Overview
This lecture explains the concept of half-life in chemical kinetics, emphasizing its calculation and significance for first-order reactions, with examples and comparisons to zero- and second-order reactions.
Definition of Half-Life
- Half-life is the time for the concentration of a reactant to decrease to half its initial value.
- For first-order reactions, half-life is independent of the initial concentration.
- Half-life is used to measure how fast a reaction progresses.
Mathematical Proof and Formula
- The integrated rate law for first-order reactions: ln[A]_t = ln[A]_0 - k·t.
- Setting [A]_t = 0.5[A]_0 and solving for t gives t_1/2 = (ln 2)/k or 0.693/k.
- Higher rate constants result in shorter half-lives.
Graphical Interpretation
- Plotting ln[reactant] vs. time yields a straight line, with each half-life period reducing concentration by half.
- After n half-lives, [A]_t = [A]_0 × (1/2)^n.
- After one half-life: 1/2 initial; two: 1/4; three: 1/8 remaining.
Comparison of Reaction Orders
- First-order: half-life is independent of concentration; t_1/2 = (ln 2)/k.
- Zero-order: half-life is directly proportional to initial concentration.
- Second-order: half-life is inversely proportional to initial concentration.
Example Calculations
- Given k = 9.2 s⁻¹ for a first-order reaction: t_1/2 = 0.693/9.2 = 0.075 s (2 sig figs).
- Time for [A] to reach 1/10 initial: Use ln([A]_t/[A]_0) = -kt; solve to find t = 0.25 s (2 sig figs).
Key Terms & Definitions
- Half-life (t_1/2) — Time required for a reactant's concentration to halve.
- Rate constant (k) — Proportionality factor in the rate law, determines reaction speed.
- First-order reaction — Reaction rate proportional to reactant’s concentration; half-life is independent of concentration.
Action Items / Next Steps
- Review and memorize the half-life formulas for different reaction orders.
- Practice solving half-life problems using the integrated rate law.
- Understand how to apply significant figures in calculations.