What are isotopes? What is a radioisotope? Define the half-life of a radioisotope - Leaving Cert Chemistry - Question c - 2016

A radioisotope is an isotope that has an unstable nucleus. Such isotopes spontaneously decay, emitting radiation in the form of alpha, beta, or gamma rays.

The half-life of a radioisotope is the time required for half of the sample to decay, which results in the activity of the sample reducing (or decaying) to half its original amount.

^{226}_{88} \text{Ra} \rightarrow \; ^{222}_{86} \text{Rn} + \, ^{4}_{2} \text{He}

Initially, we start with 1.0 × 10^4 moles of radium-223. To find the remaining sample after 87.5% decayed, we can use the formula:

Remaining moles = Initial moles × (1 - Decay fraction)

where Decay fraction = 0.875. Thus,

Remaining moles = 1.0 × 10^4 × (1 - 0.875) = 1.0 × 10^4 × 0.125 = 1.25 × 10^3 moles.
The number of atoms can be calculated using Avogadro's number:

e.g. Number of atoms = moles × Avogadro's number = 1.25 × 10^3 × 6.022 × 10^{23} \approx 7.5 × 10^{25} \text{ atoms}.