Exponential decay is usually represented by an exponential function of time with base e and a negative exponent increasing in absolute value as the time passes:
F(t)=A⋅e−K⋅t
where K is a positive number characterizing the speed of decay. Obviously, this function is descending from some initial value at t=0 down to zero as time increases towards infinity.
For example, this function can represent a radioactive decay of certain quantity of plutonium-239 and describes the amount of plutonium-239 left after a time period t.
Half life is the value of t when there will be left only half of what was in the beginning at t=0.
At t=0 the value of our function equals to
F(0)=A⋅e−K⋅0=A⋅e0=A⋅1=A
If at time t=T there is only half of the initial amount that is left, we have an equation:
A2=F(T)=A⋅e−K⋅T
The above represents an equation with T being an unknown.
Solution is:
A2=A⋅e−K⋅T (reduce by A)
12=e−K⋅T (take natural logarithm)
−K⋅T=ln(12)=−ln(2) (now we can resolve for T)
T=ln(2)K
So, all we need to know to find half life is the speed of a decay K. It can be determined experimentally for most practical situations since it depends on inner physical and chemical characteristics of a decaying substance.
For instance, half life of plutonium-239 is 24110 years, half life of caesium-135 is 2.3 million years, half life of radium-224 is only a few days.
