Half-Life Explained: Zero Order vs First Order Reactions
Radioactive carbon-14 has a half-life of 5,730 years regardless of how much you have. But the alcohol in your bloodstream disappears at a fixed rate per hour — meaning the time to halve your blood alcohol level depends entirely on how much you started with. These two behaviours represent the fundamental difference between first order and zero order half-lives.
Understanding this distinction is one of the most important concepts in chemical kinetics.
What Is Half-Life in Chemistry?
The half-life (t½) of a reaction is the time required for the concentration of a reactant to fall to exactly half of its initial value.
Half-life is most famous from nuclear physics and radioactive decay, but it applies equally to chemical reactions of all kinds — drug metabolism, industrial reactions, food spoilage, and any process that can be described by a rate law.
The half-life depends on two things:
- The order of the reaction (zero, first, or second)
- The rate constant k (and sometimes the initial concentration)
First Order Half-Life — The Constant Half-Life
For a first order reaction where rate = k[A], the integrated rate law is:
To find the half-life, set [A] = [A]₀/2:
| Time elapsed | Fraction remaining | % remaining |
|---|---|---|
| 0 | 1 | 100% |
| 1 × t½ | 1/2 | 50% |
| 2 × t½ | 1/4 | 25% |
| 3 × t½ | 1/8 | 12.5% |
| 4 × t½ | 1/16 | 6.25% |
| 5 × t½ | 1/32 | 3.125% |
| n × t½ | (1/2)ⁿ | (0.5)ⁿ × 100% |
The general formula for concentration after n half-lives: [A] = [A]₀ × (1/2)ⁿ = [A]₀ × (0.5)ⁿ
This exponential decay pattern is mathematically identical to radioactive decay, compound interest in reverse, and the discharge of a capacitor.
Zero Order Half-Life — The Concentration-Dependent Half-Life
For a zero order reaction where rate = k, the integrated rate law is:
To find the half-life, set [A] = [A]₀/2:
| Half-life period | Starting [A] | t½ for this period |
|---|---|---|
| 1st | [A]₀ | [A]₀/(2k) |
| 2nd | [A]₀/2 | [A]₀/(4k) — half as long |
| 3rd | [A]₀/4 | [A]₀/(8k) — quarter as long |
| 4th | [A]₀/8 | [A]₀/(16k) — eighth as long |
| Complete | 0 | Total time = [A]₀/k |
A zero order reaction reaches complete consumption at a finite time — unlike a first order reaction, which theoretically approaches zero asymptotically and never quite gets there.
Use our Reaction Order Calculator to calculate half-life for both zero and first order reactions with any initial concentration and rate constant.
Side-by-Side Comparison — Zero Order vs First Order Half-Life
Successive half-lives get shorter
Reaction reaches zero at finite time
All half-lives identical
Asymptotically approaches zero
Each bar = duration of that half-life period
All bars identical — constant duration
| Property | Zero Order | First Order |
|---|---|---|
| Rate law | rate = k | rate = k[A] |
| Integrated law | [A] = [A]₀ − kt | ln[A] = ln[A]₀ − kt |
| Half-life formula | t½ = [A]₀/(2k) | t½ = 0.693/k |
| Depends on [A]₀? | YES | NO |
| Successive half-lives | Getting shorter | All identical |
| Reaches zero? | YES — finite time | NO — asymptotic |
| Units of k | M·s⁻¹ | s⁻¹ |
| Linear graph | [A] vs t | ln[A] vs t |
| Conceptual basis | Constant amount consumed per time | Constant fraction consumed per time |
Worked Examples
Problem: A first order reaction has k = 0.0347 min⁻¹. Find the half-life.
Check — concentration after 60 minutes (3 half-lives):
[A] = [A]₀ × (0.5)³ = [A]₀ × 0.125 → 12.5% of initial concentration remains ✓
Problem: A zero order reaction has k = 0.050 M/min and [A]₀ = 1.20 M. Find: (a) the first half-life, (b) the second half-life, (c) when all reactant is consumed.
After 12 min: [A] = 0.60 M
(b) New [A]₀ = 0.60 M → t½ = 0.60/(2 × 0.050) = 6 min
After 6 more min (18 min total): [A] = 0.30 M
(c) Total time = [A]₀/k = 1.20/0.050 = 24 min
Notice: total time = 24 min = 2 × first half-life (12 min). This is always true for zero order reactions.
Problem: A drug follows first order elimination kinetics. Its plasma concentration halves every 4.5 hours. Find k. How much drug remains after 24 hours if [A]₀ = 100 ng/mL?
[A] = [A]₀ × e^(−kt) = 100 × e^(−0.154 × 24)
[A] = 100 × e^(−3.696) = 100 × 0.0248 = 2.48 ng/mL
Check using half-lives: 24 hr / 4.5 hr = 5.33 half-lives → [A] = 100 × (0.5)^5.33 = 2.49 ng/mL ✓
Problem: An experimentalist measures these half-lives at different initial concentrations. What is the reaction order?
| [A]₀ (M) | Measured t½ (min) |
|---|---|
| 0.20 | 5.0 |
| 0.40 | 10.0 |
| 0.80 | 20.0 |
| 1.60 | 40.0 |
This proportionality is the signature of zero order kinetics.
Find k: k = [A]₀/(2t½) = 0.20/(2 × 5.0) = 0.020 M/min
Check: k = 0.80/(2 × 20.0) = 0.020 M/min ✓
Problem: A first order reaction starts at [A]₀ = 0.500 M. How many half-lives are needed to reach [A] = 0.031 M?
Taking log: n × log(0.5) = log(0.062)
n = log(0.062)/log(0.5) = (−1.208)/(−0.301)
Check: (0.5)⁴ = 0.0625 × 0.500 = 0.031 M ✓
The Biological Significance of Constant Half-Lives
The fact that first order reactions have a constant half-life has profound implications in medicine and biology.
Drug Dosing and Pharmacokinetics
Most drugs are eliminated from the body by first order kinetics — enzymatic metabolism and renal filtration both follow first order rate laws at therapeutic concentrations. A drug with a 6-hour half-life will be at 50% of its initial level after 6 hours, 25% after 12 hours, 12.5% after 18 hours, and less than 2% after 42 hours (7 half-lives). Clinicians use the rule of thumb that a drug is essentially eliminated after 5 half-lives (3.125% remains). Fluoxetine (Prozac) has a half-life of 1–4 days and an active metabolite with 4–16 days — which is why it takes weeks to reach steady state and weeks to wash out.
Radiocarbon Dating
Carbon-14 has a half-life of 5,730 years and decays by first order kinetics. Living organisms continuously exchange carbon with their environment, maintaining a constant ratio of ¹⁴C to ¹²C. When an organism dies, the exchange stops and the ¹⁴C decays away at its constant first order rate. By measuring the fraction of ¹⁴C remaining, scientists can calculate the age. A sample with 25% of its original ¹⁴C has gone through 2 half-lives: age ≈ 2 × 5,730 = 11,460 years. This method works reliably for samples up to about 50,000 years old (approximately 8–9 half-lives).
Why Alcohol Elimination Differs
Alcohol (ethanol) follows zero order kinetics in the liver because alcohol dehydrogenase (ADH) is saturated at typical drinking concentrations. The practical consequence: a person with a BAC of 0.16% (double the legal driving limit) takes exactly twice as long to sober up as someone at 0.08% — because the elimination rate is fixed at approximately 0.015% BAC per hour, regardless of concentration. This is fundamentally different from caffeine, which follows first order kinetics with a half-life of about 5 hours. Two beers and one beer do NOT leave the body proportionally — the extra alcohol adds extra time because the rate is fixed, not fractional.
The Graph Comparison — What Each Looks Like
Understanding the shape of the concentration-vs-time graph helps you immediately identify reaction order and half-life behaviour:
Zero order [A] vs t graph: Straight line declining at constant slope −k. Equal time intervals show equal drops in concentration. Line reaches the x-axis at time t = [A]₀/k. Half-life intervals get shorter and shorter as the line approaches zero.
First order [A] vs t graph: Exponential decay curve. Equal time intervals show equal fractional drops (not equal absolute drops). Curve approaches the x-axis asymptotically — never quite reaches zero. Half-life intervals are all identical widths.
Common Mistakes with Half-Life Calculations
❌ Applying the Constant Half-Life Rule to Zero Order Reactions
t½ = 0.693/k applies ONLY to first order reactions. For zero order, t½ = [A]₀/(2k) and changes with initial concentration. Using the wrong formula gives a completely wrong answer.
❌ Forgetting to Identify Reaction Order First
Before calculating any half-life, you must know the reaction order. Ask: does the half-life depend on concentration? If yes → zero order (or second order). If no → first order.
❌ Using t/t½ for Zero Order When t½ Is Not Constant
For zero order reactions, you cannot simply divide total time by t½ to find concentration — because t½ changes with each period. Use the integrated rate law [A] = [A]₀ − kt directly instead.
❌ Confusing Half-Life with Mean Lifetime
The mean lifetime τ (tau) is the average time a single molecule survives before reacting. For first order: τ = 1/k = t½/0.693 = 1.443 × t½. Half-life and mean lifetime are related but not equal.
❌ Wrong Units for k When Calculating t½
If k is in min⁻¹, the half-life comes out in minutes. If k is in s⁻¹, t½ is in seconds. Always check that k and t½ are in consistent time units.
Frequently Asked Questions
t½ = [A]₀/(2k), where [A]₀ is the initial concentration and k is the rate constant. Unlike first order, this half-life changes with concentration — larger initial concentration means longer first half-life.
t½ = ln(2)/k = 0.693/k, where k is the first order rate constant in units of s⁻¹ (or min⁻¹, hr⁻¹ etc.). This half-life is constant — it does not depend on initial concentration.
First order reactions consume a constant fraction of reactant per unit time, so halving takes the same time regardless of starting amount. Zero order reactions consume a constant amount per unit time, so halving a larger amount takes longer than halving a smaller amount.
For first order kinetics: after n half-lives, fraction remaining = (0.5)ⁿ. For 99% elimination (1% remaining): (0.5)ⁿ = 0.01 → n = log(0.01)/log(0.5) = 6.64 half-lives. Practically, 7 half-lives removes more than 99% of a first order reactant.
Radioactive decay is always first order — the decay rate is proportional to the number of atoms present. Each nucleus decays independently with a fixed probability per unit time, giving a constant half-life regardless of how much material is present. This is why radiocarbon dating works — the half-life of ¹⁴C is always 5,730 years.
Yes — second order reactions have a half-life of t½ = 1/(k[A]₀), which also depends on initial concentration (like zero order), but successive half-lives get progressively longer rather than shorter. Each subsequent half-life doubles in duration as concentration decreases.
⚗️ Calculate Half-Life Instantly
Our Reaction Order Calculator calculates the half-life for zero order, first order, and second order reactions. Enter your rate constant and initial concentration (where needed) and it shows the half-life formula, the calculation, and a table of concentration at each successive half-life period. For exponential decay calculations including radioactive decay, see our Exponential Growth and Decay Calculator.