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

💡 Solution & Explanation

# Finding the Order of Reaction from Half-Life Relationship **Given:** Half-life is inversely proportional to the cube of initial concentration $$t_{1/2} \propto \frac{1}{[A_0]^3}$$ **Step 1: Write the general half-life formula** For a reaction of order $n$: $$t_{1/2} = \frac{K}{[A_0]^{n-1}}$$ where $K$ is a constant that depends on the rate constant $k$ and order $n$. **Step 2: Compare with the given relationship** From the given condition: $$t_{1/2} \propto \frac{1}{[A_0]^3}$$ This means: $$t_{1/2} = \frac{\text{constant}}{[A_0]^3}$$ **Step 3: Match the exponents** Comparing the exponent of $[A_0]$ in both expressions: $$n - 1 = 3$$ $$n = 4$$ **Conclusion:** The reaction is **fourth order (or 4th order)**, so the correct answer is **D**. *This relationship is characteristic of fourth-order reactions, where the half-life depends strongly on the initial concentration.*