For a generic order chemical reaction, the half-life evaluates strictly to when applied specifically — Chemical Kinetics Chemistry Question
Question
For a generic $n^{\text{th}}$ order chemical reaction, the half-life $t_{1/2}$ evaluates strictly to $Y / (2 k a^2)$ when applied specifically to a third-order reaction ($n=3$). Using the integrated constraints for non-first order kinetics, deduce the exact integer value of $Y$.
Answer: 3
💡 Solution & Explanation
The general non-first order half-life formula is $t_{1/2} = \frac{2^{n-1} - 1}{k(n-1)a^{n-1}}$. Substituting $n=3$ for a third-order reaction: $t_{1/2} = \frac{2^{3-1} - 1}{k(3-1)a^{3-1}} = \frac{2^2 - 1}{k(2)a^2} = \frac{4-1}{2ka^2} = \frac{3}{2ka^2}$. Thus, the numerator constant $Y$ is exactly $3$.
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