Concept:The mathematical relationship between half-life (\( t_{1/2} \)) and the initial concentration (\( a \)) is unique to each reaction order.
Formula:$$ t_{1/2} \propto \frac{1}{a^{n-1}} $$
Solution:- We are given the expression: \( t_{1/2} = \frac{1}{ka} \).
- This shows that the half-life is inversely proportional to the first power of the initial concentration (\( a^1 \)).
- Using the general proportionality formula, if \( a^{n-1} = a^1 \), then \( n - 1 = 1 \).
- Solving for \( n \) yields \( n = 2 \).
- Therefore, this formula strictly governs a second-order reaction.
Why other options are incorrect:Zero order: \( t_{1/2} = a / 2k \). First order: \( t_{1/2} = 0.693 / k \) (independent of \( a \)). Third order: \( t_{1/2} \propto 1/a^2 \).
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