How do you calculate nuclear half life?

1 Answer
Mar 13, 2014

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.