Answer:-

i. For the gas-phase reaction:

A(g) → B(g) + C(g)

Let the initial pressure of A be P_i, which decreases by x within time t.

ii. Pressure of reactant A at time t:

P_A = P_i - x ....(1)

The pressures of the products B and C at time t:

P_B = P_C = x

iii. The total pressure at time t is then:

P = P_i - x + x + x = P_i + x

Hence, x = P - P_i ...(2)

The pressure of A at time t is obtained by substituting Eq. (1) into Eq. (2).

Thus:

P_A = P_i - (P - P_i) = P_i - P + P_i = 2P_i - P

iv. The integrated rate law turns out to be:

k = 2.303/t log₁₀([A]₀/[A]_t)

The concentration is now expressed in terms of pressures:

Thus, [A]₀ = P_i and [A]_t = P_A = 2P_i - P

Substitution gives in the above equation:

k = 2.303/t log₁₀(P_i / (2P_i - P)) ...(3)

Here, P is the total pressure of the reaction mixture at time t.