The decomposition of `N_(2)O` into `N_(2)` and `O_(2)` in presence of gaseous argon follows second order kinetics with rate constant, `K=5.0 xx10^(11) e^(-30000K//T)mol^(-1)s^(-1)`.
The enrgy of activation are respectively `(R=2 cal K^(-1) mol^(-1))`

Options

A. `2.0xx10^(8)L mol^(-1)s^(-1),30Kcal mol^(-1)`
B. `2.0xx10^(8)L mol^(-1)s^(-1),60Kcal mol^(-1)`
C. `2.0xx10^(11)L mol^(-1)s^(-1),60Kcal mol^(-1)`
D. None of these

Explanation

To solve the problem, we need to determine the activation energy (Ea) for the decomposition of N₂O into N₂ and O₂, given the rate constant (K) and the universal gas constant (R). The rate constant is expressed in the Arrhenius equation form.

### Step-by-Step Solution:

1. **Identify the given data:**
- Rate constant, \( K = 5.0 \times 10^{11} e^{-\frac{30000}{T}} \, \text{mol}^{-1} \text{s}^{-1} \)
- Universal gas constant, \( R = 2 \, \text{cal K}^{-1} \text{mol}^{-1} \)
- The term in the exponent, \( -\frac{30000}{T} \), is related to the activation energy.

2. **Relate the given rate constant to the Arrhenius equation:**
The Arrhenius equation is given by:
\[ K = A e^{-\frac{E_a}{RT}} \]
where:
- \( K \) is the rate constant,
- \( A \) is the pre-exponential factor,
- \( E_a \) is the activation energy,
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin.

3. **Extract the activation energy from the exponent:**
From the form of the rate constant given, we can see that:
\[ -\frac{E_a}{R} = -\frac{30000}{T} \]
This implies:
\[ E_a = 30000 \, \text{cal/mol} \]

4. **Convert the activation energy to the appropriate units:**
Since \( R \) is given in cal, we can directly use the value of \( E_a \):
\[ E_a = 30000 \, \text{cal/mol} \]

5. **Final calculation of activation energy:**
To express \( E_a \) in a more standard form, we can convert it to kilojoules per mole if needed:
\[ E_a = 30000 \, \text{cal/mol} \times \frac{4.184 \, \text{J}}{1 \, \text{cal}} \times \frac{1 \, \text{kJ}}{1000 \, \text{J}} = 125.52 \, \text{kJ/mol} \]
However, since the question does not specify the need for conversion and asks for the activation energy directly, we can state:
\[ E_a = 30000 \, \text{cal/mol} \]

6. **Conclusion:**
The activation energy for the decomposition of N₂O is \( 30000 \, \text{cal/mol} \).