How to Calculate Half-life

Half-life is the time period taken by a chemical substance that decays to decrease to half of its initial amount at a given time period. Half-life is usually calculated for radioactive elements. This article on how to calculate half-life explains calculation of half-life using a formula and simple steps.

Half lives are calculated for exponential as well as non-exponential decays. In case of exponential decay the half-life is constant throughout the life of decay while in case of non-exponential decay half-life of the substances varies throughout the decay process. It is usually seen that radioactive elements undergo exponential decay.

The half-life of a substance is denoted by t1/2. The half-life period of a substance was initially called as ‘dating’. Well, it has nothing to do with ‘dating’ a boy or girl!

Calculating half-life of a substance using formula:

The original formula for calculating half-life of a substance is given as,

In the above formula,

  • “N0” is the original amount of radioactive isotope that is present before any decay has taken place.
  • “Nt” it the amount of radioactive isotope that is left after a certain time period t.
  • “t” is the time period after which the number of atoms within a radioactive element changes from N0 to Nt.
  • “t1/2” is the half-life of the decaying substance

Using the above formula you can easily calculate the amount of radioactive left after a certain time period t. You can also calculate the half-life of the radioactive decay, but to make it simple, let us evaluate the above formula.

Simplifying the above formula to calculate half-life we get,

t1/2 = (t × log 2 )/ log ( N0/Nt)

  • To calculate the half-life of a substance you need to know the initial quantity of the radioactive substance and the amount of substance left after a time period t.
  • You should also know the time period after which the substance has decayed to a particular quantity.
  • Substituting the above values in the formula you can easily calculate the half-life of any radioactive substance.
  • You also need a logarithmic chart and a calculator to perform the calculations.
  • If you are not familiar with finding the logs using the charts then you can either use scientific calculators or online calculators to find the logarithmic values.

Let us illustrate this using a simple example.

  • Suppose the number of atoms of a radioactive isotope is 1676 at time 0 and after a certain time, say 4 minutes, the number of atoms left behind are 1358.

From the above information,

  • N0 = 1676
  • Nt = 1358
  • t = 4 minutes

Substituting the above vales in the formula for calculating half-life,

t1/2 = (t × log 2 )/ log ( N0/Nt)

t1/2 = (4 × log 2 / log(1676/1358)

t1/2 = 4 × 0.3010/ 0.09131

t1/2 = 13.1858 minutes

Therefore, half-life of this radioactive isotope i.e. the time taken by the isotope to decay half of its atoms every time is 13.1858 minutes.

Alternative methods for calculating half-life of a substance:

Instead of using the formula for calculating half-life of a substance you can use a manual method for calculating half-life of a substance. This method is explained using an example below.

Suppose the initial number of atoms of a radioactive isotope is 2016 and after 35 days it decays to 63. Calculate the half-life of the substance.

Here,

  • N0 = 2016
  • Nt = 63
  • t = 35 days

Now, divide the initial amount of the substance (2016) by 2 until it is reduced to the 63.

Doing this we get the values as,

  • First half-life = 2016/2 = 1008
  • Second half-life = 1008/2 = 504
  • Third half-life = 504/2 = 252
  • Fourth half-life = 252/ 2 = 126
  • Fifth half-life = 126/2 = 63

Therefore, the substance takes five half lives to undergo a decay from 2016 to 63. This means that it takes 35 days to complete 5 half lives.

Therefore,

Each half-life = t1/2 = 35/ 5

Therefore,

t1/2 = 7 days

Tips:

  • Above methods for calculating half lives can be used only for the elements which undergo exponential decay i.e. radioactive elements.

You can also use online calculators for calculating the half-life of any substance.


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