For the decomposition of `N_(2)O_(5)((g))`, it is given that:
`2N_(2)O_(5g)rarr 4NO_(2(g))+O_(2(g))`,
Activation energy `E'_(a)` then ,
`E_(a), N_(2)O_(5(g))rarr 2NO_(2(g))+1//2O_(2(g))`,
Activation energy `E_(a)^(')` then:

Options

A. `E_(a)=E'_(a)`
B. `E_(a) gt E'_(a)`
C. `E_(a) lt E'_(a)`
D. `E_(a)=2E'_(a)`

Explanation

To solve the problem regarding the activation energies for the decomposition of \( N_2O_5 \), we need to analyze the two reactions provided and their respective activation energies.

### Step-by-Step Solution:

1. **Identify the Reactions**:
- The first reaction is:
\[ 2N_2O_5(g) \rightarrow 4NO_2(g) + O_2(g) \]
with activation energy \( E_a' \).
- The second reaction is:
\[ N_2O_5(g) \rightarrow 2NO_2(g) + \frac{1}{2}O_2(g) \]
with activation energy \( E_a \).

2. **Understand Activation Energy**:
- Activation energy (\( E_a \)) is the minimum energy required for a reaction to occur. It is a characteristic of the reaction and is influenced by the reaction mechanism, not by the stoichiometric coefficients of the reactants or products.

3. **Relate the Two Reactions**:
- The second reaction is essentially half of the first reaction. If we divide the first reaction by 2, we get:
\[ N_2O_5(g) \rightarrow 2NO_2(g) + \frac{1}{2}O_2(g) \]
- However, the activation energy remains unchanged because it is a property of the reaction mechanism and not dependent on the stoichiometric coefficients.

4. **Conclusion**:
- Since the activation energy does not depend on the stoichiometric coefficients, we can conclude that:
\[ E_a = E_a' \]

### Final Answer:
The activation energy \( E_a \) for the reaction \( N_2O_5(g) \rightarrow 2NO_2(g) + \frac{1}{2}O_2(g) \) is equal to the activation energy \( E_a' \) for the reaction \( 2N_2O_5(g) \rightarrow 4NO_2(g) + O_2(g) \).