Exponential Decay of Activity
Quantum and Nuclear

Some useful equations for half-lives

Teaching Guidance for 14-16 PRACTICAL PHYISCS

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

dNdt = -λN

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.

N = N0e -λt (Equation 1)

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:

N1N2 = e -λt1e -λt2

N1N2 = e -λ(t1-t2)

In a fixed time interval, t2t1 is a constant. Therefore the ratio

N1N2 = a constant

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.


N = N0e -λt

Taking natural logs of both sides:


Therefore a graph of N against t will be a straight line with a slope of QuantitySymbol{-λ}.

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:

N = N02 when t=T½

By substituting this expression in equation (see above),

N02 = N0e -λT½

Taking natural logs of both sides gives:


ln2 = +λT½

T½ = 0.693λ

Exponential Decay of Activity
can be analysed using the quantity Activity
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