Mass Defect and Binding Energy , relation between mass defect and binding energy class 12

know all define Mass Defect and Binding Energy , relation between mass defect and binding energy class 12 also relate the two formula ?

Composition of the Nucleus

The nucleus of an atom contains protons and neutrons which are collectively called nucleons. The total number of nucleons is called the mass number and its denoted by A. The number of protons in a nucleus is called its atomic number Z. The number of neutrons is denoted by N, so that

A = Z + N

 The Atomic Nucleus
The radius R of a nucleus of mass number A is given by the relation

where R₀ is a quantity which varies slightly from one nucleus to another

Mass Defect and Binding Energy

The mass of a nucleus which contains Z protons and (A – Z) neutrons is always less than the sum of the masses of these particles in the free state. The difference is called the mass defect of the given nucleus and is given by

where m = mass of the nucleus, m_p = mass of a proton and m_n  = mass of a neutron. The binding energy of the nucleus is given by

where c is the speed of light in free space. Since A is the total number of nucleous

Nuclear Fission

The splitting of a heavy nucleus into two or more fragments of moderate and comparable sizes is called nuclear fission. The fission of uranium-235 is represented by the reaction.

The energy released per fission is about 200 MeV which is much more than the energy released in the usual nuclear reactions. This makes the fission reaction a particularly suitable source of energy. The fission reaction given above has a unique feature. Apart from the fission fragments, the reaction results in the release of 2 to 3 neutrons—the very particles that initiated the reaction. So fission after fission, the neutrons present in a bulk sample of uranium increase in geometric ratio. The rate of energy release also increases similarly in a geometric ratio. The fission reaction is thus a self sustaining chain reaction. When the number of neutrons released per fission is limited to one per fission by absorption of excess neutrons, the chain reaction is a controlled one and is used in nuclear reactors. When there is no such control on the number of released neutrons, we have an uncontrolled chain reaction and this is the source of energy in the atom bomb. An essential part of a controlled fission reaction is known as the moderator. The role of the moderator is to slow down the neutrons released in fissions so that they may be easily absorbed by another ₉₂U²³⁵ nucleus. Media which contain nuclei of masses comparable to the neutron are found to act as efficient moderators.

Nuclear Fusion

The process of nuclear fusion consists in the ‘combination’ of two light nuclei to form a stable nucleus of mass less than the total initial mass. It is believed to be the main source of energy for the sun and the stars. The fusion reaction in stars is believed to occur either via the proton-proton cycle or the carbon-cycle. The proton-proton cycle is as follows:

The energy released in this sequence works out to be 24.7 MeV. Nuclear fusion occurs at very high temperatures of about 10⁷ K and under extremely high pressures.

Radioactivity

The phenomenon of self-emission of radiations from a nucleus is called radioactivity and substances which emit these radiations are called radioactive substances. The radiations emitted from a radioactive element are of three types.

1. Alpha rays: These rays consist of a-particles. An alpha particle is a helium nucleus having two protons and two neutrons. It has a positive charge equal to the charge of two protons. It has an initial speed of about 10⁷ ms^-1. They have very little penetrating power.

2. Beta rays: These rays consist of electrons. Their speed is very nearly equal to the speed of light. They have more penetrating power than alpha particles.

3. Gamma rays: These are high frequency electromagnetic waves having a high penetrating power

Alpha Decay The process of emission of an alpha particle from a nucleus is called alpha decay. When a nucleus emits an alpha particle ₂⁴He , it loses two protons and two neutrons which means that the daughter nucleus has its mass number reduced by 4 and its atomic number reduced by 2. When a nucleus _Z^AX of mass number A and atomic number Z emits an a-particle ₂⁴He , it is transformed into a nucleus _Z-2^A-4Y whose mass number is (A – 4) and atomic number is (Z – 2). Alpha decay is represented by

‘

Beta Decay The process of the emission of an electron from a nucleus is called beta decay. In this process also, the nucleus achieves greater stability by emitting an electron. A neutron inside the nucleus decays into a proton with the emission of an electron (e) and a particle called antineutrino (v)  . Because the mass of an electron is negligibly small, the mass number of the resulting nucleus remains unaltered but its atomic number is increased by one. For example, when a radium nucleus ₈₈²²⁸Ra emits a β-particle, the resulting element is an isotope of actinium ₈₉²²⁸Ac. Thus in β-decay also, a new element is formed. The transformation of a nucleus _Z^AX into the nucleus _Z+1^AY by β-decay is represented by an equation

Gamma Decay

Gamma rays are high-frequency electromagnetic radiations (i.e. photons) which do not carry any charge. Hence in γ-decay, the mass number and atomic number of the nucleus remain unchanged so that no new element is formed.

Radioactive Decay

Law If N is the number of radioactive nuclei present in a sample at a given instant of time, then the rate of decay at that instant is proportional to N, i.e.

The proportionality constant λ is called the disintegration constant. If N₀ is the number of radioactive nuclei at time t = 0, then the number of radioactive nuclei at a later time t is given by

Half life : The half life of a radioactive element is the time in which half the number of nuclei decay. It is given by

Average life : The average life of a radioactive sample is the reciprocal to its disintegration constant, i.e.

Radioactivity decay rate or Activity: It is useful to use the concept of the decay rate R which is defined as the number of radioactive disintegrations taking place in a sample per second, which is given by

As N decreases exponentially with time, R will also decrease exponentially with time. The SI unit of the decay rate R is called curie (symbol Ci) in honour of Madame M.S. Curie (1867–1934). It is defined as the decay rate of 3.7 x 10¹⁰ disintegrations per second, i.e.

1 Ci (curie) = 3.7 x 10¹⁰ disintegrations/s

1 mCi (millicurie) = 3.7 x 10⁷ disintegrations/s

1 μCi (microcurie) = 3.7 x 10⁴ disintegrations/s