User:Inconspicuum/Physics (A Level)/Radioactive Decay

Decay Constant
We can model radioactive decay by assuming that the probability that any one nucleus out of N nuclei decays in any one second is a constant &lambda;. &lambda; is known as the decay constant, and is measured in s-1 (technically the same as Hz, but it is a probability, not a frequency, so we use s-1).

Activity
As our N nuclei decay, the number of nuclei decreases. The activity of the N nuclei we have left is, on average, the probability that any one nucleus will decay per. unit time multiplied by the number of nuclei. If we have 200 nuclei, and the decay constant is 0.5, we would expect, on average, 100 nuclei to decay in one second. This rate would decreases as time goes by. This gives us the following formula for the activity A of a radioactive sample:

$$A = -\frac{dN}{dt} = \lambda N$$

Activity is always positive, and is measured in becquerels (Bq). It is easy to see that the rate of change of the number of nuclei is -A = -&lambda;N.

Decay
The solution of the differential equation for activity given above is an exponential relationship:

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

where N is the number of nuclei present at a time t, and N0 is the number of nuclei present at time t=0. You can define t=0 to be any point in time you like, provided you are consistent. Since A = &lambda;N and therefore A0 = &lambda;N0:

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

where A is the activity of the sample at a time t, and A0 is the activity at time t=0.