Question 5: Define and explain half-life of a radioactive element.

### Definition of the half-life of a radioactive substance

Time during which half of the number of nuclei of a radioactive material decays is called half-life of the material. It is denoted by T.

Explanation

Suppose there are 10000 atoms of a radioactive substance present at a certain instant of time. Then,

1. 5000 atoms of the sample will decay in the first half-life. 5000 will remain.
2. 2500 atoms of the sample will decay in the second half-life and 2500 will remain.
3. 1250 atoms of the sample will decay in the third half-life and 1250 will remain.

And so on and so forth.

Generally, if there are N0 number of the radioactive atoms of the material at a certain time, t = 0, then
• number of atoms will decay in the first half-life and N0/2 number will survive.
• of atoms will decay in the second half-life and the same number will survive.
• number of atoms will decay in the third half-life and the same number will survive.
Generally, after n half-lifes the number that decay (or survive) is #### Mathematical Expression for half-life

From the law of radioactivity, we know that N0 is the initial number of radioactive atoms and N is the number of atoms at any time t.

Now if during a half-life, N = N0/2 and t = T, then put in the above equation, Cancel N0 on both sides,    Now taking logarithm on both sides, Now log2 = 0.693 and loge = 1, therefore, the above equation becomes, This equation gives the half-life of a radioactive substance. Thus if we know the value of λ for a certain radioactive material we can determine the half-life of it.

#### Measurement of half-life

The half-life of a radioactive material can be determined by finding the activity of the given radioactive sample. A count-rate meter (such as GM counter) is used to do this. When a nuclear radiation enters the tube of the counter, a pulse is recorded. Let N pulses are recorded in time t, then the rate of activity, Now using equation (1), the half-life of the sample can be found.