# Some useful equations for half-lives

The rate of decay of a radioactive source is proportional to the number of radioactive atoms (N) which are present.

is the decay constant, which is the chance that an atom will decay in unit time. It is constant for a given isotope.

The solution of this equation is an exponential one where N0 is the initial number of atoms present.

Constant ratio
This equation shows one of the properties of an exponential curve: the constant ratio property.

The ratio of the value, N1, at a time t1 to the value, N2, at a time t2 is given by:

In a fixed time interval, t2 – t1 is a constant. Therefore the ratio

So, in a fixed time interval, the value will drop by a constant ratio, wherever that time interval is measured.

Straight line log graph
Another test for exponential decay is to plot a log graph, which should be a straight line.

Since

Taking natural logs of both sides:

Therefore a graph of N against t will be a straight line with a slope of

Half-life and decay constant
The half-life is related to the decay constant. A higher probability of decaying (bigger λ) will lead to a shorter half-life.

This can be shown mathematically.

After one half life, the number, N of particles drops to half of N0 (the starting value). So:

By substituting this expression in equation (1) above,

Taking natural logs of both sides gives:

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