Atoms and Nuclei Summary

• Dalton had postulated that matter is made up of atoms, which are indivisible. Thomson was the first to suggest a structure for an atom. According to him, an atom is a positively charged sphere of radius=10^-10 m in which the mass and the positive charge of the atom are uniformly distributed. Inside the sphere, electrons carrying equal negative charge are embedded like seeds in a watermelon. This model failed, as it could not explain the origin of spectral series of hydrogen atom.

• Large angle scattering of alpha particles lead Rutherford to the discovery of atomic nucleus, the central core of every atom. The entire positive charge and almost entire mass of the atom are concentrated in the nucleus. Electrons carrying equal negative charge revolve around the nucleus in circular orbits. This model of atom postulated by Rutherford failed because an electron revolving around the nucleus would lose energy continuously and radius of its orbit would go on decreasing and ultimately electron would fall into the nucleus which is not the case. Further, as electron would lose energy continuously, the atoms should emit electro-magnetic waves over a continuous range. But atoms emit only discrete frequency wavelengths only.

• From alpha ray scattering experiment, Rutherford calculated distance (r_o) of closest approach of a - particle to the nucleus. This gave him the order of size of nucleus. At r = r_o, whole of K.E of alpha particle is converted into potential energy i.e.,

Where Ze is charge on nucleus and (2e) is charge on an alpha particle.

Yet another parameter that determined the angle of scattering was impact parameter (b) - which is the perpendicular distance of the velocity vector of alpha particle from the central line of the nucleus, when particle is far away from the atom.

• Bohr improved upon Rutherford model postulating that electrons can revolve around the nucleus in certain discrete, non-radiating orbits, called stationary orbits, for which total angular momentum of revolving electron is an integral multiple of h / 2p where h is Planck's constant.

Where n = 1,2,3… is called principal quantum number.

Further, when an electron jumps from an outer stationary orbit to an inner stationary orbit, difference in the total energies of electron in the two orbits is radiated in the form of a spectral line.

Also, the force of attraction of nucleus on the electron provides the centripetal force required by electron in moving in a circular orbit.

From equation (i) (ii) and (iii) we can calculate radius of stationary orbit as

This shows that r a n² i.e., radii of stationary orbits are in the ratio 1 : 4 : 9 : 16 …….

Velocity of electron in Bohr's orbit

Total energy of electron in an orbit

The origin of spectral lines is explained in terms of the formula

Where  = wave number of radiation emitted = number of complete waves in unit length.

Lyman series is obtained when an electron jumps to first stationary orbit from any subsequent stationary orbit ( i.e., n₁= 1, n₂ = 2,3,4…).

For Balmer series, n₁ = 2, n₂ = 3,4,5,…

For Paschen series, n₁=3, n₂ = 4,5,6…

For Brackett series, n₁ = 4, n₂ = 5,6,7….

For P fund series, n₁ = 5, n₂ = 6,7,8….

Total energy of an electron in a stationary orbit is calculated from the relation

Atomic masses are measured in a.m.u (atomic mass unit)

1 a.m.u = 1.66 x 10^-27 kg.

Electron volt is unit of energy.

1 eV = 1.6 x 10^-19 joule.

1 MeV = 10 eV = 1.6 x 10^-13 MeV.

• Atomic nucleus is represented as _zX^A where z, the charge number = number of protons.

A, the mass number = number of protons + number of neutrons.

Number of neutrons in the nucleus = (A-Z) Nuclear radius (R) is given by R = R₀ A^1/3, When R₀ is a constant whose value is 1.1 x 10^-15 m.

This shows that that heavier nuclei are bigger in size.

Density of nuclear matter

This value is constant for all nuclei.

• Binding energy of a nucleus is the energy with which nucleons are bound in the nucleus. It is measured by the work required to be done to separate the nucleons an infinite distance apart from the nucleus, so that they may not interact with each other.

It was discovered that mass of a nucleus (m_N) is always less than the calculated mass of nucleons in the nucleus. This difference in masses is called man defect (Dm). It can be written as

Dm = {zm_p + {A-Z}m_n - m_N}

Total B.E = (Dm) C²

= [zm_p + ((A-Z)m_n) - m_N]c²

Average B.E per nucleon

• Natural radioactivity is the phenomenon of emission of active radiations by heavy nuclei, on their own, without any external provocation.

The three types of radiations are alpha, beta and gamma radiations. An alpha particle has 4 units of mass and 2 units of positive charge. A b - particle is nothing but an electron. A g-ray photon has no rest mass and no charge.

According to radioactive decay law,

Where l is disintegration constant.

Half life of a radioactive element is the time during which half the total number of atoms in the element (N₀) disintegrate, i.e., in

Average life or mean life of radioactive element is

= 1.44 T

Units of radioactivity are

1 curie = 3.7 x 10¹⁰ disintegration/sec.

1 rutherford = 10⁶ disintegration/sec.

In alpha decay, mass number decreases by 4 and charge number decreases by 2. In beta decay, mass number remains unaffected and charge number increases by one. In gamma decay, the mass number and charge number both remain unaffected, only the energy changes.

• Artificial radioactivity is the phenomenon of disintegration of an otherwise stable nucleus by bombarding it with a suitably accelerated projectile.

• A nuclear reaction represents transformation of one stable nucleus into another. In all nuclear reactions, linear momentum, total energy, charge and nucleon number are conserved. Q value of nuclear reaction is calculated from Einstein's mass energy equivalence

E = (Dm)C²

• Nuclear fission is the phenomenon of splitting a heavy nucleus into two or more smaller nuclei. Mass defect in this process appears in the form of energy. The fission ₉₂U²³⁵ by thermions is neutron as:

The Q value of the reaction is 200 MeV.

Under suitable conditions, the three secondary neutrons may cause further fission of U²³⁵ nuclei and start what is known as nuclear chain.

reaction. The nuclear chain reaction is controlled by

Neutron reproduction factor (K)

K = 1 represents critical stage K > 1 represents supercritical stage and K < 1 represents sub critical stage.

• A nuclear reactor uses nuclear energy for peaceful purposes. It is based on the phenomenon of controlled chain reaction. Moderators like heavy water, graphite, paraffin and deuterium slow down neutrons. Rods of cadmium or boron serve as control rods. Ordinary water and heavy water serve as coolants.

• Nuclear fusion is the phenomenon of fusing of two or more lighter nuclei to form a single heavy nucleus. Mass defect in the process appears as energy. For example,

Temperatures = 10⁷ K are required for fusion to take place.

• Stellar energy is the energy obtained from the sun and stars. Our sun is radiating energy at the rate of 3.8 x 10²⁶ joule per second. Nuclear fusion is said to be responsible for stellar energy. The fusion reaction can be represented as

Calculations show that Q = 26.7 MeV.

The wide spread destruction caused by nuclear weapons like atom bomb, hydrogen bomb etc., is named nuclear holocaust.