Integrated rate laws: first, second, and zeroth order reactions and half-lives

Intuition Beginner

Unit 14.08.01 introduced the rate law: $\text{Rate}=k[A{]}^m$. The rate law tells you how fast a reaction proceeds at any given concentration. But chemists usually measure concentrations at several times and need to extract the order $m$ and the rate constant $k$ from that data. The integrated rate law gives concentration as a function of time, which is the form you need.

Each reaction order produces a distinct mathematical relationship between concentration and time. For a zeroth-order reaction, concentration decreases linearly with time. For a first-order reaction, the natural log of concentration decreases linearly with time. For a second-order reaction, the reciprocal of concentration increases linearly with time.

The half-life is the time it takes for the concentration of a reactant to drop to half its starting value. First-order reactions have a special property: the half-life depends only on the rate constant, not on the starting concentration. This is why radioactive decay has a characteristic half-life that never changes regardless of how much material is present.

Visual Beginner

Each reaction order produces a characteristic graphical signature when you plot the data the right way:

Order	Rate law	Linear plot	Slope	Half-life
0	Rate $=k$	$[A]$ vs. $t$	$-k$	$[A{]}_0/(2k)$
1	Rate $=k[A]$	$\ln [A]$ vs. $t$	$-k$	$\ln 2/k$
2	Rate $=k[A{]}^2$	$1/[A]$ vs. $t$	$+k$	$1/(k[A{]}_0)$

The key experimental technique is to try all three linearisations. Whichever plot gives a straight line tells you the reaction order, and the slope gives you $k$.

Worked example Beginner

The decomposition of ${\text{N}}_2{\text{O}}_5$ at 45${}^{\circ }$C is first-order with $k=6.2\times 10^{-4}{\text{s}}^{-1}$. Starting from $[A{]}_0=0.500\text{M}$:

Half-life. $t_{1/2}=\ln 2/k=0.693/6.2\times 10^{-4}=1118\text{s}\approx 18.6\text{min}$.

Concentration after 30 minutes. Thirty minutes is 1800 seconds.

$$[A{]}_t=[A{]}_0\cdot e^{-kt}=0.500\cdot e^{-6.2\times 10^{-4}\times 1800}=0.500\cdot e^{-1.116}=0.500\times 0.328=0.164\text{M}$$

About 33% remains after 30 minutes.

Time to reach 0.100 M.

$$\ln (0.100/0.500)=-kt⟹\ln (0.200)=-6.2\times 10^{-4}t⟹-1.609=-6.2\times 10^{-4}t$$

$t=1.609/6.2\times 10^{-4}=2595\text{s}\approx 43.2\text{min}$.

Check your understanding Beginner

Formal definition Intermediate+

The general integration

For a reaction $A\to$ products with rate law $v=k[A{]}^m$, the differential rate equation is:

$$-\frac{d[A]}{dt}=k[A{]}^m.$$

Separating variables and integrating from $t=0$ to $t$ with corresponding concentrations $[A{]}_0$ and $[A{]}_t$:

$${\int }_{[A{]}_0}^{[A{]}_t}\frac{d[A]}{[A{]}^m}=-k{\int }_0^{t}d{t}^{\prime }=-kt.$$

The integral on the left evaluates differently depending on $m$.

Zeroth order ($m=0$)

$${\int }_{[A{]}_0}^{[A{]}_t}d[A]=-kt⟹[A{]}_t=[A{]}_0-kt.$$

The concentration decreases linearly. The reaction proceeds at a constant rate regardless of how much reactant is left. This behaviour is observed in surface-catalysed reactions and enzyme-saturated reactions where the catalyst is the limiting factor, not the reactant concentration. The reactant is exhausted at $t=[A{]}_0/k$.

Setting $[A{]}_t=[A{]}_0/2$:

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

First order ($m=1$)

$${\int }_{[A{]}_0}^{[A{]}_t}\frac{d[A]}{[A]}=-kt⟹\ln \frac{[A{]}_t}{[A{]}_0}=-kt⟹[A{]}_t=[A{]}_0{e}^{-kt}.$$

A plot of $\ln [A]$ vs. $t$ is a straight line of slope $-k$. The fraction remaining at time $t$ is $[A{]}_t/[A{]}_0=e^{-kt}$, which depends only on $kt$ -- a dimensionless combination. Setting $[A{]}_t=[A{]}_0/2$:

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

The half-life is constant: every successive half-life reduces the concentration by the same factor of 2, regardless of how much reactant is present. This is the unique property of first-order kinetics and explains why radioactive isotopes have a characteristic half-life.

Second order ($m=2$)

$${\int }_{[A{]}_0}^{[A{]}_t}[A{]}^{-2}d[A]=-kt⟹{\left[-\frac{1}{[A]}\right]}_{[A{]}_0}^{[A{]}_t}=-kt⟹\frac{1}{[A{]}_t}=\frac{1}{[A{]}_0}+kt.$$

A plot of $1/[A]$ vs. $t$ is a straight line of slope $k$. Setting $[A{]}_t=[A{]}_0/2$:

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

The half-life depends on the initial concentration: starting from a lower concentration gives a longer half-life. Each successive half-life doubles (because $[A{]}_0$ halves each time, and $t_{1/2}\propto 1/[A{]}_0$).

Graphical determination of order

Given time-concentration data for a reaction of unknown order, the standard procedure is:

1. Plot $[A]$ vs. $t$. If linear, the reaction is zeroth order.
2. Plot $\ln [A]$ vs. $t$. If linear, the reaction is first order.
3. Plot $1/[A]$ vs. $t$. If linear, the reaction is second order.

Only one of these three plots will be a straight line (for reactions of order 0, 1, or 2). The slope of the linear plot gives $k$ (with appropriate sign).

Pseudo-first-order kinetics

For a bimolecular reaction $A+B\to$ products with rate law $v=k[A][B]$, the integrated rate law has no simple closed form. If one reactant is present in large excess (typically at least 10-fold, preferably 20-fold or more), its concentration remains approximately constant throughout the reaction: $[B]\approx [B{]}_0$. The rate law becomes:

$$v=k[A][B{]}_0=k^{\prime }[A],k^{\prime }=k[B{]}_0.$$

The reaction follows first-order kinetics with an effective rate constant $k^{\prime }$ that depends on the concentration of the excess reactant. Varying $[B{]}_0$ changes $k^{\prime }$, and plotting $k^{\prime }$ vs. $[B{]}_0$ recovers the true bimolecular rate constant $k$ from the slope.

This technique is ubiquitous in experimental kinetics because first-order data analysis is far simpler than the integrated bimolecular law.

Radioactive decay

Radioactive decay is a first-order process. Each radioactive nucleus decays independently with a constant probability per unit time $\lambda$ (the decay constant). The number of nuclei remaining after time $t$ is:

$$N(t)=N_0{e}^{-\lambda t}.$$

The half-life is $t_{1/2}=\ln 2/\lambda$, and the mean lifetime is $\tau =1/\lambda =t_{1/2}/\ln 2$. The activity (decays per second) is $A=\lambda N$, which also decays exponentially. Carbon-14 dating uses the known half-life of ${}^{14}\text{C}$ (5730 years) to determine the age of organic materials by measuring the remaining ${}^{14}\text{C}$ fraction.

Worked example: determining order from data

The following data were collected for the decomposition of a compound $A$ at constant temperature:

$t$ (s)	$[A]$ (M)
0	0.800
100	0.600
200	0.450
400	0.253
600	0.142

Testing first order: compute $\ln [A]$ and check linearity.

$t$ (s)	$\ln [A]$
0	$-0.223$
100	$-0.511$
200	$-0.799$
400	$-1.374$
600	$-1.951$

$\ln [A]$ vs. $t$ is linear with slope $\approx -2.87\times 10^{-3}{\text{s}}^{-1}$, so the reaction is first order with $k\approx 2.87\times 10^{-3}{\text{s}}^{-1}$. Half-life: $t_{1/2}=0.693/2.87\times 10^{-3}=241\text{s}$.

Key result Intermediate+

Summary of integrated rate laws and half-lives

Theorem (Integrated rate laws). For a reaction $A\to$ products with rate law $v=k[A{]}^m$:

$m$	Integrated law	Linear plot	$t_{1/2}$
0	$[A{]}_t=[A{]}_0-kt$	$[A]$ vs. $t$, slope $-k$	$[A{]}_0/(2k)$
1	$\ln [A{]}_t=\ln [A{]}_0-kt$	$\ln [A]$ vs. $t$, slope $-k$	$\ln 2/k$
2	$1/[A{]}_t=1/[A{]}_0+kt$	$1/[A]$ vs. $t$, slope $+k$	$1/(k[A{]}_0)$

First-order half-life is independent of $[A{]}_0$. Zeroth-order and second-order half-lives depend on $[A{]}_0$.

Proof. All three cases follow from separating variables in $-d[A]/dt=k[A{]}^m$ and integrating. The zeroth-order integral is straightforward. The first-order integral gives $\ln ([A{]}_t/[A{]}_0)=-kt$ by the fundamental theorem of calculus applied to $1/[A]$. The second-order integral gives $-1/[A{]}_t+1/[A{]}_0=-kt$ by integrating $[A{]}^{-2}$. Half-life formulas follow by substituting $[A{]}_t=[A{]}_0/2$ and solving for $t$.

Fraction remaining for first-order reactions

For a first-order reaction, the fraction remaining after $n$ half-lives is $(1/2{)}^n$:

Half-lives elapsed	Fraction remaining
1	0.500
2	0.250
3	0.125
4	0.0625
5	0.03125

After 10 half-lives, less than 0.1% of the original reactant remains.

Exercises Intermediate+

Generalised integration and the $n$th-order case Master

The three integrated rate laws treated in the Intermediate tier are the special cases $m=0,1,2$ of the general ODE $-d[A]/dt=k[A{]}^m$. For arbitrary integer or non-integer order $m\ne 1$, separation of variables gives:

$${\int }_{[A{]}_0}^{[A{]}_t}[A{]}^{-m}d[A]=-kt⟹\frac{[A{]}_t^{1-m}-[A{]}_0^{1-m}}{1-m}=-kt.$$

Rearranging:

$$[A{]}_t^{1-m}=[A{]}_0^{1-m}+(m-1)kt,m\ne 1.$$

The first-order case $m=1$ is singular because $\int [A{]}^{-1}d[A]=\ln [A]$, not $[A{]}^{1-m}/(1-m)$. This is the standard pole at $m=1$ that separates the power-law family from the exponential family. For fractional orders (which arise in heterogeneous catalysis and radical chain mechanisms), the power-law expression above applies directly.

The half-life for general $m\ne 1$ is:

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

This expression recovers $t_{1/2}=[A{]}_0/(2k)$ for $m=0$ (with $2^{-1}-1=-1$, but the overall sign from $(m-1)$ corrects it) and $t_{1/2}=1/(k[A{]}_0)$ for $m=2$ (with $2^1-1=1$ and $(m-1)=1$). The dependence on $[A{]}_0$ is always through $[A{]}_0^{1-m}$, so only $m=1$ gives a concentration-independent half-life.

The full bimolecular integrated law and Guggenheim's method Master

The pseudo-first-order approximation of the Intermediate tier avoids the full integrated bimolecular law because that law is algebraically more involved. For $A+B\to$ products with $v=k[A][B]$ and $[A{]}_0\ne [B{]}_0$, let $x$ be the extent of reaction ($[A{]}_t=[A{]}_0-x$, $[B{]}_t=[B{]}_0-x$). The rate equation becomes:

$$\frac{dx}{dt}=k([A{]}_0-x)([B{]}_0-x).$$

Separating and integrating by partial fractions:

$${\int }_0^x\frac{dx}{([A{]}_0-x)([B{]}_0-x)}=kt.$$

The partial-fraction decomposition uses $1/(([A{]}_0-x)([B{]}_0-x))=[1/([B{]}_0-[A{]}_0)][1/([A{]}_0-x)-1/([B{]}_0-x)]$, giving:

$$\frac{1}{[B{]}_0-[A{]}_0}\ln \frac{[A{]}_0([B{]}_0-x)}{[B{]}_0([A{]}_0-x)}=kt,$$

or equivalently:

$$\frac{1}{[B{]}_0-[A{]}_0}\ln \frac{[B{]}_t/[B{]}_0}{[A{]}_t/[A{]}_0}=kt.$$

A plot of $\ln ([B{]}_t\cdot [A{]}_0)/([A{]}_t\cdot [B{]}_0)$ vs. $t$ is linear with slope $k([B{]}_0-[A{]}_0)$. When $[A{]}_0=[B{]}_0$, the formula degenerates (the partial-fraction decomposition divides by zero), and the system reduces to second order in a single variable: $1/[A{]}_t-1/[A{]}_0=kt$.

Guggenheim's method. When the initial concentrations are not known precisely (for instance, when the reaction is initiated by rapid mixing and the first measurement comes seconds later), Guggenheim (1924, Phil. Mag. 47) proposed measuring $[A]$ at times $t$ and $t+\tau$ where $\tau$ is a fixed interval. For a first-order reaction:

$$\ln ([A{]}_t-[A{]}_{t+\tau })=-kt+\text{const}.$$

A plot of $\ln ([A{]}_t-[A{]}_{t+\tau })$ vs. $t$ is linear with slope $-k$, and the initial concentration never appears. The method eliminates one of the largest sources of systematic error in kinetics experiments: uncertainty in the mixing-time zero point. Guggenheim's method extends to second-order reactions with $[A{]}_0=[B{]}_0$ by replacing $\ln$ with $1/[A]$, but the resulting expressions are less convenient. Mangelsdorf's variant (1955, J. Appl. Phys. 26) and the Kezky-Bruice method extend the idea to arbitrary-order reactions by using numerical differentiation of the Guggenheim plot.

Pseudo-first-order kinetics: the Tikhonov singular-perturbation foundation Master

The Intermediate tier introduced pseudo-first-order kinetics through the practical approximation $[B]\approx [B{]}_0$ when $[B{]}_0\gg [A{]}_0$. The rigorous mathematical basis is Tikhonov's theorem on singular perturbations of ODE systems. For the bimolecular system $d[A]/dt=-k[A][B]$, $d[B]/dt=-k[A][B]$, define $\epsilon =[A{]}_0/[B{]}_0\ll 1$. Rescaling $A^{\prime }=[A]/[A{]}_0$ and $B^{\prime }=[B]/[B{]}_0$ with dimensionless time $\tau =k[A{]}_{0}t$ gives:

$$\frac{d{A}^{\prime }}{d\tau }=-A^{\prime }{B}^{\prime },\epsilon \frac{d{B}^{\prime }}{d\tau }=-A^{\prime }{B}^{\prime }.$$

On the fast time scale $\tau /\epsilon$, $B^{\prime }$ relaxes to a quasi-steady state. On the slow time scale $\tau$, $B^{\prime }\approx 1$ to leading order in $\epsilon$, and $A^{\prime }$ evolves as $d{A}^{\prime }/d\tau =-A^{\prime }$, recovering the pseudo-first-order law. Tikhonov's theorem guarantees that the error between the true solution and the pseudo-first-order approximation is $O(\epsilon )$ on time scales of order $1/(k[B{]}_0)$, and the approximation improves as $\epsilon \to 0$. This is the same singular-perturbation framework that underlies the steady-state approximation in enzyme kinetics (see 14.08.01 and 17.04.01).

The practical consequence: pseudo-first-order conditions are justified when $[B{]}_0\ge 10[A{]}_0$ (error below 5% throughout the reaction) and are excellent when $[B{]}_0\ge 20[A{]}_0$ (error below 2.5%). Below 10-fold excess, the approximation degrades and the full bimolecular integrated law should be used.

Radioactive decay and the stochastic-deterministic connection Master

Radioactive decay provides the cleanest physical realisation of first-order kinetics. Each unstable nucleus decays independently with a constant probability $\lambda dt$ in an infinitesimal time interval $dt$. The number of nuclei $N(t)$ in a sample of initial size $N_0$ follows:

$$N(t)=N_0{e}^{-\lambda t}.$$

This deterministic equation is the law-of-large-numbers limit of an underlying stochastic process. Individual decay events are independent and exponentially distributed with mean waiting time $\tau =1/\lambda$. The total count $N(t)$ is a binomial random variable with $N_0$ trials and success probability $p=e^{-\lambda t}$, giving standard deviation $\sigma =\sqrt{N_{0}p(1-p)}$. For macroscopic samples ($N_0\sim 10^{23}$), the relative fluctuation $\sigma /N$ is of order $10^{-12}$, so the deterministic law is effectively exact.

Three time scales are used:

• Half-life: $t_{1/2}=\ln 2/\lambda$. The time for half the sample to decay.
• Mean lifetime: $\tau =1/\lambda =t_{1/2}/\ln 2\approx 1.443t_{1/2}$. The average time a nucleus survives before decaying.
• Tenth-life: $t_{1/10}=\ln 10/\lambda \approx 3.322t_{1/2}$. The time for 90% of the sample to decay.

Successive radioactive decay. Many decay chains involve a parent isotope decaying to a radioactive daughter: $A\stackrel{{\lambda }_A}{\to }B\stackrel{{\lambda }_B}{\to }C$ (stable). The ODE system is:

$$\frac{d{N}_A}{dt}=-{\lambda }_A{N}_A,\frac{d{N}_B}{dt}={\lambda }_A{N}_A-{\lambda }_B{N}_B.$$

The solution for $N_A$ is $N_A(t)=N_{A,0}{e}^{-{\lambda }_{A}t}$. Substituting into the equation for $N_B$ and solving (by integrating factor or Laplace transform):

$$N_B(t)=\frac{{\lambda }_A{N}_{A,0}}{{\lambda }_B-{\lambda }_A}\left(e^{-{\lambda }_{A}t}-e^{-{\lambda }_{B}t}\right),{\lambda }_A\ne {\lambda }_B.$$

Three regimes emerge:

1. Secular equilibrium (${\lambda }_A\ll {\lambda }_B$, parent much longer-lived): After a transient of order $1/{\lambda }_B$, $N_B$ reaches a plateau where $N_B/N_A\approx {\lambda }_A/{\lambda }_B$. The daughter activity equals the parent activity.

2. Transient equilibrium (${\lambda }_A<{\lambda }_B$, but comparable): The ratio $N_B/N_A$ approaches a constant ${\lambda }_A/({\lambda }_B-{\lambda }_A)$, but both activities decay measurably.

3. No equilibrium (${\lambda }_A>{\lambda }_B$, parent shorter-lived): The daughter outlives the parent. $N_B$ rises to a maximum at $t_{\max }=\ln ({\lambda }_A/{\lambda }_B)/({\lambda }_A-{\lambda }_B)$ and then decays with its own half-life.

The ${}^{238}\text{U}$ decay chain to ${}^{206}\text{Pb}$ involves 14 successive alpha and beta decays. At secular equilibrium (achieved after $\sim 10^6$ years in a closed system), every isotope in the chain has equal activity. This principle underlies uranium-thorium dating and uranium-lead geochronology, which can date rocks from the early solar system with uncertainties below 1 Myr on ages of 4.5 Gyr.

Connections Master

• Chemical kinetics rate laws and Arrhenius 14.08.01. This unit is the direct continuation. Unit 14.08.01 introduced the differential rate law; this unit integrates it. The Arrhenius temperature dependence of $k$ from the prerequisite applies to the integrated forms without modification.

• Reaction mechanisms 14.08.03. Determining the rate law from experimental data (via the graphical methods of this unit) is the first step in proposing and testing a reaction mechanism. The steady-state approximation in mechanism analysis reduces complex mechanisms to effective rate laws of order 0, 1, or 2.

• Electrochemistry 14.11.01. Electrochemical cell potentials depend on concentrations via the Nernst equation. When a cell reaction is first-order, the concentration decay measured by the cell potential follows $E(t)=E^{\circ }+(RT/nF)\ln ([A{]}_0{e}^{-kt})$, a linear function of $t$.

• Enzyme kinetics 17.04.01. The Michaelis-Menten mechanism produces a rate law that is first-order in substrate at low $[S]$ and zeroth-order at high $[S]$. The pseudo-first-order technique developed here applies directly to initial-rate measurements in enzyme assays, where $[S]\gg [E{]}_0$ is the standard condition.

• Statistical mechanics 14.07.01. The exponential decay law for first-order reactions is the macroscopic manifestation of the Poisson process at the molecular level. The connection between stochastic single-molecule kinetics and deterministic rate laws is a law-of-large-numbers result.

• Acid-base chemistry 14.10.01. Pseudo-first-order conditions appear naturally in acid-catalysed reactions where $[{\text{H}}^{+}]$ is buffered to a constant value, making the reaction effectively first-order in the substrate.

Historical notes Master

The study of reaction rates dates to Wilhelmy (1850), who measured the acid-catalysed inversion of sucrose and found it followed a first-order law. Harcourt and Esson (1866-1867) established the first second-order rate law from the oxidation of oxalic acid by potassium permanganate. Arrhenius (1889) provided the temperature-dependence framework. The systematic classification into zeroth, first, and second order came from Ostwald and van't Hoff in the 1880s. Guggenheim (1926) introduced his method for analysing first-order reactions without needing the infinite-time concentration.

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