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

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**Rate of radioactive decay**** **

** 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,

** 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.

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**Units
of Radioactivity**

** 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 × 10

^{10}dps).
** Rutherford is
a more recent unit: 1 Rutherford = 10

^{6}dps
** The S.I.
unit is Becquerel: 1 Bq = 1 dps

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**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 × 10

^{10}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.

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**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).

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**Calculation of sample left
after time T**

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

** Knowing the value of λ, the
ratio of N

_{0}/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.

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**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.

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**Solved
problem**

**Problem (1): Calculate the half-life of radium-226 if 1 g of it emits 3.7 × 10**

^{10}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 × 10

^{10}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^{133}I.**Solution:**

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

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