Modelling radioactive decay

In a nutshell

While decay is completely random, there is still a way to use the exponential equations to predicted numbers of undecayed nuclei after a certain time. This lesson will discuss how to use iterations to model exponential radioactive decay, and how this is put in practice with radioactive dating.

Definitions

Key word	Definition
Iteration	The process of repeating a calculation using the results of a previous calculation.
Radioactive dating	A technique used to estimate the age of objects by looking at the activity of certain elements within them.

Iteration

The key to predicting radioactive decay comes in the form of the decay equation and its exact solution. The decay equation is

$\dfrac{\Delta N}{\Delta t} = -\lambda N$

and its exact solution is

$N = N_0e^{-\lambda t}$

The iteration approach is used to calculate changes in the number of undecayed nuclei over very small periods of time and then using those in subsequent calculations. When doing these calculations, the value for $\Delta t$ is chosen to be a very small interval compared to the half-life $t_{1/2}$ . This is done so that it can be assumed that activity is not changing significantly.

procedure

1.	Start with a known number of undecayed nuclei $N_0$ .
2.	Calculate the number of decaying nuclei $\Delta N$ within a time interval $\Delta t$ using the decay equation.
3.	Subtract the decaying nuclei $\Delta N$ from the number of undecayed nuclei $N_0$ , to get a new value for the number of undecayed nuclei. Use this new value of $N_0$ in the next calculation.
4.	Repeat the previous step with multiples of the time interval $\Delta t$ .

These calculations are inputted into a table for the time interval and number of undecayed nuclei. A graph is then plotted to show the model of radioactive decay.

Carbon-dating

All living things contain carbon atoms. The carbon from the atmosphere is absorbed into plants through photosynthesis, then is taken into animals when they eat the plants and is released again during respiration.

The stable isotope of carbon is known as carbon-12, but there is another isotope of carbon that is radioactive, called carbon-14. Carbon-14 is continuously produced in the upper atmosphere when cosmic rays interact with it, and has a half-life of $5700\ years$ . This corresponds to a decay constant of $3.84\times10^{-12}$ .

The ratio of carbon-14 to carbon-12 is $1.3\times 10^{-12}$ . This ratio is the same everywhere, including within living plants and animals, as they all take in both isotopes of carbon from the atmosphere.

When an organism dies, it stops taking in or releasing carbon. The amount of carbon-12 within the organism will therefore stay the same but the amount of the radioactive carbon-14 will decrease due to decay, causing this ratio to decrease.

Measuring this ratio in a sample of organic material and comparing it to the known ratio of carbon in similar living material allows scientists to determine when the sample died. This process is called carbon-dating.

Example

A wooden artefact is measured to have an activity of $8.2\,Bq$ . A similar piece of living wood is measured to have an activity of $11.6\,Bq$ . Calculate the age, in years, of the wooden artefact.

First, write down the known values:

$A = 8.2\,Bq$

$A_0 = 11.6\,Bq$

$\lambda_{carbon-14} = 3.84\times10^{-12}\,s^{-1}$

Next, write down the equation for activity:

$A = A_0e^{-\lambda t}$ 

Rearrange the equation to find the time:

$t = -\dfrac{\ln(A/A_0)}{\lambda}$ 

Substitute the values in and calculate the final answer:

$t = -\dfrac{\ln(8.2/11.6)}{3.84\times10^{-12}}$

$t = 9.03\times10^{10}\,s$

$t = 2900\,years$

The age of the wooden artefact is $\underline{2900\ years}$ .

Limitations to carbon-dating

The technique behind carbon-dating assumes that the ratio between carbon-14 and carbon-12 has remained constant over time. This may not be true, due to increased emissions of carbon dioxide through fossil fuels having the unintended effect of reducing this ratio further. More extreme events, such as solar flares, may also have an effect on the ratio.

In addition to this, the activity of carbon-14 within organic matter is incredibly small - about $15$ counts per minute - which is comparable to regular background count rate. Overall, carbon-dating isn't used for anything estimated to be over $50\,000\ years$ old, as the quantity of carbon-14 within these samples is far too small.