Second Order Half-life

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Half-life for second-order (A → products) with rate = k[A]² is:

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1. **Integrated Rate Law (Second Order)**: $\frac{1}{[A]} - \frac{1}{[A]_0} = kt$.
2. **Substitute Half-life**: At $t = t_{1/2}, [A] = \frac{[A]_0}{2}$.
3. **Calculation**:
   - $\frac{2}{[A]_0} - \frac{1}{[A]_0} = k t_{1/2}$
   - $\frac{1}{[A]_0} = k t_{1/2}$
4. **Result**: **$t_{1/2} = \frac{1}{k[A]_0}$**.
5. **Key Observation**: For second-order reactions, the half-life is **inversely proportional** to the initial concentration.

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Second Order Half-life

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