4.4 Half-life and its applications

Half-life is a crucial concept in chemical kinetics, measuring the time it takes for a reactant's concentration to halve. For first-order reactions, it's constant and independent of initial concentration, making it a handy tool for predicting reaction progress.

Understanding half-life helps us grasp how quickly reactions occur and how long substances persist. It's super useful in real-world applications like drug elimination in the body and radioactive decay, helping us make sense of complex chemical processes.

Half-Life and Its Applications

Definition and significance of half-life

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• Time required for reactant concentration to decrease to half its initial value
• Measures reaction rate and remains constant for first-order reactions
  • Shorter half-life indicates faster reaction rate (radioactive decay of uranium-235)
  • Longer half-life indicates slower reaction rate (decomposition of sucrose in solution)
• Independent of initial concentration for first-order reactions
• Crucial for predicting reaction progress and determining time required for reactant to reach specific concentration (drug elimination in the body)

Half-life vs rate constant relationship

• First-order reaction rate law: $\frac{-d[A]}{dt} = k[A]$
  • $k$: rate constant
  • $[A]$: reactant A concentration
• Integrated rate law: $\ln\frac{[A]_t}{[A]_0} = -kt$
  • $[A]_0$: initial concentration of A
  • $[A]_t$: concentration of A at time $t$
• At half-life, $[A]_t = \frac{1}{2}[A]_0$, substituting into integrated rate law:
  • $\ln\frac{\frac{1}{2}[A]_0}{[A]_0} = -kt_{1/2}$
  • $\ln\frac{1}{2} = -kt_{1/2}$
• Solving for $t_{1/2}$: $t_{1/2} = \frac{\ln 2}{k}$
  • Equation relates half-life to rate constant for first-order reactions (decomposition of hydrogen peroxide)

Calculation of first-order reaction half-life

• Using equation $t_{1/2} = \frac{\ln 2}{k}$, calculate half-life if rate constant is known
  • Example: $k = 0.05 \text{ s}^{-1}$, then $t_{1/2} = \frac{\ln 2}{0.05 \text{ s}^{-1}} \approx 13.9 \text{ s}$
• Half-life independent of initial concentration for first-order reactions
  • Changing initial concentration does not affect half-life (decomposition of nitrogen pentoxide)
• If initial concentration and concentration at specific time are known, calculate half-life using integrated rate law
  • Example: $[A]_0 = 1.0 \text{ M}$, $[A]_t = 0.25 \text{ M}$ at $t = 20 \text{ s}$, then $\ln\frac{0.25 \text{ M}}{1.0 \text{ M}} = -k(20 \text{ s})$, solving for $k$ and using $t_{1/2} = \frac{\ln 2}{k}$ gives $t_{1/2} \approx 13.9 \text{ s}$

Applications of half-life concept

• Radioactive decay follows first-order kinetics, half-life is time required for half of original amount of isotope to decay
  • Amount of radioisotope remaining after certain number of half-lives: $A_t = A_0(\frac{1}{2})^n$
    • $A_0$: initial amount of radioisotope
    • $A_t$: amount remaining after time $t$
    • $n$: number of half-lives elapsed (carbon-14 dating)
• Drug elimination in body often follows first-order kinetics, half-life is time required for drug concentration in body to decrease by half
  • Elimination rate constant determined from half-life: $k = \frac{\ln 2}{t_{1/2}}$
  • Drug concentration in body after certain time calculated using integrated rate law: $[D]_t = [D]_0e^{-kt}$
    • $[D]_0$: initial drug concentration
    • $[D]_t$: drug concentration at time $t$ (caffeine metabolism)