First Order Half-Life Equation
The half-life (\(t_{1/2}\)) of a first-order reaction is strictly constant and independent of initial concentration, related directly to the rate constant via \(t_{1/2} = \dfrac{\ln 2}{k} \approx \dfrac{0.693}{k}\).
What Is Half-Life?
The half-life (t½) is the time required for the concentration of a reactant to decrease to exactly half of its current value.
The Key Equation
This equation applies only to first-order reactions. Notice that t½ depends only on k, not on [A]₀. This means every successive half-life has the same duration.
First-Order Decay with Constant Half-Life
Worked Example
Calculating First-Order Half-Life
Problem: The radioactive decay of Iodine-131 (a first-order process) has a rate constant \(k = 0.138\text{ days}^{-1}\). Calculate its half-life (\(t_{1/2}\)).
Calculation:
\(t_{1/2} = \dfrac{\ln 2}{k} = \dfrac{0.693}{0.138} = \mathbf{5.02\text{ days}}\)
Graphically Measuring First Order Half-Life
To demonstrate first order kinetics from a concentration-time graph in an exam: explicitly mark and label the time taken for \([A]\) to fall from \(1.0 ightarrow 0.5 ext{ M}\), then from \(0.5 ightarrow 0.25 ext{ M}\). Equal intervals confirm first order.
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