# 8.2.3 Half-Life

### Half-Life

• Half-life is defined as:

The time taken for the initial number of nuclei to halve for a particular isotope

• This means when a time equal to the half-life has passed, the activity of the sample will also half
• This is because the activity is proportional to the number of undecayed nuclei, AN 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)

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

N = N0 e–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)
• 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

• t½ = 28 years = 28 × 365 × 24 × 60 × 60 = 8.83 × 108 s

Step 2: Write the equation for half-life #### Exam Tip

Although you may not be expected to derive the half-life equation, make sure you’re comfortable with how to use it in calculations such as that in the worked example.

### Half-Life from Decay Curves

• The half-life of a radioactive substance can be determined from decay curves and log graphs
• Since half-life is the time taken for the initial number of nuclei, or activity, to reduce by half, it can be found by
• Drawing a line to the curve at the point where the activity has dropped to half of its original value
• Drawing a line from the curve to the time axis, this is the half-life

#### Log Graphs

• Straight-line graphs tend to be more useful than curves for interpreting data
• Nuclei decay exponentially, therefore, to achieve a straight line plot, logarithms can be used
• Take the exponential decay equation for the number of nuclei

N = N0 e–λt

• Taking the natural logs of both sides

ln N = ln (N0) − λt

• In this form, this equation can be compared to the equation of a straight line

y = mx + c

• Where:
• ln (N) is plotted on the y-axis
• t is plotted on the x-axis
• gradient = −λ
• y-intercept = ln (N0)
• Half-lives can be found in a similar way to the decay curve but the intervals will be regular as shown below:

#### Worked Example

The radioisotope technetium is used extensively in medicine. The graph below shows how the activity of a sample varies with time.

Determine:

a) The decay constant for technetium

b) The number of technetium atoms remaining in the sample after 24 hours

Part (a)

Step 1: Draw lines on the graph to determine the time it takes for technetium to drop to half of its original activity

Step 2: Read the half-life from the graph and convert to seconds

• t ½ = 6 hours = 6 × 60 × 60 = 21 600 s

Step 3: Write out the half life equation Step 4: Calculate the decay constant Part (b)

Step 1: Draw lines on the graph to determine the activity after 24 hours

• At t = 24 hours, A = 0.5 × 107 Bq

Step 2: Write out the activity equation

A = λN

Step 3: Calculate the number of atoms remaining in the sample Close Close

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