# Rate of radioactive decay and calculation of Half-life time

** The decay of a radioactive isotope takes place by disintegration of the atomic nucleus. It is not influenced by any external conditions. Therefore the rate of decay is characteristic of an isotope and depends only on the number of atoms present.

** If N be the number of undecayed atoms of an isotope present in a sample of the isotope, at time t,

where  – dN/dt means the rate of decrease in the number of radioactive atoms in the sample; and λ is the proportionality factor. This is known as the decay constant or disintegration constant. Putting dt = 1 in equation (1) we have:

** Thus decay constant may be defined as the proportion of atoms of an isotope decaying per second.

** The standard unit of radioactivity (i.e. rate of disintegration) is Curie (c).
** A curie is a quantity of radioactive material decaying at the same rate as 1 g of Radium (3.7 × 1010 dps).
** Rutherford is a more recent unit: 1 Rutherford = 106 dps
** The S.I. unit is Becquerel: 1 Bq = 1 dps

#### Half-Life time

** The half-life or half-life period of a radioactive isotope is the time required for one-half of the isotope to decay. Or, it may be defined as the time for the radioactivity of an isotope to be reduced to half of its original value.
**  Half-life period is characteristic of a radioactive element.
** For example, the half-life of radium is 1620 years. This means that 1g of radium will be reduced to 0.5 g in 1620 years and to 0.25 g in further 1620 years; and so on. Some other radioactive elements may have half-life of a fraction of a second and for others it may be millions of years.
** The unit of half-life period is time– 1.

#### The activity of a Radioactive Substance

** It is defined as the rate of decay or the number of disintegrations per unit time.
** The activity of a sample is denoted by A. It is given by the expression:

** The unit of activity is the curie (Ci) which is the rate of decay of 3.7 × 1010 disintegrations per second. The SI unit of activity is becquerel (Bq) which is defined as one disintegration per second.
** The activity of a radioactive sample is usually determined experimentally with the help of a Geiger-Muller counter.

#### Calculation of Half-Life time

The value of λ can be found experimentally by finding the number of disintegrations per second with the help of a Geiger-Muller counter. Hence, half-life of the isotope concerned can be calculated by using the relation (5).

#### Calculation of sample left after time T

** It follows from equation (4) stated earlier that

** Knowing the value of λ, the ratio of N0/N can be calculated.

** If the amount of the sample present to start with is given, the amount left after lapse of time t can be calculated.

#### Average life

** In a radioactive substance, some atoms decay earlier and others survive longer.

** The statistical average of the lives of all atoms present at any time is called the Average life. It is denoted by the symbol τ and has been shown to be reciprocal of decay constant, λ.

** The average life of a radioactive element is related to its half-life by the expression:

** The average life is often used to express the rate of disintegration of a radioactive element. The average life of radium is 2400 years.

#### Solved problem

Problem (1): Calculate the half-life of radium-226 if 1 g of it emits 3.7 × 1010 alpha particles per second.
Solution:

Problem (2): Calculate the disintegration constant of cobalt 60 if its half-life to produce nickel–60 is 5.2 years.
Solution:

Problem (3): The half-life period of radon is 3.825 days. Calculate the activity of radon. (atomic weight of radon = 222)
Solution:

we know that:
dN = λN

where dN is the number of atoms disintegrating per second, λ is the decay constant and N is
the number of atoms in the sample of radon.

By definition, the activity of radon is its mass in grams which gives 3.7 × 1010 disintegrations
per second. Therefore activity of radon = 6.51 × 10– 6 g curie.

Problem (4): Cobalt-60 disintegrates to give nickel-60. Calculate the fraction and the percentage of the sample that remains after 15 years. The disintegration constant of cobalt-60 is 0.13 yr– 1.
Solution:

Hence the fraction remaining after 15 years is 0.14 or 14 per cent of that present originally.

Problem (5): How much time would it take for a sample of cobalt-60 to disintegrate to the extent that only 2.0 per cent remains ? The disintegration constant λ is 0.13 yr– 1.
Solution:

Problem (6): A sample of radioactive 133 I gave with a Geiger counter 3150 counts per minute at a certain time and 3055 counts per unit exactly after one hour later. Calculate the half life period of 133I.

Solution:

Reference: Essentials of Physical Chemistry /Arun Bahl, B.S Bahl and G.D. Tuli / multicolour edition.