# Radiation Calculations

Hey guys! Welcome to this video over measuring nuclear radiation.

Radiation can affect people in various ways, and can even result in death. Because of this effect, we must be able to measure the amount of radiation present.

There are a few things you need to be aware of when making radiation calculations: Decay and Half-life, and source strength.

Radioactive decay, also commonly known as radioactivity, is a set of processes that allow unstable atomic nuclei, or nuclides to emit radiation. This happens randomly, and can not be predicted. However, on a larger scale, radioactive material does behave in a way that is possible to predict. Radioactive decay always occurs at an exponential rate.

The amount of radioactive material, N, present after time (t) is given by the equation:

\(N=N_0\times e^{-\lambda t}\)

Where \(N_0\) is the amount of material present at time \(t=0\), and \(\lambda\) is the decay constant of the material.

The time required for half the mass of a radioactive material to decay is known as its half-life. Half-life is the most commonly used measure of a material’s rate of decay, and it can be short as a fraction of a second or as long as millions of years.

If the half-life of the material is known, lambda, the decay constant can be calculated as \(\lambda=\frac{\text{ln}(2)}{t_{\frac{1}{2}}}\), where \(t_{\frac{1}{2}}\) is the half-life.

For example, suppose a sample of radium-224 starts at a mass of 10 grams. Its half-life is 3.63 days. The remaining mass of radium-224 after 8 days can be calculated by:

\(\lambda=\frac{\text{ln}(2)}{3.63\text{ days}}=0.191\text{ disintegrations per day}\).

The source strength of a radioactive material is defined as the number of disintegrations per second, and is measured in becquerels (Bq) or curies (Ci). One becquerel is one disintegration per second; one curie is \(3.7×10^{10}\text{ disintegrations per second}\). Source strength is given by the equation:

\(Q= \frac{\lambda m N_A}{W}\)

- \(Q\) is the number of disintegrations per second
- \(\lambda\) is the decay constant
- \(M\) is the mass of the radioisotope
- \(N_A\) is Avogadro’s number, \(6.02×10^{23}\frac{\text{ g}}{\text{g-atoms}}\)
- \(W\) is the atomic weight of the radioisotope

For example, if radium-224 has decay constant of \(2.0289×10^{-6}\) disintegrations per second and an atomic mass of 224.02, a \(5.8×10^{12}\) becquerel source of Ra-224 can be calculated to have a mass of

\(m=\frac{QW}{\lambda N_A}\)

\(=\frac{(3.8×10^{12})\times224.02}{2.2089\times10^{-6}\times6.02×10^{23}} = 640 μg\)

I hope that this video was helpful.