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

Exam code: H556

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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, A space proportional to space N

Half-Life Graph, downloadable AS & A Level Physics revision notes

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 space equals space N subscript 0 space e to the power of negative lambda t end exponent

  • 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 t subscript 1 divided by 2 end subscript, the activity N of the sample will be half of its original value, so N space equals space 1 half N subscript 0

1 half N subscript 0 space equals space N subscript 0 space e to the power of negative lambda t subscript 1 divided by 2 end subscript end exponent

  • The formula can then be derived by first dividing both sides by N subscript 0:

1 half space equals space e to the power of negative lambda t subscript 1 divided by 2 end subscript end exponent

  • Then, taking the natural log of both sides:

ln space open parentheses 1 half close parentheses space equals space minus lambda t subscript 1 divided by 2 end subscript

  • Finally, applying properties of logarithms:

lambda t subscript 1 divided by 2 end subscript space equals space ln space 2

  • Therefore, half-life t subscript 1 divided by 2 end subscript can be calculated using the equation:

t subscript 1 divided by 2 end subscript space equals space fraction numerator ln space 2 over denominator lambda end fraction space asymptotically equal to space fraction numerator 0.693 over denominator lambda end fraction

  • This equation shows that half-life t subscript 1 divided by 2 end subscript 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 lambda, in s–1, of strontium-90.

Answer:

Step 1: Convert the half-life into seconds

  • t subscript 1 divided by 2 end subscript = 28 years = 28 × (365 × 24 × 60 × 60) = 8.83 × 108 s

Step 2: Write the equation for half-life

t subscript 1 divided by 2 end subscript space equals space fraction numerator ln space 2 over denominator lambda end fraction

Step 3: Rearrange for λ and calculate

t subscript 1 divided by 2 end subscript space equals space fraction numerator ln space 2 over denominator 8.83 cross times 10 to the power of 8 end fraction space equals space 7.85 cross times 10 to the power of negative 10 end exponent space straight s to the power of negative 1 end exponent

Examiner Tips and Tricks

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. 

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, t (s)

  • Dependent variable = corrected count rate, C

  • Control variables:

    • Radioactive source

    • Distance of GM tube to source

    • Location / background radiation

Equipment list

Equipment

Purpose

sealed bottle containing uranium salt solution and protactinium-234

to use as a source of radioactive emission

Geiger-Muller tube and counter or datalogger

to measure the count rate of the radioactive source

stopwatch

to measure the same time interval for each count rate reading

tray lined with paper towel

to avoid contamination should the bottle leak

Method

Apparatus for determining the half-life of protactinium-234

A setup with a sealed bottle containing an organic layer with protactinium on the top and a uranyl nitrate layer on the bottom. A GM tube connected to a Geiger counter is aligned with the organic layer of the bottle.

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:

Time, t space divided by space straight s

Count rate divided by space counts space straight s to the power of negative 1 end exponent

Corrected count rate, C space divided by space counts space straight s to the power of negative 1 end exponent

0

30

60

90

120

150

Analysis of results

  • The count rate of the source decreases exponentially according to

C space equals space C subscript 0 space e to the power of negative lambda t end exponent

  • Taking the natural logs of both sides

ln space C space equals space ln space open parentheses C subscript 0 close parentheses space minus space lambda t

  • Compared to the equation of a straight line open parentheses y space equals space m x space plus space c close parentheses:

    • y-axis variable, y space equals space ln space C

    • x-axis variable, x space equals space t

    • gradient = negative lambda space equals space minus fraction numerator ln space 2 over denominator t subscript 1 divided by 2 end subscript end fraction

    • y-intercept = ln space open parentheses C subscript 0 close parentheses

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

  2. Plot a graph of ln space C against time t

  3. Determine the half-life t subscript 1 divided by 2 end subscript from the gradient of the best-fit line

t subscript 1 divided by 2 end subscript space equals space minus fraction numerator ln space 2 over denominator gradient end fraction

Log graph of corrected count rate against time

Graph of ln C versus time, showing a negative linear slope. The initial value is ln C₀ and the gradient is labelled as negative lambda.

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

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Katie M

Author: Katie M

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Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.