Half-Life (OCR A Level Physics): Revision Note

Half-Life

• Half life is defined as:

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

• This means that when a time equal to the half-life has passed, the activity of the sample will be half of its original value

• This is because activity is proportional to the number of undecayed nuclei,

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:

• 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 is equal to the half-life , the activity of the sample will be half of its original value, so

• The formula can then be derived by first dividing both sides by :

• Then, taking the natural log of both sides:

• Finally, applying properties of logarithms:

• Therefore, half-life can be calculated using the equation:

• This equation shows that half-life 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

Determining the Half-Life of an Isotope

Aim of the experiment

• The aim of this experiment is to determine the half-life of an isotope such as protactinium

Variables:

• Independent variable = time, (s)

• Dependent variable = corrected count rate,

• Control variables:

  • Radioactive source

  • Distance of GM tube to source

  • Location / background radiation

Equipment list

Method

Apparatus for determining the half-life of protactinium-234

In the lower layer of the sealed bottle, uranium-238 decays to thorium-234 and then to protactinium-234. Shaking the bottle separates the isotopes as protactinium dissolves in the organic solvent, whereas thorium does not.

1. Connect the Geiger-Müller tube to the counter and, without any sources present, measure background radiation over a period of one minute

2. Repeat this three times, and take an average. Subtract this value from all subsequent readings.

3. Shake the bottle gently for 15 seconds to dissolve the protactinium in the organic layer, then wait for it to float to the top

   • This is done because the uranium salt in the aqueous layer decays to form protactinium, which is soluble in the organic solvent but insoluble in the aqueous layer

4. Align the GM tube with the organic layer and immediately start a stopwatch

5. Record the number of counts in 10 seconds, and repeat every 30 seconds

• A suitable table of results might look like this:

Analysis of results

• The count rate of the source decreases exponentially according to

• Taking the natural logs of both sides

• Compared to the equation of a straight line :

  • y-axis variable,

  • x-axis variable,

  • gradient =

  • y-intercept =

1. Subtract the background radiation from each count rate reading to give the corrected count rate

2. Plot a graph of against time

3. Determine the half-life from the gradient of the best-fit line

Log graph of corrected count rate against time

By plotting a log graph of ln C against time, the decay constant and half-life can be found directly from the gradient

Evaluating the experiment

Systematic Errors:

• Position the GM tube close to the bottle, but make sure they don't touch

  • This ensures that only the beta particles from the protactinium are detected, and not the alpha particles from the uranium sample (which are absorbed by the bottle)

Random Errors:

• Radioactive decay is random, so repeat readings are vital in this experiment

• The half-life of protactinium is short, so more frequent readings of count rate may be necessary

Safety considerations

• Protactinium-234 is safe to use due to its short half-life

• Ensure the bottle remains sealed at all times to prevent leakage

• Handle the bottle with long tongs

• Safety clothing such as a lab coat, gloves and goggles must be worn