Half-Life

Learning Objectives (OCR Spec 5.1.1)

By the end of this lesson you should be able to:

• Define the half-life of a reaction
• Recognise that constant half-life is evidence for a first-order reaction
• Use the relationship k = ln 2 / t(1/2) for first-order reactions
• Apply half-life to calculate rate constants and time-dependent concentrations

Definition

The half-life t(1/2) of a reactant is the time taken for its concentration to fall to half of its initial value.

For example, if [A] starts at 0.80 mol dm^-3 and after 50 s has dropped to 0.40 mol dm^-3, the half-life of A at that stage is 50 s.

Half-Life and Order

Half-life behaves very differently for different orders. This makes it one of the clearest diagnostic tools at A-Level.

Order	How t(1/2) varies with [A]0
0	t(1/2) decreases as [A]0 decreases (t(1/2) = [A]0 / 2k)
1	t(1/2) is constant — independent of [A]0
2	t(1/2) increases as [A]0 decreases (t(1/2) = 1 / (k[A]0))

The most important result: constant half-life proves a reaction is first order. This is a specification bullet point and very common in exams.

Why First Order Has Constant Half-Life

For a first-order reaction, [A] decays exponentially: [A] = [A]0 e^(-kt). Setting [A] = [A]0/2:

[A]0/2 = [A]0 e^(-k t(1/2))
1/2 = e^(-k t(1/2))
ln(1/2) = -k t(1/2)
-ln 2 = -k t(1/2)
t(1/2) = ln 2 / k

The half-life depends only on k, not on [A]0. So every successive half-life is the same length of time. After one half-life, [A] has fallen to 1/2 of its start. After two half-lives, 1/4. After three half-lives, 1/8.

This is why radioactive decay (which is first order) has a single, well-defined half-life.

The Key Equation

k = ln 2 / t(1/2) = 0.693 / t(1/2)

With t(1/2) in seconds, k has units of s^-1. With t(1/2) in minutes, k comes out in min^-1. Always convert to seconds for the final answer unless the question specifies otherwise.

Worked Example 1 — Finding k from Half-Life

A first-order reaction has a half-life of 120 s. Calculate k.

k = ln 2 / t(1/2)
  = 0.693 / 120
  = 5.78 x 10^-3 s^-1

Worked Example 2 — Finding Half-Life from k

A first-order reaction has rate constant k = 2.0 x 10^-4 s^-1. Calculate the half-life.

t(1/2) = ln 2 / k
       = 0.693 / (2.0 x 10^-4)
       = 3465 s
       approx 58 minutes

Worked Example 3 — Using Multiple Half-Lives

A first-order reaction starts with [A]0 = 0.60 mol dm^-3. The half-life is 25 s. What is [A] after 100 s?

100 s = 4 half-lives.