# Episode 515: The radioactive decay formula

Lesson for 16-19

- Activity time 50 minutes
- Level Advanced

Here, the key idea is the random nature of the decay. Avoid simply pulling pull equations out of the air – at least make them plausible.

Lesson Summary

- Discussion: The meaning of the decay constant l (15 minutes)
- Discussion: The link with half-life (15 minutes)
- Student experiments: Analogue experiments linking probability with decay rates (20 minutes)

## Discussion: The meaning of the decay constant l

Start from the definition of the decay constant λ: the probability or chance that an *individual* nucleus will decay per second. (You may like to comment on the problem with notation in physics. λ is used for wavelength as well as the decay constant. The context should make it unambiguous.)

Units: λ is measured in s^{-1} (or h^{-1} , year^{-1} , etc).

If you have a sample of *N* undecayed nuclei, what will its activity *A* be? In other words, how many of the *N* will decay in a second?

*A* = λ × *N* (because the probability for each of the *N* is λ).

As time passes, *N* will get smaller, so *A* represents a *decrease* in *N* . To make the formula reflect this, it needs a minus sign.

*A* = − λ × *N*

Plausibility: The more undecayed nuclei you have, and the greater the probability that an individual one will decay, the greater the activity of the sample.

In calculus notation, this is

d *N* d *t* = − λ × *N*

Ask your students to put this equation into words. (The rate of decay of undecayed nuclei *N* is proportional to the number N of undecayed nuclei present.) This is the underlying relationship in any process that follows *exponential decay* .

More generally: if the rate of change is proportional to what is left to change, then exponential decay follows. Conversely if data gives an exponential decay graph, then you know something about the underlying process.

Episode 515-1: Smoothed out radioactive decay (Word, 43 KB)

Episode 515-2: Half-life and time constant (Word, 35 KB)

## Discussion: The link with half-life

How are λ and *T*_{ ½ } linked?

If a nuclide has a large value of λ, will it have a long or short *T*_{ ½ } ? (A high probability of decay implies it will decay quite quickly whihc leads to a short half life, and vice versa.)

What type of proportionality does this suggest? (Inverse proportion.)

Thus λ ~ 1*T*_{ ½ }.

In fact,

*T*_{ ½ } = ln(2)λ

*T*_{ ½ } = 0.693λ

This is a very important and useful formula. Depending which of l or *T*_{ ½ } is easiest to measure experimentally, the other can be determined. In particular it allows very long half lives (sometimes millions of years) to be determined. How could we measure the 4.5 billion year half-life of uranium-238? (Take a sample of a known number of U-238 atoms; count how many decay in one second. This gives l, and we can calculate half-life.)

## Student experiments (or demonstrations): Analogue experiments linking probability with decay rates.

Explore some analogue systems to reinforce the way in which decay probability is related to half-life. Each gives a good exponential graph. For example:

Throw a large number of dice. A 6 represents decayed

, and this dice is removed. Each throw represents the same time interval. Each face has a probability of 16 of being upwards on each throw. (A quick way to find out how many remain to decay is to weigh them.)

Episode 515-3: Modelling radioactive decay (Word, 36 KB)

Cubes with differently coloured faces instead of dice can incorporate 3 different decay constants

e.g. colouring three faces red gives a chance of a red face being uppermost representing decay has a probability of decay of ½; colour two faces blue (chance of decay = 2 × 16, so 13), and one face yellow (chance of decay 16). Sugar cubes can be used, painted with food colouring.

Throw drawing pins: decay → point upwards. Safety: beware sharp points!

Is the chance of decay 12?

The drop in height of the head on a glass of beer usually shows exponential behaviour – opportunity for a field trip?

Water flowing out through a restriction: at any instant the water remaining represents the undecayed nuclei; the water that has flowed out represents the decayed nuclei. Flow rate depends upon the pressure head ~ the height of water remaining ~ quantity of water remaining.