What is the half life of a radioactive isotope if 8g of a 32g sample remains unchanged after 12 hours?

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

Explanation:

Half-life (t½) is the time required for the nuclei to decay to half of the original amount.

Radioactive nuclei decay according to the equation:

Equation 1: Nt/N0=e^-λt, where

N0 is the initial number of nuclei at time t=0

Nt 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: t1/2=in2/λ

We can combine these two equations to get

Equation 3: Nt/N0=0.5^t/t½

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:

Nt/N0=0.5^t/t½

5.7g/50g=0,5^5000yr/t1/2

0.114=0.5^5000yr/t1/2

Take the natural logarithm of each side

In 0.114= 5000yr/t1/2 × In 0.5

-2.7= 5000yr/t1/2 × (-0.693)

t1/2= (5000yr × 0.693/2.17) = 1600yr

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

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