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**Question 13.1** (a) Two stable isotopes of lithium 6
3 Li and 7
3 Li 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.

(b) Boron has two stable isotopes, 10
5B and 11
5B. 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 10
5B and 11
5 B .

**Question 13.2** The three stable isotopes of neon: 20 21 22
10 10 10 Ne, Ne and Ne 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.

**Question 13.3** Obtain the binding energy (in MeV) of a nitrogen nucleus (14 )
7N ,
given m (14 )
7N =14.00307 u

**Question 13.4** Obtain the binding energy of the nuclei 56
26Fe and 209
83 Bi in units of
MeV from the following data:

m ( 56
26Fe ) = 55.934939 u m ( 209
83 Bi ) = 208.980388 u

**Question 13.5** 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
63
29Cu atoms (of mass 62.92960 u).

**Question 13.6** Write nuclear reaction equations for
(i) α-decay of 226
88 Ra (ii) α-decay of 242
94 Pu
(iii) β–-decay of 32
15 P (iv) β–-decay of 210
83 Bi
(v) β+-decay of 11
6 C (vi) β+-decay of 97
43 Tc
(vii) Electron capture of 120
54 Xe

**Question 13.7** 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?

**Question 13.8** 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 14
6C present with the
stable carbon isotope 12
6C . 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 14
6C , and the measured activity, the age of the specimen
can be approximately estimated. This is the principle of 14
6C 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 civilisation.

**Question 13.9** Obtain the amount of 60
27Co necessary to provide a radioactive source
of 8.0 mCi strength. The half-life of 60
27Co is 5.3 years.

**Question 13.10** The half-life of 90
38Sr is 28 years. What is the disintegration rate of
15 mg of this isotope?

**Question 13.11** Obtain approximately the ratio of the nuclear radii of the gold isotope
197
79 Au and the silver isotope 107
47 Ag .

**Question 13.12** Find the Q-value and the kinetic energy of the emitted α-particle in
the α-decay of (a) 226
88 Ra and (b) 220
86 Rn .
Given m ( 226
88 Ra ) = 226.02540 u, m ( 222
86 Rn ) = 222.01750 u,
m ( 222
86 Rn ) = 220.01137 u, m ( 216
84 Po ) = 216.00189 u.

**Question 13.13** The radionuclide 11C decays according to
11 11 +
6C → 5 B+e +ν : T1/2=20.3 min
The maximum energy of the emitted positron is 0.960 MeV.
Given the mass values:
m ( 11
6C) = 11.011434 u and m ( 11
6B ) = 11.009305 u,
calculate Q and compare it with the maximum energy of the positron
emitted.

**Question 13.14** The nucleus 23
10 Ne decays by β– emission. Write down the β-decay
equation and determine the maximum kinetic energy of the
electrons emitted. Given that:
m ( 23
10 Ne ) = 22.994466 u
m ( 23
11 Na ) = 22.089770 u.

**Question 13.15** The Q value of a nuclear reaction A + b → C + d is defined by
Q = [ mA + mb – mC – md]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) 1 3 2 2
1 1 1 1 H+ H → H+ H
(ii) 12 12 20 4
6 6 10 2 C+ C → Ne+ He
Atomic masses are given to be
m ( 2
1H) = 2.014102 u
m ( 3
1H) = 3.016049 u
m ( 12
6C ) = 12.000000 u
m ( 20
10 Ne ) = 19.992439 u

**Question 13.16** Suppose, we think of fission of a 56
26Fe nucleus into two equal
fragments, 28
13 Al . Is the fission energetically possible? Argue by
working out Q of the process. Given m ( 56
26Fe ) = 55.93494 u and
m ( 28
13 Al ) = 27.98191 u.

**Question 13.17 **The fission properties of 239
94 Pu are very similar to those of 235
92 U. 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 239
94 Pu undergo
fission?

**Question 13.18** A 1000 MW fission reactor consumes half of its fuel in 5.00 y. How
much 235
92 U did it contain initially? Assume that the reactor operates
80% of the time, that all the energy generated arises from the fission
of 235
92 U and that this nuclide is consumed only by the fission process.

**Question 13.19** How long can an electric lamp of 100W be kept glowing by fusion of
2.0 kg of deuterium? Take the fusion reaction as
2 2 3
1H+ 1H→ 2He+n+3.27 MeV?

**Question 13.20** 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.)

**Question 13.21** From the relation R = R0A1/3, 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).

**Question 13.22** 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).
1
A A
Z Z e+ X Y ν
− + → +
Show that if β+ emission is energetically allowed, electron capture
is necessarily allowed but not vice–versa.

ADDITIONAL EXERCISES QUESTIONS

**Question 13.23** 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 24
12Mg
(23.98504u), 25
12Mg (24.98584u) and 26
12Mg (25.98259u). The natural
abundance of 24
12Mg is 78.99% by mass. Calculate the abundances
of other two isotopes.

**Question 13.24** 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 41
20Ca and 27
13 Al from the following data:
m( 40
20Ca ) = 39.962591 u
m( 41
20Ca ) = 40.962278 u
m( 26
13 Al ) = 25.986895 u
m( 27
13 Al ) = 26.981541 u?

**Question 13.25** A source contains two phosphorous radio nuclides 32
15P (T1/2 = 14.3d)
and 33
15P (T1/2 = 25.3d). Initially, 10% of the decays come from 33
15P .
How long one must wait until 90% do so?

**Question 13.26** Under certain circumstances, a nucleus can decay by emitting a
particle more massive than an α-particle. Consider the following
decay processes:
223 209 14
88 82 6 Ra→ Pb + C 223 219 4
88Ra→ 86Rn + 2He
Calculate the Q-values for these decays and determine that both
are energetically allowed.

**Question 13.27** Consider the fission of 238
92U 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 140
58Ce and 99
44Ru . Calculate Q
for this fission process. The relevant atomic and particle masses
are
m( 238
92U ) =238.05079 u
m( 140
58Ce ) =139.90543 u
m( 99
44Ru ) = 98.90594 u

**Question 13.28** Consider the D–T reaction (deuterium–tritium fusion)
2 3 4
1 1 2 H+ H→ He + n
(a) Calculate the energy released in MeV in this reaction from the
data:
m( 2
1H )=2.014102 u
m( 3
1H ) =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 = Boltzman’s constant, T = absolute temperature.)

**Question 13.29** Obtain the maximum kinetic energy of β-particles, and the radiation
frequencies of γ decays in the decay scheme shown in Fig. 13.6. You
are given that
m(198Au) = 197.968233 u
m(198Hg) =197.966760 u

**Question 13.30** 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.

**Question 13.31** 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 235U to be about 200MeV.

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