To calculate the rate constant (k) for a first-order reaction, we can use the following formula derived from the half-life (t_1/2):

k = ln(2) / t_1/2

Where:

• k is the rate constant
• ln(2) is the natural logarithm of 2, approximately 0.693
• t_1/2 is the half-life of the reaction

Given that the half-life (t_1/2) for the decomposition of P₂O₅ at 400°C is 346 minutes, we can substitute this value into the formula:

k = 0.693 / 346 minutes

k ≈ 0.002003 minutes^-1

Therefore, the rate constant for the first-order reaction of P₂O₅ at 400°C is approximately 0.002003 min^-1.