Class 12 Physics CHAPTER 13 NUCLEI

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NCERT Notes For Class 12 Physics CHAPTER 13 NUCLEI

Class 12 Physics CHAPTER 13 NUCLEI

 

Atomic number (Z)

• It is the number of protons in the nucleus.
• It is denoted by Z.

Mass number (A)

• It is the total number of nucleons
• Total no. of nucleons = no. of protons + number of neutrons
• Mass number is denoted by A.

Neutron number (N)

• It is the total number of neutrons.
• Denoted by N and N= A-Z.

Representation of nuclei

• An atom is represented as _ZX^A.
• A- mass number, Z- atomic number

Atomic mass

• Accurate measurement of atomic masses is carried out with a mass spectrometer.
• Atomic mass unit (u), is used for expressing atomic masses.
• It is defined as 1/12th of the mass of the carbon (¹²C) atom.

Composition of nucleus

• Nucleus contains protons and neutrons
• The mass of a proton is

• James Chadwick-discovered neutrons
• Mass of a neutron is

• A free neutron is unstable.
• It decays into a proton, an electron and a antineutrino (another elementary particle), and has a mean life of about 1000s.
• It is stable inside the nucleus

Isotopes

• Atomic species with same atomic number but different mass number are called isotopes.
• Hydrogen has three isotopes having masses 1.0078 u (protium), 2.0141 u (deuterium), and 3.0160 u (tritium).
• Tritium nuclei, being unstable, do not occur naturally and are produced artificially in laboratories

Isobars

• All nuclides with same mass number A and different atomic number are called isobars. Eg:

Isotones

• Nuclides with same neutron number N but different atomic number Z are called isotones

 

SIZE OF THE NUCLEUS

• The radius of a nucleus with mass number A is given by

• Where

• Thus the density of nucleus is a constant, independent of A, for all nuclei.
• The density of nuclear matter is

Mass – Energy

• Einstein showed that mass is another form of energy and one can convert massenergy into other forms of energy, say kinetic energy and vice-versa.

• In a reaction the conservation law of energy states that the initial energy and the final energy are equal provided the energy associated with mass is also included.

Mass Defect

• The difference in mass of a nucleus and its constituents, ΔM, is called the mass defect, and is given by

• The atomic mass of ¹⁶ ₈O found from mass spectroscopy experiments is seen to be 15.99493 u.

• Substracting the mass of 8 electrons (8 × 0.00055 u) from this, we get the experimental mass of O nucleus to be

15.99053 u.

Nuclear Binding Energy

• It is the energy equivalent of mass defect.

• If a certain number of neutrons and protons are brought together to form a nucleus of a certain charge and mass, an energy E_b will be released in the process.
• The ratio of the binding energy E_b of a nucleus to the number of the nucleons, A, in that nucleus is called binding energy per nucleon

Plot of the binding energy per nucleon E_bn versus the mass number A

Features of the Graph

• The binding energy per nucleon, E_bn, is practically constant, i.e. practically independent of the atomic number for nuclei of middle mass number ( 30 < A < 170).
• The curve has a maximum of about 8.75 MeV for A = 56 and has a value of 7.6 MeV for A = 238
• E_bn is lower for both light nuclei (A<30) and heavy nuclei (A>170).

Conclusions:

• A very heavy nucleus, say A = 240, has lower binding energy per nucleon compared to that of a nucleus with A = 120. Thus if a nucleus A = 240 breaks into two A = 120 nuclei, nucleons get more tightly bound.

• Thus energy would be released when a heavy nucleus is broken into light nucleus- the process- nuclear fission
• Similarly when two light nuclei (A≤ 10) are joined together to form a heavy nucleus , energy is released- nuclear Fusion

NUCLEAR FORCE

• Force that binds the nucleons together.
• Strongest force in nature.
• Short range force.
• Does not depend on charge.
• The property that a given nucleon influences only nucleons close to it is also referred to as saturation property of the nuclear force.
• The nuclear force between two nucleons falls rapidly to zero as their distance is more than a few femtometres
• Acts through the exchange of π-mesons

Plot of the potential energy between two nucleons as a function of distance

• The potential energy is a minimum at a distance r0 of about 0.8 fm.
• This means that the force is attractive for distances larger than 0.8 fm and repulsive if they are separated by distances less than 0.8 fm.

RADIOACTIVITY

• H. Becquerel discovered radioactivity in 1896.
• Radioactivity is a nuclear phenomenon in which an unstable nucleus undergoes a decay. This is referred to as radioactive decay.
• Three types of radioactive decay occur in nature :
• α-decay in which a helium nucleus (He) is emitted;
• β-decay in which electrons or positrons (particles with the same mass as electrons, but with a charge exactly opposite to that of electron) are emitted;
• γ-decay in which high energy (hundreds of keV or more) photons are emitted.

Law of radioactive decay

• This law states that the number of nuclei undergoing the decay per unit time is proportional to the total number of nuclei in the sample.
• If a sample contains N undecayed nuclei and let dN nuclei disintegrate in dt second, thus the rate of disintegration

• The negative sign shows that the number of nuclei decreases with time.
• Thus

• Where λ is called the radioactive decay constant or disintegration constant.

• Now, integrating both sides of the above equation, we get

• Here N₀ is the number of radioactive nuclei in the sample at some arbitrary time t₀ and N is the number of radioactive nuclei at any subsequent time t.
• Setting t₀ = 0

• Thus

N = N_o e^{– λt}

Decay Rate

• It gives the number of nuclei decaying per unit time

• Here R₀ is the radioactive decay rate at time t = 0, and R is the rate at any subsequent time t.

Thus

• The total decay rate R of a sample of one or more radionuclide’s is called the activity of that sample.
• The SI unit for activity is becquerel, named after the discoverer of radioactivity.
• 1 becquerel = 1Bq = 1 decay per second
• An older unit, the curie, is still in common use.

Half life period (T_1/2)

• It is the time in which the number of undecayed nuclei falls into half of its original number.
• Thus it is the time at which both N and R have been reduced to one-half their initial values.

Mean life (τ)

• It is the average life of all the nuclei in a radioactive sample.
• Mean life = total life time of all nuclei / total number of nuclei present initially

• The number of nuclei which decay in the time interval t to t + Δt is

• Each of them has lived for time t. Thus the total life of all these nuclei would be

• Therefore mean life is given by

Alpha decay

• When a nucleus undergoes alpha-decay, it transforms to a different nucleus by emitting an alpha-particle (a helium nucleus)

• The difference between the initial mass energy and the final mass energy of the decay products is called the Q value of the process or the disintegration energy.

• This energy is shared by the daughter nucleus and the alpha particle,in the form of kinetic energy
• Alpha-decay obeys the radioactive law
• Alpha particles are positively charged particles
• Can be deflected by electric and magnetic fields.
• Can affect photographic plates.

Beta decay

• A nucleus that decays spontaneously by emitting an electron or a positron is said to undergo beta decay.
• In beta-minus decay, a neutron transforms into a proton within the nucleus according to

• Where ν is the antineutrino
• In beta minus (β ⁻) decay, an electron is emitted by the nucleus.
• Eg:

• When β ^– particles are emitted, the atomic number increases by one.
• In beta-plus decay, a proton transforms into neutron (inside the nucleus)

• Where ν is the neutrino
• In beta plus (β+ ) decay, a positron is emitted by the nucleus,
• Eg:

• When β⁺ particles are emitted the atomic number decreases by one.

Neutrinos and Antineutrinos

• The particles which are emitted from the nucleus along with the electron or positron during the decay process.
• Neutrinos interact only very weakly with matter; they can even penetrate the earth without being absorbed.

Gamma decay

• There are energy levels in a nucleus, just like there are energy levels in atoms.
• When a nucleus is in an excited state, it can make a transition to a lower energy state by the emission of electromagnetic radiation.
• As the energy differences between levels in a nucleus are of the order of MeV, the photons emitted by the nuclei have MeV energies and are called gamma rays.

• Most radionuclides after an alpha decay or a beta decay leave the daughter nucleus in an excited state.
• The daughter nucleus reaches the ground state by a single transition or sometimes by successive transitions by emitting one or more gamma rays.

NUCLEAR ENERGY

• In conventional energy sources like coal or petroleum, energy is released through chemical reactions.
• One kilogram of coal on burning gives 10⁷ J of energy, whereas 1 kg of uranium, which undergoes fission, will generate on fission 10¹⁴ J of energy.

Nuclear Fission

• Enrico Fermi found that when neutrons bombard various elements, new radioactive elements are produced.
• Eg:

• The fragment nuclei produced in fission are highly neutron-rich and unstable.
• They are radioactive and emit beta particles in succession until each reaches a stable end product.
• The energy released (the Q value ) in the fission reaction of nuclei like uranium is of the order of 200 MeV per fissioning nucleus.
• The disintegration energy in fission events first appears as the kinetic energy of the fragments and neutrons.
• Eventually it is transferred to the surrounding matter appearing as heat.
• The source of energy in nuclear reactors, which produce electricity, is nuclear fission.
• The enormous energy released in an atom bomb comes from uncontrolled nuclear fission.

Nuclear reactor

• Neutrons liberated in fission of a uranium nucleus were so energetic that they would escape instead of triggering another fission reaction.
• Slow neutrons have a much higher intrinsic probability of inducing fission in U (235) than fast neutrons.

• The average energy of a neutron produced in fission of U (235) is 2 MeV.
• In reactors, light nuclei called moderators are provided along with the fissionable nuclei for slowing down fast neutrons.
• The moderators commonly used are water, heavy water (D2O) and graphite.
• The Apsara reactor at the Bhabha Atomic Research Centre (BARC), Mumbai, uses water as moderator.
• The other Indian reactors, which are used for power production, use heavy water as moderator.

Multiplication factor

• It is the ratio of number of fission produced by a given generation of neutrons to the number of fission of the preceding generation.
• It is the measure of the growth rate of the neutrons in the reactor.
• For K = 1, the operation of the reactor is said to be critical, which is what we wish it to be for steady power operation.
• If K becomes greater than one, the reaction rate and the reactor power increases exponentially.
• Unless the factor K is brought down very close to unity, the reactor will become supercritical and can even explode.
• The explosion of the Chernobyl reactor in Ukraine in 1986 is a sad reminder that accidents in a nuclear reactor can be catastrophic.
• The reaction rate is controlled through control-rods made out of neutronabsorbing material such as cadmium.
• In addition to control rods, reactors are provided with safety rods which, when required, can be inserted into the reactor and K can be reduced rapidly to less than unity.
• The abundant U(238) isotope, which does not fission, on capturing a neutron leads to the formation of plutonium.

• Plutonium is highly radioactive and can also undergo fission under bombardment by slow neutrons

Pressurized-water reactor

• In such a reactor, water is used both as the moderator and as the heat transfer medium

• In the primary-loop, water is circulated through the reactor vessel and transfers energy at high temperature and pressure (at about 600 K and 150 atm) to the steam generator, which is part of the secondaryloop.
• In the steam generator, evaporation provides high-pressure steam to operate the turbine that drives the electric generator.
• The low-pressure steam from the turbine is cooled and condensed to water and forced back into the steam generator.
• A kilogram of U(235) on complete fission generates about 3 × 10⁴ MW.
• in nuclear reactions highly radioactive elements are continuously produced.
• Therefore, an unavoidable feature of reactor operation is the accumulation of radioactive waste, including both fission products and heavy transuranic elements such as plutonium and americium.

Nuclear fusion

• Energy can be released if two light nuclei combine to form a single larger nucleus, a process called nuclear fusion.

• The fusion reaction in the sun is a multistep process in which hydrogen is burned into helium, hydrogen being the ‘fuel’ and helium the ‘ashes’.
• The proton-proton (p, p) cycle by which this occurs is represented by the following sets of reactions:.

• The combined reaction is

• In sun it has been going on for about 5 × 10⁹ y, and calculations show that there is enough hydrogen to keep the sun going for about the same time into the future.
• In about 5 billion years, however, the sun’s core, which by that time will be largely helium, will begin to cool and the sun will start to collapse under its own gravity.
• This will raise the core temperature and cause the outer envelope to expand, turning the sun into what is called a red giant.
• If the core temperature increases to 10⁸ K again, energy can be produced through fusion once more – this time by burning helium to make carbon.

Controlled thermonuclear fusion

• The first thermonuclear reaction on earth occurred at Eniwetok Atoll on November 1, 1952, when USA exploded a fusion device, generating energy equivalent to 10 million tons of TNT (one ton of TNT on explosion releases 2.6 × 10’22 MeV of energy).
• A sustained and controllable source of fusion power is considerably more difficult to achieve.