## 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.

*dN**dt* = *- 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 *N*_{0} is the initial number of atoms present.

*N* = *N*_{0}*e*^{ -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, *N*_{1}, at a time *t*_{1} to the value, *N*_{2}, at a time *t*_{2} is given by:

*N*_{1}*N*_{2} = *e*^{ -λt1}*e*^{ -λt2}

*N*_{1}* N_{2}* =

*e*^{ -λ(t1-t2)}In a fixed time interval, *t*_{2} – *t*_{1} is a constant. Therefore the ratio

*N*_{1}* N_{2}* =

*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* = *N*_{0}*e*^{ -λt}

Taking natural logs of both sides:

ln*N*=ln*N*_{0}-*λ**t*

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 *N*_{0} (the starting value). So:

*N* = *N*_{0}*2* when t=*T*_{½}

By substituting this expression in equation (see above),

*N*_{0}*2* = *N*_{0}*e*^{ -λT½}

Taking natural logs of both sides gives:

ln½=-*λ**T*_{½}

ln2 = +*λ**T*_{½}

*T*_{½} = 0.693*λ*