depends solely on \( [N_2O_5] \). Therefore, this mechanism is not consistent with the given rate law.

4. **Conclusion**:
Based on the analysis, the correct answer is:
- **Option (a)** is consistent with the observed rate law.