## Half-Life: Break down and its calculation

The half-life of an element is the time it takes for half the atoms of a radioisotope to decay into other atoms. From the equation

Nt = mass of radioactive material at time interval (t)

N0 = mass of the original amount of radioactive material

k = decay constant

t = time interval (t1/2 for the half-life) Half-life of some elements

Let us look at Sr-90, from the table above, it will take 28 days for half of the atoms to decay into other atoms. It will take another 28 days for half of the remaining atoms to decay. Let’s assume that we have a sample of strontium that weighs 8g. After the ﬁrst 28 days there will be:

1/2 x 8 = 4 g Sr-90 left

After 56 days, there will be:

1/2 x 4 g = 2 g Sr-90 left

After 84 days, there will be:

1/2 x 2 g = 1 g Sr-90 left

If we convert these amounts to a fraction of the original sample, then after 28 days 1/2 of the sample remains undecayed.

After 56 days 1/4 is undecayed and after 84 days, 1/8 and so on.

For more explanations and calculations visit the classroom section

How ever, let us take a look at some other examples below

Question: A 100 g sample of Cs-137 is allowed to decay. Calculate the mass of Cs-137 that will be left after 90 years

Answer Step 1 : You need to know the half-life of Cs-137 The half-life of Cs-137 is 30 years.

Step 2 : Determine how many times the quantity of sample will be halved in 90 years. If the half-life of Cs-137 is 30 years, and the sample is left to decay for 90 years, then the number of times the quantity of sample will be halved is 90/30 = 3.

Step 3 : Calculate the quantity that will be left by halving the mass of Cs-137 three times

1. After 30 years, the mass left is 100 g × 1/2 = 50 g
2. After 60 years, the mass left is 50 g × 1/2 = 25 g
3. After 90 years, the mass left is 25 g × 1/2 = 12.5 g

Note that a quicker way to do this calculation is as follows: Mass left after 90 years = (1/2)3 × 100 g = 12.5 g (The exponent is the number of times the quantity is halved)

Exercise on half-life

1. Imagine that you have 100 g of Na-24.
(a) What is the half life of Na-24?

(b) How much of this isotope will be left after 45 hours?

(c) What percentage of the original sample will be left after 60 hours?

2. A sample of Sr-90 is allowed to decay. After 84 days, 10 g of the sample remains.
(a) What is the half life of Sr-90?

(b) How much Sr-90 was in the original sample?

(c) How much Sr-90 will be left after 112 days

For more explanations and calculations visit the classroom section

Recommended: Nuclear structure and stability