Some useful equations for half-lives
Teaching Guidance
for 14-16

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, t2 – t1 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.
Since
N = N0e -λt
Taking natural logs of both sides:
lnN=lnN0-λt
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:
ln½=-λT½
ln2 = +λT½
T½ = 0.693λ