For the reaction `2N_(2) O_(5) (g) overset(k)rarr 4NO_(2) (g) + O_(2) (g)` having rate law. Rate =`k [N_(2)O_(5)]=R_(1)`
Select correct statement.

Options

A. Rate constant of the reaction would be 0.25L `"mole"^(-1) sec^(-1)`
B. Rate of disappearance of `N_(2)O_(5) (g) = R_(1)`
C. Rate of formation of `NO_(2) (g)= 4R_(1)`
D. Arrhenius constant for the reaction is `10^(6)` mole `L^(-1) "min"^(-1)`

Explanation

To solve the question regarding the reaction

\[ 2N_2O_5 (g) \overset{k}{\rightarrow} 4NO_2 (g) + O_2 (g) \]

with the rate law

\[ \text{Rate} = k [N_2O_5] = R_1, \]

we need to analyze the statements provided and determine which one is correct.

### Step-by-Step Solution:

1. **Understanding the Rate Law**:
The rate law indicates that the rate of the reaction depends on the concentration of \(N_2O_5\). The rate constant \(k\) is a proportionality constant that relates the rate of the reaction to the concentration of the reactants.

2. **Identifying the Order of the Reaction**:
Since the rate law is given as \( \text{Rate} = k [N_2O_5] \), this indicates that the reaction is first order with respect to \(N_2O_5\). The overall order of the reaction is 1.

3. **Rate of Disappearance of \(N_2O_5\)**:
The rate of disappearance of a reactant is given by the negative change in concentration over time. For \(N_2O_5\), this can be expressed as:
\[ -\frac{d[N_2O_5]}{dt} = R_1 \]
Thus, the rate of disappearance of \(N_2O_5\) is equal to \(R_1\) but with a negative sign, indicating a decrease in concentration.

4. **Rate of Formation of \(NO_2\)**:
The stoichiometry of the reaction shows that for every 2 moles of \(N_2O_5\) that react, 4 moles of \(NO_2\) are formed. Therefore, the rate of formation of \(NO_2\) can be expressed as:
\[ \frac{d[NO_2]}{dt} = \frac{4}{2} \cdot R_1 = 2R_1 \]
However, since we need to account for the stoichiometric coefficients correctly, we find:
\[ \frac{d[NO_2]}{dt} = 4R_1 \]
This means that the rate of formation of \(NO_2\) is indeed \(4R_1\).

5. **Evaluating the Arrhenius Constant**:
The Arrhenius constant (or pre-exponential factor) is not provided in the problem statement, and without specific data, we cannot determine its value. Thus, any statement regarding the Arrhenius constant would be incorrect.

### Conclusion:
Based on the analysis, the correct statement is that the rate of formation of \(NO_2\) is equal to \(4R_1\).