Half-Life (Zero Order Reaction) Equation, Derivation & Rearrangements

Core idea

Overview

The half-life (t₁/₂) of a zero-order reaction is the time required for the concentration of a reactant to decrease to half its initial value. Unlike higher-order reactions, for a zero-order reaction, the half-life is directly proportional to the initial concentration of the reactant and inversely proportional to the rate constant. This means that as the initial concentration increases, the half-life also increases, which is a unique characteristic of zero-order kinetics.

When to use: Use this equation when analyzing the kinetics of a zero-order reaction and needing to determine the time it takes for half of the initial reactant to be consumed. It's particularly useful in scenarios where the reaction rate is independent of reactant concentration, such as enzyme-catalyzed reactions at saturation or reactions occurring on a surface.

Why it matters: Understanding the half-life of zero-order reactions is crucial in fields like pharmacology (drug degradation), environmental science (pollutant breakdown), and industrial chemistry (catalytic processes). It allows for prediction of reactant depletion over time, aiding in shelf-life determination, process optimization, and understanding reaction mechanisms.

Symbols

Variables

[A]_0 = Initial Concentration of A, k = Rate Constant, $t_{1/2}$ = Half-Life

[A]_{0}

Initial Concentration of A

mol/L

k

Rate Constant

m o l L^{- 1} s^{- 1}

t_{1/2}

Half-Life

s

Walkthrough

Derivation

Formula: Half-Life (Zero Order Reaction)

The half-life for a zero-order reaction is the time required for the reactant concentration to decrease to half its initial value.

The reaction strictly follows zero-order kinetics.
The rate constant 'k' remains constant throughout the reaction.

1

Start with the Integrated Rate Law for a Zero-Order Reaction:

This equation describes the concentration of reactant A at time t, [A]t, in terms of its initial concentration [A]₀, the rate constant k, and time t.

[A]_{t} = [A]_{0} - k t

2

Define Half-Life Conditions:

By definition, at the half-life (t₁/₂), the concentration of the reactant has decreased to half of its initial concentration.

A tt = t_{1/2}, [A]_{t} = \frac{[ A ] _{0}}{2}

3

Substitute Half-Life Conditions into the Integrated Rate Law:

Substitute the half-life conditions into the integrated rate law equation.

\frac{[ A ] _{0}}{2} = [A]_{0} - k t_{1/2}

4

Rearrange to Solve for t_{1/2}:

Move the kt₁/₂ term to the left and [A]₀/2 to the right.

k t_{1/2} = [A]_{0} - \frac{[ A ] _{0}}{2}

5

Simplify the Right Side:

Combine the terms on the right side: [A]₀ - [A]₀/2 = [A]₀/2.

k t_{1/2} = \frac{[ A ] _{0}}{2}

6

Isolate t_{1/2}:

Divide both sides by k to obtain the formula for the half-life of a zero-order reaction.

t_{1/2} = \frac{[ A ] _{0}}{2 k}

Result

t_{1/2} = \frac{[ A ] _{0}}{2 k}

Source: Atkins' Physical Chemistry, 11th Edition — Chapter 20: Chemical Kinetics

Free formulas

Rearrangements

Solve for $[A]_{0}$

Half-Life (Zero Order Reaction): Make [A]₀ the subject

[A]_{0} = 2 k t_{1/2}

To make [A]₀ (Initial Concentration) the subject, multiply both sides by 2k.

Difficulty: 2/5

Solve for $k$

Half-Life (Zero Order Reaction): Make k the subject

k = \frac{[ A ] _{0}}{2 t _{1/2}}

To make k (Rate Constant) the subject, first multiply by 2k, then divide by t₁/₂.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph follows a hyperbolic curve where the half-life decreases as the rate constant increases, approaching an asymptote at the horizontal axis. For a chemistry student, this shape illustrates that a larger rate constant results in a significantly shorter half-life, meaning the reaction reaches its midpoint much faster. The most important feature of this curve is the inverse relationship between the variables, which ensures that the half-life never reaches zero regardless of how high the rate constant becomes. The domain remains restricted to positive values because a rate constant cannot be zero or negative in this kinetic context.

Graph type: hyperbolic

Why it behaves this way

Intuition

Picture a fixed-rate process, like a constant-speed conveyor belt removing items from a pile; the half-life is the time it takes to remove half the initial items, and thus depends directly on the initial pile size.

t_{1/2}

The time required for the concentration of a reactant to decrease to half its initial value.

It's a direct measure of how long it takes to consume half of the starting material. For zero-order reactions, this time is longer if you start with more reactant.

[A]_{0}

The initial concentration of reactant A at the beginning of the reaction (time t=0).

Represents the total amount of reactant available at the start. For zero-order kinetics, a higher initial concentration means a proportionally longer half-life because the consumption rate is constant.

k

The rate constant for the zero-order reaction, which is equal to the reaction rate itself.

Quantifies the intrinsic speed of the reaction. A larger 'k' means the reaction proceeds faster, leading to a shorter half-life.

Free study cues

Insight

Canonical usage

Units for half-life, initial concentration, and the zero-order rate constant must be consistent to yield a half-life in a standard time unit.

Common confusion

A common mistake is using inconsistent units for concentration (e.g., mol L-1 for [A]0 and mol dm-3 for k) or time (e.g., seconds for k and minutes for t1/2), leading to incorrect half-life values.

Unit systems

$t_{1/2}$ s · Commonly expressed in seconds (s), minutes (min), or hours (h), depending on the reaction rate.

$[A]_{0}$ mol L^-1 · Typically expressed in moles per liter (mol L-1) or molarity (M).

$k$ mol L^-1 s^-1 · For a zero-order reaction, the rate constant has units of concentration per time, consistent with the reaction rate.

One free problem

Practice Problem

A drug degrades in the body via a zero-order reaction. If its initial concentration is 0.5 M and the rate constant for its degradation is 0.02 M s⁻¹, what is the half-life of the drug?

Initial Concentration of A0.5 mol/L

Rate Constant0.02 mol L^{-1} s^{-1}

Solve for: $t_{h} a l f$

Hint: Remember to use the formula t₁/₂ = [A]₀ / (2k) and ensure units are consistent.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Determining the shelf-life of certain pharmaceuticals that degrade via zero-order kinetics.

Study smarter

Tips

Ensure units for concentration and rate constant are consistent (e.g., mol L⁻¹ and mol L⁻¹ s⁻¹).
Remember that for zero-order reactions, t₁/₂ is NOT constant but depends on initial concentration.
The rate constant 'k' for a zero-order reaction has units of concentration per unit time (e.g., mol L⁻¹ s⁻¹).
This formula is specific to zero-order reactions; do not apply it to first or second-order reactions.

Avoid these traps

Common Mistakes

Applying the formula to reactions that are not zero-order.
Incorrectly using units for k or [A]₀, leading to incorrect t₁/₂ units.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The half-life for a zero-order reaction is the time required for the reactant concentration to decrease to half its initial value.

Use this equation when analyzing the kinetics of a zero-order reaction and needing to determine the time it takes for half of the initial reactant to be consumed. It's particularly useful in scenarios where the reaction rate is independent of reactant concentration, such as enzyme-catalyzed reactions at saturation or reactions occurring on a surface.

Understanding the half-life of zero-order reactions is crucial in fields like pharmacology (drug degradation), environmental science (pollutant breakdown), and industrial chemistry (catalytic processes). It allows for prediction of reactant depletion over time, aiding in shelf-life determination, process optimization, and understanding reaction mechanisms.

Applying the formula to reactions that are not zero-order. Incorrectly using units for k or [A]₀, leading to incorrect t₁/₂ units.

Determining the shelf-life of certain pharmaceuticals that degrade via zero-order kinetics.

Ensure units for concentration and rate constant are consistent (e.g., mol L⁻¹ and mol L⁻¹ s⁻¹). Remember that for zero-order reactions, t₁/₂ is NOT constant but depends on initial concentration. The rate constant 'k' for a zero-order reaction has units of concentration per unit time (e.g., mol L⁻¹ s⁻¹). This formula is specific to zero-order reactions; do not apply it to first or second-order reactions.

References

Sources

Atkins' Physical Chemistry
Wikipedia: Zero-order reaction
McQuarrie's Physical Chemistry: A Molecular Approach
IUPAC Gold Book
Atkins, P. W., & de Paula, J. (2014). Atkins' Physical Chemistry (10th ed.). Oxford University Press. (Chapter on Chemical Kinetics)
IUPAC Gold Book. (2019). 'Half-life'. Retrieved from https://goldbook.iupac.org/terms/view/H02700
IUPAC Gold Book. (2019). 'Order of reaction'. Retrieved from https://goldbook.iupac.org/terms/view/O04322
Atkins' Physical Chemistry, 11th Edition — Chapter 20: Chemical Kinetics

Half-Life (Zero Order Reaction)

Overview

Variables

Derivation

Start with the Integrated Rate Law for a Zero-Order Reaction:

Define Half-Life Conditions:

Substitute Half-Life Conditions into the Integrated Rate Law:

Rearrange to Solve for t_{1/2}:

Simplify the Right Side:

Isolate t_{1/2}:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Integrated Rate Law (Zero Order)

Half-Life (1st Order)

Frequently Asked Questions

Sources