Radioactivity
Radioactive decay is a random process. Each decay is an independent event, and one cannot tell when a particular nucleus will decay. When a given nucleus decays, it is transformed another nuclide, which may or may not be radioactive. When there is a very large number of nuclei in a sample, the rate of decay is proportional to the number of nuclei N that are present
Where
to yield
where No is the initial number of parent nuclei at t = 0. The number that survive at time t is therefore
This function is plotted in the following figure.The time required for the number of parent nuclei to fall to 50% is called the half-life, T, and may be related to
we have
It takes one half-life to drop to 50% of any starting value. The half-life for the decay of the free neutron is 12.8 min. Other half-lives range from about 10-20 s to 1016 years.
Since the number of atoms is not directly measurable, we measure the decay rate or activity (A)
where
1 becquerel (Bq) = 1 disintegration per second (dps)
1 curie = 3.7 x 1010 dps
1 rutherford = 106 dps
Mean life of a radioactive sample is defined as the average of the lives of all nuclei.
i.e.
Example 1
The half-life of Cobalt - 60 is 5.25 years. How long after its activity have decreased to about one-eight of its original value ?
Solution:
The activity is proportional to the number of undecayed atoms.
In each half-life, half the remaining sample decays.
Since
Example 2
A count rate-meter is used to measure the activity of a given sample. At one instant the meter shows 4750 counts per minute. Five minutes later it shows 2700 counts per minute.
(a) Find the decay constant
(b) Also, find the half life of the sample
Solution:
Initial velocity
Final velocity
Dividing (i) by (ii), we get
The decay constant is given by
Half life of the sample is
Example 3
The mean lives of a radio active substance are 1600 and 400 years for
Solution:
When an substance decays by a and b emission simultaneously, the average disintegration constant
where
Mean life is given by
Example 4
The half-life of radium is 1620 years. How many radium atoms decay in 1s in a 1g sample of radium. The atomic weight of radium is 226 g/mol.
Solution:
Number of atoms in 1 g sample is
The decay constant is
Taking one year
Now,
Thus
Properties of Radioactive Processes
(1)
(2)
Both processes are accompanied by
In a magnetic field, a beam of
Nuclei possessing the artificial radioactivity are obtained by bombarding stable nuclei of heavy elements by
Radioactive processes occur in accordance with the laws of conservation of energy, momentum, angular momentum, electric charge, and mass number (amount of nucleons).
In
In
ATOMIC NUCLEUS
The atomic nucleus consists of two types of elementary particles, viz. protons and neutrons. These particles are called nucleons.
The proton (denoted by p) has a charge +e and a mass
The neutron (denoted by n) is an electrically neutral particle (its charge is zero). The neutron mass
The most important characteristics of the nucleus are the charge number Z (coinciding with the atomic number of the element) and the mass number A. The charge number Z is equal to the number of protons in the nucleus, and hence it determines the nuclear charge equal to Ze. The mass number A is equal to the number of nucleons in the nucleus (i.e. to the total number of protons and neutrons).
Nuclei are symbolically designated as
where X stands for the symbol of a chemical element. For example, the nucleus of the oxygen atom is symbolically written as
Most of the chemical element have several types of atoms differing in the number of neutrons in their nuclei. These varieties are called isotopes. For example, oxygen has three stable isotopes:
Atomic masses are specified in terms of the atomic mass unit or unified mass unit (u). The mass of a neutral atom of the carbon isotope 6C12 is defined to be exactly 12 u.
Example 5
(a) Calculate the value of 1 u from Avogadro’s number.
(b) Determine the energy equivalent of 1u.
Solution:
(a) One mole of C12 has a mass of 12 g and contains Avogadro’s number, NA, of atoms.
By definition, each C12 has a mass of 12 u.
Thus, 12 g corresponds to 12 NA u which means
or
(b) From Einstein relation
⸫
Since
⸫
Hence
Size of the Nucleus
The shape of nucleus is approximately spherical and its radius is approximately related to the mass number by
Example 6
Find the mass density of the oxygen nucleus 8O16.
Solution:
Volume
Mass of oxygen atoms (A = 16) is approximately 16 u.
Therefore, density is
or
This is 1014 times the density of water.
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