(a) 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.

(b) 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 ²⁰⁹₈₃ Bi 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 ⁹⁷₄₃Tc

(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 (5730years) 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 Mohenjo-Daro gives an activity of 9 decays per minute per gram of carbon. Estimate the approximate age of the Indus-Valley civilisation.

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 (¹⁰⁷₄₇Ag).

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

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

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

The radionuclide ¹¹C decays according to ¹¹₁₁C à ¹¹₅B+ e⁺ + ν; T_1/2 =20.3 min. The maximum energy of the emitted positron is 0.960 MeV. Given the mass values: m (¹¹₆C) = 11.011434 u and m (¹¹₆B) = 11.009305 u,

Calculate Q and compare it with the maximum energy of the positron emitted.

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 um (²³₁₁Na) = 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 + ¹¹₆C → 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 ²³⁹₉₄ Pu are very similar to those of ²³⁵₉₂ 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 ²³⁹₉₄ Pu 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 à 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).

e⁺ + ^A_ZX à ^A_Z-1Y + ν

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