Finding half-life in chemistry means calculating how long it takes for a reactant’s concentration to drop to half its starting value. The formula you need depends on the reaction order: zero, first, or second. Each order has its own equation, and each one comes from the same basic algebraic process of plugging “half the starting concentration” into the integrated rate law.
What Half-Life Means in Chemistry
Half-life (written as t₁/₂) is the time required for the concentration of a reactant to fall to 50% of its initial value. It’s a useful shorthand for describing how fast a reaction proceeds. In a slow reaction, the half-life might be hours or days; in a fast one, milliseconds.
The key insight is that half-life behaves differently depending on the reaction order. For a first-order reaction, the half-life stays constant no matter how much reactant is left. For a zero-order reaction, the half-life gets shorter as concentration decreases. For a second-order reaction, the half-life gets longer as concentration decreases. These patterns are how you can identify reaction order from experimental data, even without knowing the rate constant.
The General Method for Any Reaction Order
Every half-life formula comes from the same three-step process:
- Start with the integrated rate law for the reaction order you’re working with.
- Substitute [A] = ½[A]₀ (half the initial concentration) for the concentration term, and t₁/₂ for the time term.
- Solve for t₁/₂ by rearranging the algebra.
That’s it. Once you understand this process, you don’t need to memorize the formulas because you can derive them on the spot. But having them memorized certainly saves time on exams.
First-Order Half-Life: t₁/₂ = 0.693 / k
First-order reactions are the most common type you’ll encounter in half-life problems, partly because radioactive decay follows first-order kinetics and partly because the math is the cleanest. The integrated rate law for a first-order reaction is:
ln[A] = −kt + ln[A]₀
To find the half-life, substitute [A] = ½[A]₀ and t = t₁/₂:
ln(½[A]₀) = −k·t₁/₂ + ln[A]₀
Rearrange by subtracting ln[A]₀ from both sides:
ln(½[A]₀) − ln[A]₀ = −k·t₁/₂
Using logarithm rules, the left side simplifies to ln(½), which equals −0.693. So:
−0.693 = −k·t₁/₂
Divide both sides by −k, and you get:
t₁/₂ = 0.693 / k
Notice that [A]₀ canceled out completely. This is why first-order half-life is constant: it doesn’t depend on how much reactant you started with. Whether you begin with 1.0 M or 0.001 M, the half-life is the same.
Worked Example
The anticancer drug cisplatin breaks down in water with a first-order rate constant of 1.5 × 10⁻³ min⁻¹ at pH 7.0 and 25°C. To find the half-life:
t₁/₂ = 0.693 / (1.5 × 10⁻³ min⁻¹) = 462 minutes, or about 7.7 hours.
That tells you roughly how long the drug remains at effective concentration in solution before half of it has reacted.
Zero-Order Half-Life: t₁/₂ = [A]₀ / 2k
Zero-order reactions have a rate that stays constant regardless of concentration (until the reactant runs out). The integrated rate law is:
[A] = −kt + [A]₀
Substitute [A] = ½[A]₀ and t = t₁/₂:
½[A]₀ = −k·t₁/₂ + [A]₀
Rearrange:
k·t₁/₂ = [A]₀ − ½[A]₀ = ½[A]₀
t₁/₂ = [A]₀ / 2k
Here, [A]₀ does not cancel. The half-life is directly proportional to the initial concentration: double the starting amount and the half-life doubles. This also means each successive half-life is shorter than the last, because each time the “starting” concentration for the next interval is smaller.
Second-Order Half-Life: t₁/₂ = 1 / (k[A]₀)
The integrated rate law for a second-order reaction is:
1/[A] = kt + 1/[A]₀
When [A] = ½[A]₀, the term 1/[A] becomes 2/[A]₀. Substitute and rearrange:
2/[A]₀ = k·t₁/₂ + 1/[A]₀
2/[A]₀ − 1/[A]₀ = k·t₁/₂
1/[A]₀ = k·t₁/₂
t₁/₂ = 1 / (k[A]₀)
The half-life is inversely proportional to the starting concentration. A higher initial concentration means a shorter half-life because more molecules are available to collide and react. As the reaction proceeds and concentration drops, each successive half-life gets longer. This is the opposite pattern from zero-order reactions.
How to Identify Reaction Order From a Graph
If you’re given a concentration-versus-time plot instead of a rate constant, you can determine the half-life visually. Find the time at which the concentration reaches half its starting value. Then find the time at which it reaches one-quarter (half of the half). The relationship between those two intervals tells you the reaction order:
- First order: The two intervals are equal. Every half-life takes the same amount of time.
- Zero order: The second interval is shorter than the first. Half-lives shrink as concentration drops.
- Second order: The second interval is longer than the first. Half-lives grow as concentration drops.
This graphical method is especially useful on exams when a problem gives you a plotted curve and asks you to determine the order before calculating anything.
Choosing the Right Formula
The most common mistake in half-life problems is using the first-order equation for every situation. Before plugging in numbers, confirm the reaction order. Your problem will tell you directly (“this is a second-order reaction”) or give you enough data to figure it out, such as units of the rate constant.
The units of k are a reliable clue. For a first-order reaction, k has units of inverse time (s⁻¹ or min⁻¹). For a second-order reaction, k has units of inverse concentration per time (M⁻¹s⁻¹). For a zero-order reaction, k has units of concentration per time (M·s⁻¹ or mol·L⁻¹·s⁻¹). If you see M⁻¹s⁻¹ on a rate constant, you know you need the second-order formula, not the first-order one.
Radioactive Decay as a Special Case
Radioactive decay always follows first-order kinetics, so the formula is always t₁/₂ = 0.693 / k. The difference is that nuclear half-lives are fixed physical constants (carbon-14 has a half-life of 5,730 years, for instance), while chemical half-lives depend on conditions like temperature, pH, and solvent. If your problem involves a decaying isotope, you can safely use the first-order equation without needing to verify the reaction order.
You can also work the formula in reverse. If you know the half-life of a radioactive substance, divide 0.693 by that half-life to get the decay constant k. From there, you can calculate how much material remains after any given time using the integrated rate law.