First Order

First-order reactions are among the most common and important reaction types in chemistry, appearing everywhere from radioactive decay to pharmacokinetics and enzyme catalysis. A reaction is first order when its rate depends linearly on the concentration of a single reactant: if you double the concentration, the rate exactly doubles. Mathematically, this means the rate law takes the form rate = k[A], where k is the rate constant with units of s⁻¹. The integrated rate law for a first-order reaction transforms this differential relationship into a directly usable equation: ln[A]_t = −kt + ln[A]_0. This equation has the form of a straight line (y = mx + b), meaning that a plot of ln[A] versus time will always produce a straight line with slope −k and y-intercept ln[A]_0 for any first-order process. This graphical test is one of the most powerful tools for identifying reaction order from experimental data. One of the most elegant consequences of first-order kinetics is the concept of a constant half-life — the time required for the concentration to fall to exactly half its current value. Unlike second-order reactions, the half-life of a first-order reaction is completely independent of the initial concentration. This property has profound practical applications: it is why doctors can predict drug clearance times, why geologists use carbon-14 dating, and why nuclear engineers can plan waste disposal schedules.

Key takeaways
  • Start with the differential rate law
  • Separate variables and integrate
  • Write the integrated rate law in two useful forms
  • Derive the constant half-life
  • Using the ln[A] vs t graph to find k

Start with the differential rate law

For a reaction \ce{A -> products}, the instantaneous rate is defined as the rate of decrease of [A]: rate = −d[A]/dt = k[A]. Here k has units of s⁻¹ (or min⁻¹, h⁻¹ etc.), confirming the reaction is first order. The negative sign indicates [A] is decreasing over time.

Separate variables and integrate

Rearranging gives d[A]/[A] = −k dt. Integrating both sides — the left from [A]_0 to [A]_t, the right from 0 to t — yields: ∫d[A]/[A] = −k∫dt, so ln[A]_t − ln[A]_0 = −kt. This is valid for any time t ≥ 0 and any positive initial concentration [A]_0.

Write the integrated rate law in two useful forms

The logarithmic form is: ln[A]_t = −kt + ln[A]_0. The exponential form, obtained by exponentiating both sides, is: [A]_t = [A]_0 · e^(−kt). The exponential form directly reveals why first-order decay is called 'exponential decay' — concentration falls as a decaying exponential function of time, never quite reaching zero.

Derive the constant half-life

At time t = t_{1/2}, [A]_t = [A]_0/2 by definition. Substituting into the exponential form: [A]_0/2 = [A]_0 · e^(−k·t_{1/2}). Dividing both sides by [A]_0: 1/2 = e^(−k·t_{1/2}). Taking the natural log: ln(1/2) = −k·t_{1/2}, so −ln 2 = −k·t_{1/2}, giving t_{1/2} = ln 2 / k ≈ 0.693/k. Crucially, [A]_0 cancelled completely, confirming the half-life is independent of initial concentration.

Using the ln[A] vs t graph to find k

Rearranging the logarithmic form as ln[A]_t = (−k)t + ln[A]_0 reveals that a plot of ln[A] (y-axis) versus t (x-axis) is linear with slope = −k and y-intercept = ln[A]_0. To find k experimentally: measure [A] at multiple time points, convert to ln[A], plot the data, fit a straight line, and the magnitude of the slope equals k. A curved ln[A] vs t plot immediately rules out first-order kinetics.

The Key Equation

[A]t=[A]0 e−kt⟺ln⁡[A]t=−kt+ln⁡[A]0⟹t1/2=ln⁡2k≈0.693k[A]_t = [A]_0 \, e^{-kt} \qquad \Longleftrightarrow \qquad \ln[A]_t = -kt + \ln[A]_0 \qquad \Longrightarrow \qquad t_{1/2} = \frac{\ln 2}{k} \approx \frac{0.693}{k}
  • [A]_t — Concentration of A at time t (mol L⁻¹ (M)): The molar concentration of reactant A remaining in solution at elapsed time t.
  • [A]_0 — Initial concentration of A (mol L⁻¹ (M)): The molar concentration of reactant A at the start of the reaction (t = 0).
  • k — First-order rate constant (s⁻¹ (or min⁻¹, h⁻¹)): A temperature-dependent proportionality constant unique to each reaction at a given temperature. Larger k means faster reaction.
  • t — Time (s, min, or h): Elapsed time since the reaction began. Units must match the units used for k.
  • t_{1/2} — Half-life (s, min, or h): The time required for [A] to fall to exactly one-half of its current value. For first-order reactions, this is constant throughout the reaction.
  • \ln 2 — Natural logarithm of 2 (dimensionless): A mathematical constant equal to approximately 0.6931, arising from the definition of half-life.

Worked Example

Problem. The decomposition of hydrogen peroxide in dilute aqueous solution, \ce{2H2O2(aq) -> 2H2O(l) + O2(g)}, is first order in \ce{H2O2} with a rate constant k = 1.06 × 10⁻³ min⁻¹ at 20 °C. (a) If the initial concentration of \ce{H2O2} is 0.400 M, what is [\ce{H2O2}] after 650 minutes? (b) What is the half-life of this reaction? (c) How long does it take for [\ce{H2O2}] to fall from 0.400 M to 0.050 M?

[A]650=0.201 M;t1/2=654 min;t0.050=1962 min[A]_{650} = 0.201\text{ M}; \quad t_{1/2} = 654\text{ min}; \quad t_{0.050} = 1962\text{ min}

Answer. After 650 min: [H₂O₂] = 0.201 M; Half-life = 654 min; Time to reach 0.050 M = 1962 min (≈ 3 half-lives)

Figure
Diagram
First-Order Kinetics: Concentration vs Time and ln[A] vs Time[A] vs Time (exponential)[A]₀t₁/₂2t₁/₂[A]₀/2[A]₀/4Time[A][A]=[A]₀e⁻ᵏᵗln[A] vs Time (linear)ln[A]₀Timeln[A]slope = −kriseruny-intercept = ln[A]₀Constant Half-Life Property[A]₀½[A]₀¼[A]₀⅛[A]₀t₁/₂t₁/₂t₁/₂Each interval= same t₁/₂
Origins · 1 minLudwig Ferdinand Wilhelmy and how this was discovered
LF
Ludwig Ferdinand Wilhelmy
1812–1864 · German

Ludwig Wilhelmy conducted the first quantitative study of chemical kinetics in 1850, investigating the acid-catalyzed hydrolysis of sucrose — a classic first-order reaction. Using polarimetry to track the rotation of polarised light as sucrose converted to glucose and fructose, he derived the very first differential rate law and integrated rate equation in chemical history, writing the equation d[Z]/dt = −MZS[Z] (in modern notation: rate = k[sucrose]). His work laid the mathematical foundation for all of chemical kinetics. Wilhelmy's groundbreaking 1850 paper was largely ignored by the scientific community for nearly 20 years until Wilhelm Ostwald rediscovered it in 1884 and recognised it as the origin of quantitative chemical kinetics — making Wilhelmy one of history's most underappreciated scientific pioneers.

  • Second-Order Integrated Rate Laws — Comparing second-order kinetics (rate = k[A]², plot of 1/[A] vs t is linear, half-life depends on [A]₀) directly against first-order behaviour reinforces how reaction order fundamentally changes concentration–time relationships.
  • Rate Laws and Reaction Order — Understanding the differential rate law (rate = k[A]^n) and how to determine reaction order experimentally is the essential prerequisite for deriving and applying integrated rate laws.
  • The Arrhenius Equation — The rate constant k in the integrated rate law is temperature-dependent. The Arrhenius equation k = Ae^(−Ea/RT) explains why k changes with temperature and connects kinetics to activation energy and transition state theory.
  • Radioactive Decay and Nuclear Half-Life — Radioactive decay is a perfect real-world example of first-order kinetics. The nuclear half-life concept is mathematically identical to the chemical half-life derived here, reinforcing the universality of first-order exponential decay.
  • Exponential Functions and Natural Logarithms — The derivation of the integrated rate law requires understanding separation of variables and integration of 1/x, and the final equations involve e^x and ln(x). A solid grasp of these mathematical functions is essential for manipulating rate law equations.