# 6.9.5 Half-Life

## Half-Life

• Half life is defined as:

The time taken for the initial number of nuclei to reduce by half

• This means when a time equal to the half-life has passed, the activity of the sample will also half
• This is because activity is proportional to the number of undecayed nuclei, A ∝ N When a time equal to the half-life passes, the activity falls by half, when two half-lives pass, the activity falls by another half (which is a quarter of the initial value)

## Determining the Half-Life of an Isotope

• To find an expression for half-life, start with the equation for exponential decay:

N = N0e–λt

• Where:
• N = number of nuclei remaining in a sample
• N0 = the initial number of undecayed nuclei (when t = 0)
• λ = decay constant (s-1)
• t = time interval (s)

• When time t is equal to the half-life t½, the activity N of the sample will be half of its original value, so N = ½ N0 • The formula can then be derived as follows:   • Therefore, half-life t½ can be calculated using the equation: • This equation shows that half-life t½ and the radioactive decay rate constant λ are inversely proportional
• Therefore, the shorter the half-life, the larger the decay constant and the faster the decay

#### Worked example

Strontium-90 is a radioactive isotope with a half-life of 28.0 years. A sample of Strontium-90 has an activity of 6.4 × 109 Bq.Calculate the decay constant λ, in s–1, of Strontium-90.

Step 1: Convert the half-life into seconds

28 years = 28 × 365 × 24 × 60 × 60 = 8.83 × 108 s

Step 2: Write the equation for half-life Step 3: Rearrange for λ and calculate #### Exam Tip

Make sure you are confident with the meanings of all the definitions and symbols in this unit. It is easy to get confused when completing an examination question. ### Get unlimited access

to absolutely everything:

• Unlimited Revision Notes
• Topic Questions
• Past Papers 