Zero-order half-life formula
A zero-order reaction uses reactant at a constant rate. Its integrated rate law is [A] = [A]0 − kt.
At the half-life, [A] = [A]0 / 2. Substitute this value and solve for t to get t1/2 = [A]0 / 2k.
The half-life is proportional to the initial concentration. Each new half-life is half as long as the one before it.
The rate constant k has units of concentration per time, such as M/s.
How to calculate half life for a zero-order reaction
- Confirm that the reaction is zero order. A plot of [A] against time is a straight line.
- Find k from the slope of that line: k = −slope.
- Find the initial concentration, [A]0.
- Calculate t1/2 = [A]0 ÷ (2k).
- Check that the units cancel to a time unit.
Zero-order half-life problems
Problem 1
A zero-order reaction has [A]0 = 0.80 M and k = 0.020 M/s. Find the half-life.
- t1/2 = [A]0 ÷ 2k.
- t1/2 = 0.80 M ÷ (2 × 0.020 M/s) = 20 s.
Answer: 20 s
Problem 2
What is the second half-life of the reaction in problem 1?
- After one half-life, [A] = 0.40 M. This value is the new starting concentration.
- t1/2 = 0.40 M ÷ (2 × 0.020 M/s) = 10 s.
Answer: 10 s
Problem 3
NH3 decomposes on a hot tungsten surface in a zero-order reaction with k = 2.5 × 10^-4 M/s. Find t1/2 when [A]0 = 0.050 M.
- t1/2 = 0.050 M ÷ (2 × 2.5 × 10^-4 M/s) = 100 s.
Answer: 100 s
Problem 4
A zero-order reaction has a half-life of 40 min when [A]0 = 1.2 M. Find the rate constant k.
- Rearrange the formula: k = [A]0 ÷ (2 × t1/2).
- k = 1.2 M ÷ (2 × 40 min) = 0.015 M/min.
Answer: 0.015 M/min
Half-life formulas by reaction order
| Order | Rate law | Half-life formula | Effect of [A]0 |
|---|---|---|---|
| Zero | rate = k | t1/2 = [A]0 / 2k | Half-life increases as [A]0 increases |
| First | rate = k[A] | t1/2 = ln 2 / k | No effect |
| Second | rate = k[A]^2 | t1/2 = 1 / (k[A]0) | Half-life decreases as [A]0 increases |
How to use this calculator
- Select zero as the kinetic model.
- Select the unknown, such as half-life or rate constant.
- Enter the initial concentration [A]0 and the rate constant k.
- Calculate and check the result with t1/2 = [A]0 / 2k.
Formula and variables
zero: t1/2=[A]0/(2k); first: t1/2=ln2/k; second: t1/2=1/(k[A]0)
| Symbol | Meaning | Unit |
|---|---|---|
| t1/2 | half-life | time |
| k | rate constant | order-specific |
| [A]0 | initial concentration | concentration |
Worked example
A zero-order reaction has [A]0 = 0.80 M and k = 0.020 M/s. Its half-life is 0.80 / (2 × 0.020) = 20 s.
Interpretation
Each kinetic order has a distinct equation and rate-constant dimension; radioactive decay uses the first-order exponential model explicitly.
Common mistakes
- Do not apply the radioactive/first-order equation to zero- or second-order reactions.
- Keep the rate-constant unit consistent with the chosen kinetic order.
Frequently asked questions
What is the half-life formula for a zero-order reaction?
The formula is t1/2 = [A]0 / 2k. [A]0 is the initial concentration and k is the rate constant.
Why does zero-order half-life depend on the initial concentration?
A zero-order reaction removes a fixed amount per unit time. A larger starting amount takes longer to lose half.
What are the units of k for a zero-order reaction?
The units are concentration per time, such as M/s or mol/(L·s).
Do all reaction orders use the same half-life formula?
No. Zero-, first-, and second-order reactions use different equations and rate-constant units.
Can the calculator find the remaining radioactive amount?
Yes. Select radioactive, then enter the initial amount, elapsed time, and half-life.
Can I use scientific notation?
Yes. Enter values such as 6.022e23 or 1.0e-6 in a numeric field.
How should I round the result?
Keep extra digits during the calculation. Round the final result to the precision required by your data.