Solution of Chapter 13. Nuclei (Physics Part-II Book)

Two stable isotopes of lithium ⁶₃Li and ⁷₃Li have respective abundances of 7.5% and 92.5%. These isotopes have masses 6.01512 u and 7.01600 u, respectively. Find the atomic mass of lithium.

Boron has two stable isotopes, ¹⁰₅B and ¹¹₅B. Their respective masses are 10.01294 u and 11.00931 u, and the atomic mass of boron is 10.811 u. Find the abundances of ¹⁰₅B and ¹¹₅B.

The three stable isotopes of neon:

²⁰₁₀Ne, ²¹₁₀Ne and ²²₁₀Ne, have respective abundances of 90.51%, 0.27%, and 9.22%. The atomic masses of the three isotopes are 19.99 u, 20.99 u and 21.99 u, respectively. Obtain the average atomic mass of neon.

Obtain the binding energy (in MeV) of a nitrogen nucleus (¹⁴₇N) given m (¹⁴₇N) = 14.00307 u

Obtain the binding energy of the nuclei ⁵⁶₂₆Fe and ²⁰⁹₈₃BI in units of MeV from the following data:

m (⁵⁶₂₆Fe) = 55.934939 u m (²⁰⁹₈₃BI) = 208.980388 u

A given coin has a mass of 3.0 g. Calculate the nuclear energy that would be required to separate all the neutrons and protons from each other. For simplicity assume that the coin is entirely made of ⁶³₂₉Cu atoms (of mass 62.92960 u).

Write nuclear reaction equations for

(i) α-decay of ²²⁶₈₈Ra

(ii) α-decay of ²⁴²₉₄Pu

(iii) β^–-decay of ³²₁₅P

(iv) β^–-decay of ²¹⁰₈₃BI

(v) β⁺ -decay of ¹¹₆C

vi) β⁺ -decay of ⁴³₉₇Tc

(vii) Electron capture of ¹²⁰₅₄Xe

A radioactive isotope has a half-life of T years. How long will it take the activity to reduce to a) 3.125%, b) 1% of its original value?

The normal activity of living carbon-containing matter is found to be about 15 decays per minute for every gram of carbon. This activity arises from the small proportion of radioactive ¹⁴₆C present with the stable carbon isotope ¹²₆C. When the organism is dead, its interaction with the atmosphere (which maintains the above equilibrium activity) ceases and its activity begins to drop. From the known half-life (5730 years) of ¹⁴₆C, and the measured activity, the age of the specimen can be approximately estimated. This is the principle of ¹⁴₆C dating used in archaeology. Suppose a specimen from Mohenjodaro gives an activity of 9 decays per minute per gram of carbon. Estimate the approximate age of the Indus-Valley civilization.

Obtain the amount of ⁶⁰₂₇Co necessary to provide a radioactive source of 8.0 mCi strength. The half-life of ⁶⁰₂₇Co is 5.3 years.

The half-life of ⁹⁰₃₈Sr is 28 years. What is the disintegration rate of 15 mg of this isotope?

Obtain approximately the ratio of the nuclear radii of the gold isotope ¹⁹⁷₇₉Au and the silver isotope.

Find the Q-value and the kinetic energy of the emitted α-particle in the α-decay of (a) ²²⁶₈₈Ra and (b) ²²⁰₈₆Rn

Given m (²²⁶₈₈Ra) = 226.02540 u, m (²²²₈₆Rn) = 222.01750 u,

m (²²²₈₆Rn) = 220.01137 u, m (²¹⁶₈₄Po) = 216.00189 u.

The radionuclide ¹¹C decays according to

The nucleus ²³₁₀Ne decays by β^– emission. Write down the β-decay equation and determine the maximum kinetic energy of the electrons emitted. Given that:

m (²³₁₀Ne) = 22.994466 u

m (²³₁₁Ne) = 22.089770 u.

The Q value of a nuclear reaction A + b → C + d is defined by

Q = [ m_A + m_b– m_C – m_d]c²

Where the masses refer to the respective nuclei. Determine from the given data the Q-value of the following reactions and state whether the reactions are exothermic or endothermic.

(i) ¹₁H + ³₁H → ²₁H + ²₁H

(ii) ¹²₆C + ¹²₆C → ²⁰₁₀Ne + ⁴₂He

Atomic masses are given to be

m (²₁H) = 2.014102 u

m (³₁H) = 3.016049 u

m (¹²₆C) = 12.000000 u

m (²⁰₁₀Ne) = 19.992439 u

Suppose, we think of fission of a ⁵⁶₂₆Fe nucleus into two equal fragments, ²⁸₁₃Al. Is the fission energetically possible? Argue by working out Q of the process. Given m (⁵⁶₂₆Fe) = 55.93494 u and m (²⁸₁₃Al) = 27.98191 u.

The fission properties of ²³⁹₉₄Pu are very similar to those of ²³⁵₉₂U. The average energy released per fission is 180 MeV. How much energy, in MeV, is released if all the atoms in 1 kg of pure ²³⁹₉₄Pu undergo fission?

A 1000 MW fission reactor consumes half of its fuel in 5.00 y. How much ²³⁵₉₂U did it contain initially? Assume that the reactor operates 80% of the time, that all the energy generated arises from the fission of ²³⁵₉₂U and that this nuclide is consumed only by the fission process.

How long can an electric lamp of 100W be kept glowing by fusion of 2.0 kg of deuterium? Take the fusion reaction as:

²₁H + ²₁H → ³₂He + n + 3.27 MeV

Calculate the height of the potential barrier for a head-on collision of two deuterons. (Hint: The height of the potential barrier is given by the Coulomb repulsion between the two deuterons when they just touch each other. Assume that they can be taken as hard spheres of radius 2.0 fm.)

From the relation R = R₀A^1/3, where R₀ is a constant and A is the mass number of a nucleus, show that the nuclear matter density is nearly constant (i.e. independent of A).

For the β + (positron) emission from a nucleus, there is another competing process known as electron capture (electron from an inner orbit, say, the K–shell, is captured by the nucleus and a neutrino is emitted).

Show that if β ⁺ emission is energetically allowed, electron capture is necessarily allowed but not vice–versa.

In a periodic table, the average atomic mass of magnesium is given as 24.312 u. The average value is based on their relative natural abundance on earth. The three isotopes and their masses are ²⁴₁₂Mg (23.98504u), ²⁵₁₂Mg (24.98584u) and ²⁶₁₂Mg (25.98259u). The natural abundance of ²⁴₁₂Mg is 78.99% by mass. Calculate the abundances of other two isotopes.

The neutron separation energy is defined as the energy required to remove a neutron from the nucleus. Obtain the neutron separation energies of the nuclei ⁴¹₂₀Ca and ²⁷₁₃Al from the following data:

m(⁴⁰₂₀Ca) = 39.962591 u

m (⁴¹₂₀Ca) = 40.962278 u

m (²⁶₁₃Al) = 25.986895 u

m (²⁷₁₃Al) = 26.981541 u

A source contains two phosphorous radio nuclides ³²₁₅P (T_1/2 = 14.3d) and ³³₁₅P (T_1/2 = 25.3d). Initially, 10% of the decays come from ³³₁₅P. How long one must wait until 90% do so?

Under certain circumstances, a nucleus can decay by emitting a particle more massive than a α-particle. Consider the following decay processes:

Calculate the Q-values for these decays and determine that both are energetically allowed.

Consider the fission by fast neutrons. In one fission event, no neutrons are emitted and the final end products, after the beta decay of the primary fragments, are ¹⁴⁰₅₈Ce and ⁹⁹₄₄Ru. Calculate Q for this fission process. The relevant atomic and particle masses are

m(²³⁸₉₂U) = 238.05079 u

m(¹⁴⁰₅₈Ce) = 139.90543 u

m(⁹⁹₄₄Ru) = 98.90594 u

Consider the D–T reaction (deuterium-tritium fusion)

(a) Calculate the energy released in MeV in this reaction from the data:

m (²₁H) = 2.014102 u

m (³₁H) = 3.016049 u

(b) Consider the radius of both deuterium and tritium to be approximately 2.0 fm. What is the kinetic energy needed to overcome the Coulomb repulsion between the two nuclei? To what temperature must the gas be heated to initiate the reaction?

(Hint: Kinetic energy required for one fusion event = average thermal kinetic energy available with the interacting particles

= 2(3kT/2); k = Boltzmann’s constant, T = absolute temperature.)

Obtain the maximum kinetic energy of β-particles, and the radiation frequencies of γ decay in the decay scheme shown in Fig. 13.6. You are given that

m(¹⁹⁸Au) = 197.968233 u

m(¹⁹⁸Hg) = 197.966760 u

Calculate and compare the energy released by a) fusion of 1.0 kg of hydrogen deep within Sun and b) the fission of 1.0 kg of ²³⁵U in a fission reactor.

Suppose India had a target of producing by 2020 AD, 200,000 MW of electric power, ten percent of which was to be obtained from nuclear power plants. Suppose we are given that, on an average, the efficiency of utilization (i.e. conversion to electric energy) of thermal energy produced in a reactor was 25%. How much amount of fissionable uranium would our country need per year by 2020? Take the heat energy per fission of ²³⁵₉₂U to be about 200MeV.