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

💡 Solution & Explanation

# Solution: Finding Order of Reaction from Half-life Relationship **Given:** Half-life $t_{1/2}$ is inversely proportional to the cube of initial concentration $[A]_0$ $$t_{1/2} \propto \frac{1}{[A]_0^3}$$ **Step 1: Recall the general half-life formula** For a reaction of order $n$: $$t_{1/2} = \frac{k}{(n-1)[A]_0^{n-1}}$$ (This comes from integrating the rate law and solving for time when $[A] = [A]_0/2$) **Step 2: Compare the given relationship with the formula** From the given condition: $$t_{1/2} \propto [A]_0^{-3}$$ From the general formula: $$t_{1/2} \propto [A]_0^{-(n-1)}$$ **Step 3: Equate the exponents** $$-(n-1) = -3$$ $$n-1 = 3$$ $$n = 4$$ **Answer: The reaction is **4th order** (Option D)** This makes sense because higher-order reactions have stronger concentration dependence of half-life—the half-life decreases dramatically as initial concentration increases.