Zero Order

In chemical kinetics, the integrated rate law for a zero-order reaction describes how the concentration of a reactant decreases linearly with time. Unlike first- or second-order reactions, a zero-order reaction proceeds at a rate that is completely independent of the concentration of the reactant. This means doubling or halving the concentration of the reactant has absolutely no effect on how fast the reaction proceeds — the rate remains constant throughout the entire reaction until the reactant is fully consumed. Zero-order kinetics are most commonly observed in enzyme-catalyzed reactions when the enzyme is saturated with substrate, in heterogeneous catalytic reactions where the surface of the catalyst is completely occupied, and in certain photochemical reactions where the rate is limited by the intensity of light rather than reactant concentration. A classic example is the decomposition of ammonia on a hot platinum surface: \ce{2NH3(g) ->[(Pt)] N2(g) + 3H2(g)}, which follows zero-order kinetics when the platinum surface is fully covered with adsorbed ammonia molecules. Understanding zero-order reactions is important not just in laboratory chemistry, but also in pharmacology and medicine. Many drug elimination processes in the body — such as the metabolism of ethanol (alcohol) by the liver — follow zero-order kinetics when the metabolic enzymes are working at maximum capacity. In these situations, the body eliminates a fixed amount of the substance per unit time, regardless of how much is present, which has significant implications for dosing and toxicity.

Key takeaways
  • The Rate Expression for Zero Order
  • Setting Up the Differential Equation
  • Integration to Obtain the Integrated Rate Law
  • Units of the Zero-Order Rate Constant
  • The Half-Life of a Zero-Order Reaction

The Rate Expression for Zero Order

For a general reaction \ce{A -> products}, the rate law for a zero-order reaction is written as: rate = k[A]^0 = k. Since any quantity raised to the power zero equals 1, the rate simplifies to just the rate constant k. This is the defining feature of zero-order kinetics: the rate of reaction equals the rate constant at all times, regardless of concentration.

Setting Up the Differential Equation

Mathematically, the rate of change of concentration with time is expressed as: -d[A]/dt = k. The negative sign indicates that [A] is decreasing over time. This is a simple first-order ordinary differential equation in the calculus sense (not to be confused with first-order kinetics). Separating variables gives: d[A] = -k\,dt.

Integration to Obtain the Integrated Rate Law

Integrating both sides — the left from [A]_0 to [A]_t, and the right from 0 to t — gives: [A]_t - [A]_0 = -kt, which rearranges to: [A]_t = [A]_0 - kt. This is the integrated rate law for a zero-order reaction. It has the form y = mx + b, meaning a plot of [A]_t (y-axis) versus time t (x-axis) produces a straight line with slope = -k and y-intercept = [A]_0. This linear graph is the key diagnostic test for zero-order behavior.

Units of the Zero-Order Rate Constant

Since rate = k for a zero-order reaction, the units of k must equal the units of reaction rate. Reaction rate is measured in mol·L⁻¹·s⁻¹ (or M/s). Therefore, the zero-order rate constant k has units of mol·L⁻¹·s⁻¹ (M s⁻¹). This is distinct from first-order (units: s⁻¹) and second-order (units: M⁻¹s⁻¹) rate constants. Always checking units is a powerful way to confirm the reaction order.

The Half-Life of a Zero-Order Reaction

The half-life (t₁/₂) is the time required for the concentration of reactant to fall to exactly half its initial value. Setting [A]_t = [A]_0/2 in the integrated rate law: [A]_0/2 = [A]_0 - kt₁/₂, which gives kt₁/₂ = [A]_0/2, so t₁/₂ = [A]_0 / (2k). This is a critical and distinctive result: the zero-order half-life is directly proportional to the initial concentration [A]_0. As the reaction proceeds and [A]_0 effectively decreases, each successive half-life becomes shorter — the opposite of first-order reactions where the half-life is constant, and very different from second-order reactions.

The Key Equation

[A]t=[A]0−kt[A]_t = [A]_0 - kt
  • [A]_t — Concentration at time t (mol·L⁻¹ (M)): The molar concentration of reactant A remaining at a given time t during the reaction.
  • [A]_0 — Initial concentration (mol·L⁻¹ (M)): The molar concentration of reactant A at the start of the reaction (t = 0).
  • k — Zero-order rate constant (mol·L⁻¹·s⁻¹ (M s⁻¹)): A proportionality constant that reflects how fast the reaction proceeds; its value depends on temperature but not on concentration.
  • t — Time (seconds (s)): The elapsed time since the reaction began. The equation is valid only while [A]_t > 0, i.e., before the reactant is completely consumed.

Worked Example

Problem. The decomposition of a drug in a saturated enzyme system follows zero-order kinetics with a rate constant k = 0.020 mol·L⁻¹·min⁻¹. The initial concentration of the drug is 1.00 mol·L⁻¹. (a) What is the concentration of the drug after 30.0 minutes? (b) What is the half-life of this reaction? (c) At what time will the drug be completely consumed?

[A]30=0.40 M,t1/2=25.0 min,tconsumed=50.0 min[A]_{30} = 0.40\text{ M},\quad t_{1/2} = 25.0\text{ min},\quad t_{\text{consumed}} = 50.0\text{ min}

Answer. After 30.0 min: [A] = 0.40 M; Half-life = 25.0 min; Drug fully consumed at t = 50.0 min.

Figure
Diagram
Zero-Order Reaction: Concentration vs. TimeTime (min)[A] (mol/L)00.250.500.751.0001020304050t½ = 25 mint = 50 min[A] = 0Slope = −k= −0.020 M/min[A]₀ = 1.00 MReactant fully consumed
Origins · 1 minLudwig Ferdinand Wilhelmy and how this was discovered
LF
Ludwig Ferdinand Wilhelmy
1812–1864 · German

Ludwig Wilhelmy was the first scientist to quantitatively study chemical reaction rates using mathematical methods. In 1850, he measured the rate of inversion of sucrose (conversion of sucrose to glucose and fructose in acidic solution) and derived the first differential rate equation in chemistry. Although his work predated the concept of reaction orders by decades, his pioneering use of calculus to describe how concentration changes over time laid the mathematical foundation for all integrated rate laws, including zero-order kinetics. Wilhelmy's groundbreaking 1850 paper on reaction kinetics was largely ignored by the scientific community for over 30 years until Wilhelm Ostwald rediscovered and championed it in the 1880s, finally establishing Wilhelmy's rightful place as a founder of chemical kinetics.

  • First-Order Integrated Rate Law — Comparing zero-order (linear [A] vs t) with first-order (linear ln[A] vs t) kinetics reinforces how reaction order fundamentally changes the mathematical relationship between concentration and time.
  • Second-Order Integrated Rate Law — Understanding all three common reaction orders together allows students to use graphical methods (choosing the linear plot) to experimentally determine reaction order from concentration-time data.
  • Rate Laws and Reaction Order — The differential rate law (rate = k[A]^n) is the starting point from which the integrated rate law is derived; understanding the difference between differential and integrated forms is essential.
  • Half-Life in Chemical Reactions — The concept of half-life ties together all reaction orders; contrasting t½ = [A]₀/2k (zero-order) with t½ = ln2/k (first-order) highlights how the dependence of half-life on initial concentration changes with reaction order.
  • Arrhenius Equation and Activation Energy — The rate constant k in the zero-order integrated rate law depends on temperature according to the Arrhenius equation, connecting integrated rate laws to the molecular energy perspective of kinetics.