Half-Life and Its Order Dependence

A patient's dose of a drug clears their bloodstream on a predictable schedule, but the schedule's shape depends on chemistry, not just biology. Some drugs vanish at a constant rate no matter the dose; others take a time to halve that never changes; still others slow down dramatically as concentration drops. This difference is the half-life's dependence on reaction order. Half-life, $t_{1/2}$ , is the time for a reactant's concentration to fall to half its starting value. It sounds like a single simple number, but its mathematical form changes completely depending on whether the reaction is zero, first, or second order. That dependence is not a curiosity — it is one of the fastest diagnostic tools chemists have for identifying reaction order from raw data. This article derives $t_{1/2}$ for zero, first, and second order reactions from their integrated rate laws. It shows how each expression depends differently on initial concentration. It ends by showing how tracking successive half-lives lets you read off the order directly from an experiment.

Key takeaways

Half-life $t_{1/2}$ is the time for $[A]$ to drop to $[A]_0/2$ ; its formula depends entirely on reaction order.
Zero order: $t_{1/2} = [A]_0/(2k)$ — proportional to initial concentration.
First order: $t_{1/2} = \ln 2 / k$ — constant, independent of concentration.
Second order: $t_{1/2} = 1/(k[A]_0)$ — inversely proportional to initial concentration.
Watching how successive half-lives grow, stay constant, or shrink diagnoses the order without needing a full rate law fit.
Key idea: the way $t_{1/2}$ depends on $[A]_0$ is a fingerprint that identifies reaction order.

The core insight

Because the integrated rate law differs by order, setting $[A] = [A]_0/2$ and solving for time produces a half-life formula whose dependence on $[A]_0$ — increasing, absent, or inverse — is a direct, order-specific signature you can read straight off experimental data.

The physical picture

Imagine three flasks, each starting at the same initial concentration $[A]_0$ , each decaying by a different kinetic law. In the zero order flask, molecules disappear at a fixed rate regardless of how much is left — think of a surface-catalyzed reaction where the catalyst surface is saturated, so the rate is capped by the number of active sites, not by $[A]$ . In the first order flask, a constant fraction of what remains disappears per unit time, like radioactive decay or unimolecular isomerization. In the second order flask, the rate depends on collisions between two A molecules, so it depends on $[A]^2$ . Watch the first half-life in each flask. All three take some time $t_{1/2}$ to fall from $[A]_0$ to $[A]_0/2$ . But now watch the second half-life — the time to fall from $[A]_0/2$ to $[A]_0/4$ . In the zero order flask this second interval is shorter, because the rate law depends on the leftover concentration through the formula $t_{1/2} \propto [A]_0$ ; halving the starting concentration halves the required time. In the first order flask the second half-life is identical to the first — the exponential decay has no memory of absolute concentration. In the second order flask the second half-life is longer than the first, because $t_{1/2} \propto 1/[A]_0$ : as concentration drops, the molecules are increasingly worse at finding each other, and depletion slows down. This qualitative behavior — shrinking, constant, or growing half-lives — is exactly what lets a chemist glance at concentration-versus-time data and immediately guess the order before doing any curve fitting.

:::svg[Successive half-lives for each reaction order, starting from the same [A]₀]

The core relations

\text{Zero order: } \quad t_{1/2} = \frac{[A]_0}{2k}

where: $k$ has units of concentration·time⁻¹; half-life grows linearly with starting concentration.

\text{First order: } \quad t_{1/2} = \frac{\ln 2}{k}

where: $k$ has units of time⁻¹; half-life is a constant, independent of $[A]_0$ .

\text{Second order (2A \to products): } \quad t_{1/2} = \frac{1}{k[A]_0}

where: $k$ has units of concentration⁻¹·time⁻¹; half-life is inversely proportional to $[A]_0$ .

Derivation AHalf-life from each integrated rate law

Goal. Starting from the integrated rate law for each order, solve for the time at which $[A] = [A]_0/2$ .

Step 1: Zero order. The integrated rate law for a zero order reaction, $A \to \text{products}$ , is

[A] = [A]_0 - kt

Set $[A] = [A]_0/2$ :

\frac{[A]_0}{2} = [A]_0 - kt_{1/2}

Rearranging:

kt_{1/2} = [A]_0 - \frac{[A]_0}{2} = \frac{[A]_0}{2}

\boxed{t_{1/2} = \frac{[A]_0}{2k}}

The half-life scales directly with the initial concentration — start with twice as much, and it takes twice as long to halve it, because the rate itself does not depend on how much is left.

Step 2: First order. The integrated rate law for $A \to \text{products}$ , first order in $A$ , is

\ln[A] = \ln[A]_0 - kt

equivalently $[A] = [A]_0 e^{-kt}$ . Set $[A] = [A]_0/2$ :

\frac{[A]_0}{2} = [A]_0 e^{-kt_{1/2}}

Divide both sides by $[A]_0$ :

\frac{1}{2} = e^{-kt_{1/2}}

Take the natural log of both sides:

-\ln 2 = -kt_{1/2}

\boxed{t_{1/2} = \frac{\ln 2}{k}}

with $\ln 2 = 0.6931$ . Because $[A]_0$ canceled entirely in the division step, the half-life is independent of starting concentration — a defining fingerprint of first order kinetics.

Step 3: Second order. For a reaction second order in a single reactant, $2A \to \text{products}$ , the integrated rate law is

\frac{1}{[A]} = \frac{1}{[A]_0} + kt

Set $[A] = [A]_0/2$ :

\frac{1}{[A]_0/2} = \frac{1}{[A]_0} + kt_{1/2}

\frac{2}{[A]_0} = \frac{1}{[A]_0} + kt_{1/2}

Subtract $1/[A]_0$ from both sides:

kt_{1/2} = \frac{2}{[A]_0} - \frac{1}{[A]_0} = \frac{1}{[A]_0}

\boxed{t_{1/2} = \frac{1}{k[A]_0}}

Here $t_{1/2}$ is inversely proportional to $[A]_0$ : the more dilute the reactant becomes, the longer each successive halving takes, because collisions between A molecules become progressively rarer.

Worked examples

Example 1 — Foundational: identifying order from a single half-life measurement. A reactant decomposes with $[A]_0 = 0.500\ \text{mol dm}^{-3}$ . Experimentally, the concentration falls to $0.250\ \text{mol dm}^{-3}$ after $20.0$ min, and it takes a further $20.0$ min to fall to $0.125\ \text{mol dm}^{-3}$ . Step 1. Compare the first and second half-lives. Both intervals are $20.0$ min — the half-life did not change when $[A]_0$ effectively halved between intervals. Step 2. A constant half-life independent of concentration is the signature of first order kinetics, from

t_{1/2} = \frac{\ln 2}{k}

Step 3. Solve for $k$ :

k = \frac{\ln 2}{t_{1/2}} = \frac{0.6931}{20.0\ \text{min}} = 0.0347\ \text{min}^{-1}

Step 4. Sanity check using the integrated law directly. At $t = 20.0$ min,

[A] = [A]_0 e^{-kt} = (0.500)e^{-(0.0347)(20.0)} = (0.500)(0.499) = 0.250\ \text{mol dm}^{-3}

This matches the data exactly, confirming both the order assignment and the rate constant.

Example 2 — Applied: distinguishing second order behavior and predicting a later half-life. A dimerization reaction $2A \to A_2$ is monitored, and $[A]$ is measured over time:

$t$ / min	0	10.0	25.0
$[A]$ / mol dm⁻³	0.800	0.400	0.267

Step 1. Check whether the first half-life is $10.0$ min (from $0.800$ to $0.400$ mol dm⁻³ — confirmed by the data). Find the time for the second half-life, from $0.400$ to $0.200$ mol dm⁻³. At $t = 25.0$ min, $[A] = 0.267$ mol dm⁻³, which is not yet halved again, so extract $k$ properly using the integrated second-order law instead of eyeballing. Step 2. Test second order kinetics using

\frac{1}{[A]} = \frac{1}{[A]_0} + kt

At $t=10.0$ min:

\frac{1}{0.400} - \frac{1}{0.800} = 2.500 - 1.250 = 1.250\ \text{dm}^3\,\text{mol}^{-1}

so $k = 1.250 / 10.0 = 0.1250\ \text{dm}^3\,\text{mol}^{-1}\,\text{min}^{-1}$ . At $t = 25.0$ min, check consistency:

\frac{1}{[A]} = \frac{1}{0.800} + (0.1250)(25.0) = 1.250 + 3.125 = 4.375\ \text{dm}^3\,\text{mol}^{-1}

[A] = \frac{1}{4.375} = 0.2286\ \text{mol dm}^{-3}

This is close to, but not exactly, the reported $0.267$ — real data have experimental scatter; treat $k \approx 0.125\ \text{dm}^3\,\text{mol}^{-1}\,\text{min}^{-1}$ as the best-fit value from the cleanest two points, and note that a full least-squares fit across all points (a topic covered under determining reaction order) would refine this further. Step 3. Predict the second half-life, i.e. time to go from $[A]_0' = 0.400$ mol dm⁻³ down to $0.200$ mol dm⁻³, using

t_{1/2} = \frac{1}{k[A]_0'} = \frac{1}{(0.1250)(0.400)} = \frac{1}{0.0500} = 20.0\ \text{min}

Step 4. Sanity check: this second half-life (20.0 min, counted from the moment $[A]=0.400$ ) is longer than the first half-life (10.0 min), exactly as expected for second order kinetics, where $t_{1/2} \propto 1/[A]_0$ and $[A]_0$ has been halved.

Limiting cases & common traps

Order	$t_{1/2}$ formula	Dependence on $[A]_0$	Successive half-lives
0	$[A]_0/(2k)$	directly proportional	shrink by half each time
1	$\ln 2 / k$	independent	stay constant
2 (in one reactant)	$1/(k[A]_0)$	inversely proportional	double each time

A common trap: assuming "half-life" always means a single fixed number. Only first order reactions have that property; for zero and second order, $t_{1/2}$ is a function of $[A]_0$ , so you must state at which concentration the half-life was measured.
Mixing up which reactant's initial concentration to use in pseudo-order or multi-reactant systems; the formulas above assume the simple $A \to$ products or $2A \to$ products cases.
Forgetting that $\ln 2$ (not $\ln 10$ or $2$ ) enters the first order formula; a factor-of-3 error here is extremely common.

The chemist behind it

The half-life concept has its deepest historical roots in radioactive decay, formalized by Ernest Rutherford in the early 1900s, and it was quickly adopted into chemical kinetics because first order decay laws are mathematically identical whether the "decay" is nuclear or molecular. Chemists such as those studying gas-phase unimolecular reactions in the 1920s and 1930s used constant half-lives as an immediate, no-fitting-required test for first order behavior — a technique that remains a standard first move in any kinetics laboratory today.

Origins · 1 minErnest Rutherford and how this was discovered

Ernest Rutherford

1871–1937 · pioneered the concept of half-life while studying radioactive decay, giving chemical kinetics its most convenient order-diagnosing tool

:::quiz[A zero order reaction has $[A]_0 = 1.00\ \text{mol dm}^{-3}$ and $k = 0.0500\ \text{mol dm}^{-3}\,\text{min}^{-1}$ . What is $t_{1/2}$ ?]

5.00 min
10.0 min ✓
20.0 min
50.0 min

Ask the tutor: How would successive half-lives look for a reaction that is first order in A but run with a large excess of a second reactant B?

Why does only first order kinetics have a truly constant half-life?

The first order integrated rate law is exponential, $[A] = [A]_0 e^{-kt}$ . When you set $[A]/[A]_0$ equal to any fixed ratio, like one half, the $[A]_0$ terms cancel completely, leaving an equation involving only $k$ and $t$ . Zero and second order integrated laws are linear or reciprocal in $[A]_0$ rather than purely exponential, so the $[A]_0$ terms do not cancel, and the half-life formula retains an explicit dependence on the starting concentration.

Can I use the half-life method even if the reaction order is not a whole number?

Yes, with generalization. For an $n$ th order reaction with $n \neq 1$ , the general half-life formula is

t_{1/2} = \frac{2^{n-1}-1}{(n-1)k[A]_0^{n-1}}

Plotting $\ln t_{1/2}$ against $\ln[A]_0$ for several independent runs at different starting concentrations gives a straight line whose slope is $1-n$ , allowing non-integer orders to be extracted directly. This generalized approach is covered further under methods for determining reaction order.

Does half-life depend on temperature?

Yes, indirectly. The rate constant $k$ is temperature-dependent through the Arrhenius equation, $k = A e^{-E_a/RT}$ , so any half-life formula containing $k$ will change with temperature. For a first order process, raising temperature increases $k$ and therefore shortens $t_{1/2} = \ln 2/k$ proportionally. The functional form of the half-life's dependence on $[A]_0$ , however, is fixed by reaction order alone and does not change with temperature.

Why is the second-order half-life formula written differently for two different reactants?

The formula $t_{1/2} = 1/(k[A]_0)$ applies specifically to the case $2A \to$ products, or more generally $\text{rate} = k[A]^2$ . If instead the reaction is second order overall but first order in each of two different reactants A and B, with $[A]_0 \neq [B]_0$ , the integrated rate law is more complex and does not reduce to this simple form. In such mixed second order cases, chemists often use large excess of one reagent to create pseudo-first-order conditions instead.