The decomposition of `N_(2)O_(5)` according to the equation: `2N_(2)O_(5)(g) to 4NO_(2)(g) + O_(2)(g)` is a first order reaction. After 30 min. from the start of the decomposition in a closed vessel, the total pressure developed is found to be 284.5 mm of Hg and on complete decomposition, the total pressure is 584.5 mm of Hg. Calculate the rate constant for the reaction.

Explanation

To solve the problem step by step, we will follow the information provided in the question and the video transcript.

### Step 1: Determine the Initial Pressure of N2O5
The total pressure at complete decomposition is given as 584.5 mm of Hg. The reaction shows that 2 moles of N2O5 produce 4 moles of NO2 and 1 mole of O2, which gives a total of 5 moles of products.

Using the stoichiometry of the reaction:
- Initial moles of N2O5 = 2
- Moles of products = 4 (NO2) + 1 (O2) = 5
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