# How is radioactive decay related to half life?

Apr 5, 2016

The faster the decay, the shorter the half life.

#### Explanation:

Half life (mathematically ${T}_{\frac{1}{2}}$) is how long it takes for half of the atoms in a substance to radioactively decay.

If you want to know the maths behind their relationship,

$N = {N}_{0} {e}^{- \lambda t}$ applies to radioactive substances, where

$N$ is the number of radioactive atoms at time $t$
${N}_{0}$ is the number of radioactive atoms at the beginning of the process, when $t = 0$
$e$ is Euler's constant, $\approx 2.71828$
$t$, as mentioned, is time, and
$l a m \mathrm{da}$ is the decay rate, which is a constant value for each isotope. It can also be thought of as the probability for an atom to decay in a unit time.

When $t = {T}_{\frac{1}{2}}$, then half the initial atoms have decayed, which means that $N = {N}_{0} / 2$.

Substituting this into the equation,

${N}_{0} / 2 = {N}_{0} {e}^{- \lambda {T}_{\frac{1}{2}}}$

$\frac{1}{2} = {e}^{- \lambda {T}_{\frac{1}{2}}}$

Taking natural logs and rearranging from there,

$\ln \left(\frac{1}{2}\right) = - \lambda {T}_{\frac{1}{2}}$
$\ln 1 - \ln 2 = - \ln 2 = - \lambda {T}_{\frac{1}{2}}$
$\ln 2 = \lambda {T}_{\frac{1}{2}}$

${T}_{\frac{1}{2}} = \ln \frac{2}{\lambda} = \frac{0.693}{\lambda}$

which is mathematically how the rate of radioactive decay is related to half life.

From this equation, we can see that if decay rate ($\lambda$) increases, ${T}_{\frac{1}{2}}$ will get shorter, and if decay slows down, half life will increase.

They are inversely proportional.