Dr Mark Marais

May 2025

Dr Mark Marais

May 2025

Radioactive Decay Law

the number of atoms likely to decay in each time interval is proportional to the number of atoms already there

in a radioactive sample

• the form of the radioactive decay law
• half-life
• activity and specific activity
• average life
• biological life

​

• apply the radioactive decay law to solve problems

let’s aim to

Dr Mark Marais

May 2025

• understand and explain…

radioactive isotope ≡ radioisotope ≡ radionuclide

Dr Mark Marais

May 2025

set of isotopes of a particular element ≡ nuclides

Dr Mark Marais

May 2025

Radioactive Decay Law

• In a sample of radioactive material, the total number, N, of radioactive nuclei remaining after time, t, is

​

N(t) = N₀(t) e ^-λt

• N = number of nuclei left after some time
• N₀ = initial number of nuclei
• e = 2.7182818285…(Euler’s number)
• λ = probability of one nucleus decaying in

the next time interval (decay constant)

• t = the time over which the isotope decays

Dr Mark Marais

May 2025

Radioactive Decay Law

time (units of time)

N₀

Half-life

Dr Mark Marais

May 2025

• Given by

T_1/2 or t_1/2

• Half-life is the time it takes for a substance to reduce to half the amount it was before.

Dr Mark Marais

May 2025

Radioactive Decay Law

This means that the mass, m,

also decreases in the same way.

​

m(t) = m₀(t) e ^-λt

Half-life

Dr Mark Marais

May 2025

E.g.

The number drops rapidly at first, then drops more slowly.

Dr Mark Marais

May 2025

Half-life

Dr Mark Marais

May 2025

• For the half-life,

• Derive the formula for λ

λ =

0.693

• Half-life is the time it takes for a substance to reduce to half the amount it was before.

Activity

Dr Mark Marais

May 2025

• Activity, A, is defined as the magnitude of the decay rate.

A(t) = A₀(t) e ^-λt

A₀

Specific Activity

Dr Mark Marais

May 2025

• Specific Activity, ‘a’, is the activity per quantity of a radionuclide and is a physical property of that radionuclide.

Dr Mark Marais

May 2025

Average life

• Average life is the average amount of time that a nucleus exists before decaying.

• Given by

T or t

Dr Mark Marais

May 2025

Biological half-life

.

Dr Mark Marais

May 2025

Effective half-life

1/t _effective = 1/t _physical + 1/t _biological

Dr Mark Marais

May 2025

“Hot” toilets

In nuclear medicine, a "hot toilet" refers to a toilet designated for patients who have been administered radioactive substances and are still considered radioactive themselves. These toilets are designed to manage and contain radioactive waste, primarily urine, with separate evacuation systems for urine and faeces. The design focuses on preventing contamination of the facility and ensuring the safe disposal of radioactive waste. 

Dr Mark Marais

May 2025

Some notes

N(t) = N₀(t) e ^-λt

m(t) = m₀(t) e ^-λt

A(t) = A₀(t) e ^-λt

I(t) = I₀(t) e ^-λt

X(t) = X₀(t) e ^-λt

D(t) = D₀(t) e ^-λt

Number of Nuclei in Sample

Mass of Sample

Activity of Sample

Ionisation Rate

Exposure Rate

Dose Rate

Dr Mark Marais

May 2025

Example

A patient is given 2.84 GBq of I¹³¹ .

Calculate the activity retained after 4 days.

(Physical half-life of I¹³¹ = 8 days, biological half-life = 4 days.)

The patient excretes 50% of the given dose in 4 days.

Dr Mark Marais

May 2025

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