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.
Understanding Chemistry: Half-Life of a Reaction
A reaction slows down or keeps a steady pace depending on what controls its rate. In a first-order reaction, the rate depends on how much reactant is present. As the amount falls, fewer reactant particles are available to change each second.
The reaction therefore becomes slower in a very regular pattern. Each successive halving takes the same amount of time.
This does not mean the same amount of material disappears each time. The first interval removes half of the original amount, while the next interval removes only one quarter of the original amount.
Other reaction orders behave differently because their rates respond differently to concentration. A zero-order reaction can proceed at a nearly constant rate for a time. This can happen when a catalyst surface is fully occupied or when light supplies energy at a fixed rate.
Since the same concentration is removed in each equal time interval, later halvings take less time than earlier ones. A second-order reaction usually depends strongly on particle encounters. When concentration is high, collisions are common and the reaction is fast.
As concentration decreases, useful collisions become much less common. Later halvings then take longer. The changing pattern of half-lives gives chemists evidence about the reaction order.
Scientists measure concentration at several times during an experiment. They may use color, gas pressure, electrical conductivity, mass, or a chemical test that detects the reactant. The useful graph depends on the suspected reaction order.
A straight line can be obtained by plotting concentration for a zero-order process, the natural logarithm of concentration for a first-order process, or one divided by concentration for a second-order process. The slope of that line gives the rate constant.
Its units matter because they change with reaction order. Checking units is a simple way to catch errors in calculations.
Half-life ideas appear whenever a substance must remain effective or be removed safely. A medicine can lose active chemical over time, so dose schedules depend on how quickly the body processes it. Pollutants in water or soil may break down through chemical reactions, though sunlight, microbes, temperature, and mixing can make the real situation more complicated than a classroom model.
Food storage involves reactions that cause oxidation and spoilage. When solving problems, identify the reactant being tracked, the reaction order, the starting concentration, and the conditions. Temperature and catalysts can change the rate constant, so a half-life measured in one setting may not apply in another.
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.