# How do you Find exponential decay half life?

Apr 27, 2015

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:
$F \left(t\right) = A \cdot {e}^{- K \cdot t}$
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
$F \left(0\right) = A \cdot {e}^{- K \cdot 0} = A \cdot {e}^{0} = A \cdot 1 = A$

If at time $t = T$ there is only half of the initial amount that is left, we have an equation:
$\frac{A}{2} = F \left(T\right) = A \cdot {e}^{- K \cdot T}$
The above represents an equation with $T$ being an unknown.

Solution is:
$\frac{A}{2} = A \cdot {e}^{- K \cdot T}$ (reduce by $A$)
$\frac{1}{2} = {e}^{- K \cdot T}$ (take natural logarithm)
$- K \cdot T = \ln \left(\frac{1}{2}\right) = - \ln \left(2\right)$ (now we can resolve for $T$)
$T = \ln \frac{2}{K}$

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.