Question

How do you calculate nuclear half life?

Answer 1

You calculate the half-life from the amount of material that disappears in a given time.

Half-life (`t_½`) is the time required for the nuclei to decay to half of the original amount.

Radioactive nuclei decay according to the equation:

Equation 1: `N_t /N_0 = e^(—λt)`, where

`N_0` is the initial number of nuclei at time `t = 0`.

`N_t` is the number of nuclei that remain after time `t`. We can also use any number that is proportional to the number of nuclei, such as mass or disintegration counts.

`λ` is a constant called the decay constant. Each nucleus has its own decay constant.

The equation for half-life is

Equation 2: `t_½ = ln2/λ`

We can combine these two equations to get

Equation 3: `N_t/N_0 = 0.5^(t/t_½)`

EXAMPLE:

A 50 g sample of radium–226 decays to 5.7 g after 5000 years. What is the half-life of radium–226?

Solution:

Let’s use Equation 3:

`N_t/N_0 = 0.5^(t/t_½)`

`(5.7 g)/(50 g) = 0.5^((5000 yr)/t_½)`

`0.114 = 0.5^((5000 yr)/t_½)`

Take the natural logarithm of each side

`ln0.114 = (5000 yr)/t_½ × ln0.5`

`-2.17 = (5000 yr)/t_½ × (-0.693)`

`t_½ = ((5000 yr × 0.693)/2.17)` = 1600 yr

The half-life of radium–226 is 1600 yr.