Half-life Revision Notes

The half-life of a radioactive isotope is the time taken for half of the nuclei in a given sample to decay. This means that if you start with a certain number of radioactive atoms, after one half-life only half of those atoms remain undecayed.

Radioactive decay is a random process at the level of individual nuclei, but half-life provides a statistical measure of how quickly a large number of nuclei decay.

Each isotope has a unique half-life, which can range from fractions of a second to millions of years, depending on its nuclear stability.

• Half-life (symbol: t₁/₂) is specific to each isotope.
• It is a constant for a given isotope, independent of the amount of substance or physical conditions.
• After one half-life, the number of undecayed nuclei reduces to half the initial number.
• After two half-lives, the number reduces to one quarter, and so on, following an exponential decay pattern.

The number of undecayed nuclei decreases exponentially over time. The relationship between the number of nuclei remaining (N) after time t and the initial number (N₀) is given by:

N = N₀ × (½)^(t / t₁/₂) where t₁/₂ is the half-life.

This formula allows calculation of the remaining nuclei after any time period, provided the half-life is known.

• Use N = N₀ × (½)^(t / t₁/₂) for calculations involving half-life.
• t is the elapsed time, t₁/₂ is the half-life, N₀ is the initial number of nuclei, N is the number remaining.
• The formula can be rearranged to find t or t₁/₂ if other quantities are known.
• Calculations exclude background radiation effects for simplicity.

Decay graphs plot the number of undecayed nuclei or activity against time. The vertical axis usually shows number of nuclei or activity (in becquerels, Bq), and the horizontal axis shows time (in seconds, minutes, hours, or years).

The half-life can be found by identifying the time interval for the quantity to reduce to half its initial value.

Tables provide discrete data points of nuclei number or activity at different times, which can be used to calculate half-life or remaining nuclei by interpolation or direct application of the decay formula.

• Vertical axis: number of nuclei or activity (Bq).
• Horizontal axis: time (s, min, h, years).
• Half-life is the time interval for the quantity to halve on the graph.
• Use data points from tables or graphs for calculations.
• Graphs show exponential decay shape, decreasing rapidly at first then more slowly.

Example: A sample contains 8000 nuclei of a radioactive isotope with a half-life of 2 hours. Calculate how many nuclei remain after 6 hours.

Step 1: Identify known values: N₀ = 8000 nuclei, t₁/₂ = 2 hours, t = 6 hours.

Step 2: Calculate the number of half-lives elapsed: number of half-lives = t / t₁/₂ = 6 / 2 = 3 half-lives, meaning the sample halves three times in 6 hours, reducing to (½)³ = 1/8 of the original amount remaining after 6 hours, so N = 8000 × 1/8 = 1000 nuclei remain.

• Formula: N = N₀ × (½)^(t / t₁/₂)
• Substitute: N = 8000 × (½)^(6/2) = 8000 × (½)^3
• Calculate: N = 8000 × 1/8 = 1000 nuclei
• Answer: 1000 nuclei remain after 6 hours.
• Common error: forgetting to divide time by half-life or miscalculating powers of ½.

Half-life is measured by recording the activity (decay rate) of a radioactive sample over time using a Geiger-Müller tube or scintillation counter.

Activity is proportional to the number of undecayed nuclei and decreases exponentially with time.

To improve accuracy, multiple measurements are taken at regular intervals, and background radiation is subtracted from readings (though background is excluded in this objective's calculations).

• Use a Geiger-Müller tube to measure count rate at intervals.
• Plot count rate against time to obtain a decay curve.
• Determine half-life from the time for count rate to halve.
• Repeat measurements to reduce random errors.
• Ensure safety by using appropriate shielding and handling protocols.

The half-life of a radioactive isotope is the time required for half of the nuclei in a given sample to decay. This means that after one half-life, the quantity of the radioactive substance remaining is half of its initial amount.

Half-life is a characteristic property of each radioactive isotope and is independent of the initial amount or physical conditions such as temperature or pressure.

The unit of half-life is seconds (s), minutes (min), hours (h), or years (y), depending on the isotope's decay rate.

• Half-life (T₁/₂) = time for quantity to reduce to half its initial value.
• Characteristic of each isotope.
• Units vary with isotope decay speed.

Decay graphs typically plot the quantity of radioactive nuclei or the activity (count rate) on the vertical axis against time on the horizontal axis.

The vertical axis is usually labelled as 'Activity (counts per second)' or 'Number of nuclei (arbitrary units)', and the horizontal axis as 'Time (s)' or appropriate time units.

To find the half-life from such a graph: identify the initial quantity at time zero, then find the time at which the quantity reduces to half that initial value. The time difference between these two points is the half-life.

• Vertical axis: quantity or activity (counts/s).
• Horizontal axis: time (s, min, h).
• Half-life = time interval for quantity to halve.
• Graph shape is a decreasing curve, not linear.

Given a table of decay data showing quantity or activity at various times, the half-life can be calculated by:

1. Identifying the initial quantity (Q₀). 2. Finding the time at which the quantity is half of Q₀ (Q₀/2). 3. The difference in time between these two points is the half-life.

If the data allows, repeat this process for subsequent halving intervals to check for consistency and improve accuracy.

• Identify initial quantity Q₀.
• Find time t when quantity = Q₀/2.
• Calculate T₁/₂ = t - 0.
• Repeat for subsequent halves to confirm.

Background radiation is the natural radiation detected by instruments even when no radioactive source is present.

When determining half-life from decay data or graphs, background radiation must be excluded because it adds a constant count rate that does not decay.

If background radiation is not subtracted, the decay curve will not reach zero and the half-life calculation will be inaccurate, typically underestimated because the apparent activity decreases more slowly near the end of the measurement period.

• Background radiation adds constant counts.
• Subtract background counts before analysis.
• Ensures accurate half-life determination.
• Avoids underestimating decay rate.

Suppose a decay graph shows an initial activity of 800 counts per second at time zero.

Find the time when activity falls to 400 counts per second (half of 800). This occurs at 3 minutes.

Therefore, the half-life T₁/₂ = 3 minutes for this isotope according to the graph data.

• Initial activity Q₀ = 800 counts/s.
• Half activity Q₀/2 = 400 counts/s.
• Time at half activity = 3 min.
• Half-life T₁/₂ = 3 min.

Radioactive isotopes emit three main types of radiation: alpha (α), beta (β), and gamma (γ) rays, each with distinct penetration abilities and biological effects.

Alpha particles are helium nuclei (2 protons and 2 neutrons) with a +2 charge. They have low penetration power and can be stopped by a sheet of paper or a few centimetres of air. They are highly ionising but cannot penetrate the skin.

Beta particles are high-speed electrons (β⁻) or positrons (β⁺) with moderate penetration, stopped by a few millimetres of aluminium. They are less ionising than alpha particles but can penetrate skin and cause damage internally if ingested or inhaled.

• Alpha: low penetration, high ionisation, stopped by paper.
• Beta: moderate penetration, moderate ionisation, stopped by aluminium.
• Gamma: electromagnetic radiation, high penetration, low ionisation, requires thick lead or concrete shielding.

The half-life of a radioactive isotope is the time taken for half the nuclei in a sample to decay. It determines how long the isotope remains active and radioactive.

Isotopes with short half-lives decay quickly, making them suitable for medical applications where rapid decay reduces radiation exposure after use.

Isotopes with long half-lives remain radioactive for extended periods, useful for applications requiring long-term activity, such as smoke alarms or food irradiation.

• Half-life affects duration of radioactivity and safety.
• Short half-life: rapid decay, less long-term hazard.
• Long half-life: sustained activity, useful for continuous applications.

Smoke alarms commonly use Americium-241, an alpha emitter with a half-life of about 432 years.

Alpha radiation ionises air inside the alarm chamber, allowing a small current to flow. Smoke particles disrupt this current, triggering the alarm.

The alpha particles' low penetration ensures safety, as the radiation cannot escape the sealed chamber easily, and the long half-life means the device works reliably for many years.

• Use alpha emitters for ionisation in smoke alarms.
• Long half-life ensures device longevity.
• Low penetration alpha radiation minimises external hazard.

Food irradiation uses isotopes emitting gamma rays, such as Cobalt-60, to kill bacteria and pests without raising food temperature.

Gamma rays have high penetration, allowing treatment of packaged food and bulk quantities.

Cobalt-60 has a half-life of about 5.27 years, balancing sufficient activity for sterilization and manageable replacement schedules for safety and cost.

• Gamma emitters kill bacteria by damaging DNA.
• High penetration treats bulk and packaged food.
• Moderate half-life balances effectiveness and safety.

Beta or gamma radiation is used to measure and control material thickness in manufacturing.

The radiation source is placed on one side of the material, and a detector on the other side measures transmitted radiation intensity.

Materials of different thickness absorb or scatter radiation differently; the detected intensity correlates with thickness, allowing real-time control during production.

• Beta radiation preferred for thin materials due to moderate penetration.
• Gamma radiation used for thicker or denser materials.
• Isotope choice depends on required penetration and absorption characteristics.

Gamma-emitting isotopes, such as Technetium-99m for diagnosis and Cobalt-60 for radiotherapy, are used in medicine.

Gamma rays penetrate body tissues, allowing imaging or targeted destruction of cancer cells.

Short half-life isotopes like Technetium-99m (6 hours) reduce radiation dose to patients after diagnosis, while longer half-life isotopes provide sustained radiation for treatment.

• Gamma rays penetrate tissues for imaging and therapy.
• Short half-life isotopes minimise patient radiation exposure.
• Longer half-life isotopes deliver therapeutic doses over time.