Half-life a reaction is found to be inversely proportional to the cube of its initial concertration. — Chemical Kinetics Chemistry Question

💡 Solution & Explanation

# Determining Reaction Order from Half-life Dependence **Given:** Half-life is inversely proportional to the cube of initial concentration $$t_{1/2} \propto \frac{1}{[A_0]^3}$$ or $$t_{1/2} = \frac{k}{[A_0]^3}$$ **Step 1: Recall the general half-life formula** For a reaction of order $n$: $$t_{1/2} = \frac{2^{n-1} - 1}{k(n-1)[A_0]^{n-1}}$$ For $n \neq 1$, this simplifies to: $$t_{1/2} \propto \frac{1}{[A_0]^{n-1}}$$ **Step 2: Match the exponent** From the given relationship: $t_{1/2} \propto [A_0]^{-3}$ Comparing with the general form: $[A_0]^{-(n-1)} = [A_0]^{-3}$ $$n - 1 = 3$$ $$n = 4$$ **Answer: The reaction is of order 4 (fourth order).** The inverse cube dependence indicates that each of the three concentration terms in the rate law contributes to the half-life, making this a **fourth-order reaction**.