The decomposition of `Cl_2O_7" at "` 400 K in the gas phase to `Cl_2 and O_2` is a first order reaction.
After 50 seconds at 400 K, the pressure of `Cl_2O_7` falls from 0.062 to 0.044 atm. Calculate the rate constant.

Explanation

To solve the problem of finding the rate constant for the decomposition of \( Cl_2O_7 \) at \( 400 \, K \), we will use the first-order reaction rate equation. Here are the steps to derive the solution:

### Step 1: Identify the initial and final pressures
- The initial pressure of \( Cl_2O_7 \) at \( t = 0 \) is \( P_0 = 0.062 \, \text{atm} \).
- The final pressure after \( 50 \, \text{s} \) is \( P_t = 0.044 \, \text{atm} \).

### Step 2: Apply the first-order rate constant formula
For a first-order reaction, the rate constant \( k \) can be calculated using the formula:
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