Zero-Order Reaction Kinetics

A zero-order reaction is one whose rate does not depend on the concentration of the reactant: rate = k, and the integrated rate law is [A] = [A]₀ − kt, so the concentration falls in a straight line. In brief: with [A]₀ = 0.800 M and k = 0.0200 M/s, the half-life is [A]₀/(2k) = 20.0 s and the reactant is gone after 40.0 s.

Zero-order kinetics is rarer than first- or second-order kinetics, but it appears in important situations: reactions on the surface of a catalyst, enzyme reactions with excess substrate and the metabolism of alcohol in the body. In such cases some other factor, such as the number of available catalytic sites, limits the rate, not the concentration of the reactant. This article gives the equations, a worked example with data, the half-life formula and the physical explanations.

What is the rate law of a zero-order reaction?

For the reaction A → products with zero order in A:

rate = −d[A]/dt = k[A]0 = k

The rate stays constant throughout the reaction, as long as some A is present, whatever the amount of A. The rate constant has the units of rate, M/s (mol L⁻¹ s⁻¹). Doubling [A] has no effect on the rate.

What is the integrated rate law?

Integrating the rate law gives:

[A] = [A]0 − kt

This is the equation of a straight line. A graph of [A] against t is a line with a slope of −k and an intercept of [A]₀. This is the test for zero order: if the plot of concentration against time is a straight line, the reaction is zero order. The time at which all of the reactant is used up is t = [A]₀/k.

How does the half-life change?

The half-life is the time needed for the concentration to fall to half of its value. Substituting [A] = [A]₀/2 gives:

t½ = [A]0 / (2k)

Unlike in first-order reactions, the half-life of a zero-order reaction depends on the initial concentration, and each successive half-life is half as long as the previous one, because the same amount is consumed per unit time and there is less to consume. The reaction ends in a finite time, whereas in other orders the concentration only approaches zero asymptotically.

Data for a zero-order reaction

The rate is the same in every row, and the concentration falls by 0.200 M every 10 s. The first half-life is 20 s (0.800 → 0.400 M) and the second is only 10 s (0.400 → 0.200 M).

Worked examples

Example 1: time to reach a concentration

How long does it take for [A] to fall from 0.800 M to 0.200 M, if k = 0.0200 M/s? t = ([A]₀ − [A]) / k = (0.800 − 0.200) / 0.0200 = 30.0 s.

Example 2: finding k from a graph

A plot of [A] against time is a straight line passing through (0 s, 1.20 M) and (60 s, 0.90 M). The slope is (0.90 − 1.20) / 60 = −0.0050 M/s, so k = 0.0050 M/s. The reaction will be complete after t = 1.20 / 0.0050 = 240 s.

Example 3: alcohol in the body

Ethanol is broken down in the liver by the enzyme alcohol dehydrogenase, which becomes saturated at moderate blood alcohol levels, so that elimination is approximately zero order. The blood alcohol concentration falls by roughly 0.015 % (by mass-volume, about 0.15 g/L) per hour in many people, with a large variation between individuals. A person who starts at 0.075 % needs approximately 0.075 / 0.015 = 5 hours to reach zero. No amount of coffee or cold shower changes the rate, as the liver enzyme sets the speed.

Why do zero-order reactions occur?

• Surface-catalyzed reactions: if the surface of the catalyst is saturated with reactant molecules, the rate is limited by the number of active sites, and not by the concentration of the gas. An example is the decomposition of ammonia on a hot tungsten surface at high pressure, 2NH₃ → N₂ + 3H₂.
• Enzyme reactions: when the substrate is in a large excess, all enzyme molecules are occupied, and the rate has reached its maximum (V_max).
• Photochemical reactions: if the rate is determined by the intensity of the light and not by the concentration of the absorbing substance.
• Dissolving of drugs from controlled-release devices: the aim is a constant rate of release.

It is important to note that a reaction can be zero order only over a certain range of concentration. When the reactant has been used up to a low level, the catalyst is no longer saturated, and the order changes to first order.

Where was zero-order kinetics first seen?

Zero-order behavior was recognized in studies of reactions on solid surfaces in the early 20th century, for example in the work of Cyril Hinshelwood on the decomposition of ammonia and other gases on metal surfaces. The theory of enzyme kinetics proposed by Leonor Michaelis and Maud Menten in 1913 explained why the rate of an enzyme reaction becomes independent of the substrate concentration at high concentrations.

Which mistakes are common?

• Thinking that zero order means no reaction: it means the rate is independent of concentration.
• Using t_½ = 0.693/k: that formula belongs to first-order reactions.
• Plotting ln[A] against t: for zero order the straight line is [A] against t.
• Forgetting the units of k: they are M/s.
• Extrapolating below zero concentration: the line stops when [A] reaches zero.