The half-life of a reaction is the time required for the concentration of a reactant to decrease to half of its current value. It is a useful way to describe how fast a reactant is being consumed without tracking the entire reaction from start to finish. On a concentration versus time graph, each half-life marks a drop from [A]0 to [A]0/2, then [A]0/4, then [A]0/8.
This idea is especially important in chemical kinetics, medicine, environmental chemistry, and radioactive decay models.
Key Facts
- Half-life means [A] changes from [A]initial to [A]initial/2.
- For a first-order reaction, [A] = [A]0e^(-kt).
- For a first-order reaction, t1/2 = ln(2)/k = 0.693/k.
- For a zero-order reaction, [A] = [A]0 - kt and t1/2 = [A]0/(2k).
- For a second-order reaction in one reactant, 1/[A] = 1/[A]0 + kt and t1/2 = 1/(k[A]0).
- Only first-order half-life is independent of starting concentration.
Vocabulary
- Half-life
- The half-life of a reaction is the time it takes for a reactant concentration to fall to one half of its current value.
- Reaction order
- Reaction order describes how the reaction rate depends on the concentration of one or more reactants.
- Rate constant
- The rate constant k is the proportionality constant in a rate law and depends on the reaction and temperature.
- First-order reaction
- A first-order reaction has a rate directly proportional to the concentration of one reactant, so rate = k[A].
- Integrated rate law
- An integrated rate law relates reactant concentration to time and is used to calculate concentrations or half-lives.
Common Mistakes to Avoid
- Assuming every reaction has a constant half-life is wrong because only first-order reactions have half-lives that do not depend on starting concentration.
- Using t1/2 = 0.693/k for every reaction order is wrong because that formula applies only to first-order kinetics.
- Forgetting the units of k is wrong because the units of the rate constant change with reaction order, such as s^-1 for first order and M^-1 s^-1 for second order.
- Reading equal vertical drops as equal half-lives is wrong because half-life is based on halving the concentration, not subtracting the same concentration amount each time.
Practice Questions
- 1 A first-order reaction has k = 0.0250 s^-1. Calculate its half-life in seconds.
- 2 A zero-order reaction has [A]0 = 0.800 M and k = 0.0400 M/s. Calculate the time needed for [A] to fall to 0.400 M.
- 3 Two reactions both start at 1.00 M. Reaction X has the same half-life after each halving, while Reaction Y has a shorter half-life when the starting concentration is higher. Identify which reaction is first order and explain your reasoning.