Answer

(a) Mass of lithium isotope , m₁ = 6.01512 u

Mass of lithium isotope , m₂ = 7.01600 u

Abundance of , η₁= 7.5%

Abundance of , η₂= 92.5%

The atomic mass of lithium atom is given as:

(b) Mass of boron isotope , m₁ = 10.01294 u

Mass of boron isotope , m₂ = 11.00931 u

Abundance of , η₁ = x%

Abundance of , η₂= (100 − x)%

Atomic mass of boron, m = 10.811 u

The atomic mass of boron atom is given as:

And 100 − x = 80.11%

Hence, the abundance of  is 19.89% and that of is 80.11%.

Answer

Atomic mass of , m₁= 19.99 u

Abundance of , η₁ = 90.51%

Atomic mass of , m₂ = 20.99 u

Abundance of , η₂ = 0.27%

Atomic mass of , m₃ = 21.99 u

Abundance of , η₃ = 9.22%

The average atomic mass of neon is given as:

Answer

Atomic mass of nitrogen, m = 14.00307 u

A nucleus of nitrogen  contains 7 protons and 7 neutrons.

Hence, the mass defect of this nucleus, Δm = 7m_H + 7m_n − m

Where,

Mass of a proton, m_H = 1.007825 u

Mass of a neutron, m_n= 1.008665 u

∴Δm = 7 × 1.007825 + 7 × 1.008665 − 14.00307

= 7.054775 + 7.06055 − 14.00307

= 0.11236 u

But 1 u = 931.5 MeV/c²

∴Δm = 0.11236 × 931.5 MeV/c²

Hence, the binding energy of the nucleus is given as:

E_b = Δmc²

Where,

c = Speed of light

∴E_b = 0.11236 × 931.5 

= 104.66334 MeV

Hence, the binding energy of a nitrogen nucleus is 104.66334 MeV.

Answer

Atomic mass of, m₁ = 55.934939 u

 nucleus has 26 protons and (56 − 26) = 30 neutrons

Hence, the mass defect of the nucleus, Δm = 26 × m_H + 30 × m_n − m₁

Where,

Mass of a proton, m_H = 1.007825 u

Mass of a neutron, m_n = 1.008665 u

∴Δm = 26 × 1.007825 + 30 × 1.008665 − 55.934939

= 26.20345 + 30.25995 − 55.934939

= 0.528461 u

But 1 u = 931.5 MeV/c²

∴Δm = 0.528461 × 931.5 MeV/c²

The binding energy of this nucleus is given as:

E_b1 = Δmc²

Where,

c = Speed of light

∴E_b1 = 0.528461 × 931.5 

= 492.26 MeV

Average binding energy per nucleon 

Atomic mass of, m₂ = 208.980388 u

 nucleus has 83 protons and (209 − 83) 126 neutrons.

Hence, the mass defect of this nucleus is given as:

Δm‘ = 83 × m_H + 126 × m_n − m₂

Where,

Mass of a proton, m_H = 1.007825 u

Mass of a neutron, m_n = 1.008665 u

∴Δm‘ = 83 × 1.007825 + 126 × 1.008665 − 208.980388

= 83.649475 + 127.091790 − 208.980388

= 1.760877 u

But 1 u = 931.5 MeV/c²

∴Δm‘ = 1.760877 × 931.5 MeV/c²

Hence, the binding energy of this nucleus is given as:

E_b2 = Δm‘c²

= 1.760877 × 931.5

= 1640.26 MeV

Average bindingenergy per nucleon = 

Answer

Mass of a copper coin, m’ = 3 g

Atomic mass of atom, m = 62.92960 u

The total number of atoms in the coin

Where,

N_A = Avogadro’s number = 6.023 × 10²³ atoms /g

Mass number = 63 g

nucleus has 29 protons and (63 − 29) 34 neutrons

∴Mass defect of this nucleus, Δm‘ = 29 × m_H + 34 × m_n − m

Where,

Mass of a proton, m_H = 1.007825 u

Mass of a neutron, m_n = 1.008665 u

∴Δm‘ = 29 × 1.007825 + 34 × 1.008665 − 62.9296

= 0.591935 u

Mass defect of all the atoms present in the coin, Δm = 0.591935 × 2.868 × 10²²

= 1.69766958 × 10²² u

But 1 u = 931.5 MeV/c²

∴Δm = 1.69766958 × 10²² × 931.5 MeV/c²

Hence, the binding energy of the nuclei of the coin is given as:

E_b= Δmc²

= 1.69766958 × 10²² × 931.5 

= 1.581 × 10²⁵ MeV

But 1 MeV = 1.6 × 10⁻¹³ J

E_b = 1.581 × 10²⁵ × 1.6 × 10⁻¹³

= 2.5296 × 10¹² J

This much energy is required to separate all the neutrons and protons from the given coin.

Answer

α is a nucleus of helium and β is an electron (e⁻ for β⁻ and e⁺ for β⁺). In every α-decay, there is a loss of 2 protons and 4 neutrons. In every β⁺-decay, there is a loss of 1 proton and a neutrino is emitted from the nucleus. In every β⁻-decay, there is a gain of 1 proton and an antineutrino is emitted from the nucleus.

For the given cases, the various nuclear reactions can be written as:

Answer

Half-life of the radioactive isotope = T years

Original amount of the radioactive isotope = N₀

(a) After decay, the amount of the radioactive isotope = N

It is given that only 3.125% of N₀ remains after decay. Hence, we can write:

Where,

λ = Decay constant

t = Time

Hence, the isotope will take about 5T years to reduce to 3.125% of its original value.

(b) After decay, the amount of the radioactive isotope = N

It is given that only 1% of N₀ remains after decay. Hence, we can write:

Since, λ = 0.693/T

Hence, the isotope will take about 6.645T years to reduce to 1% of its original value.

Answer

Decay rate of living carbon-containing matter, R = 15 decay/min

Let N be the number of radioactive atoms present in a normal carbon- containing matter.

Half life of, = 5730 years

The decay rate of the specimen obtained from the Mohenjodaro site:

R‘ = 9 decays/min

Let N’ be the number of radioactive atoms present in the specimen during the Mohenjodaro period.

Therefore, we can relate the decay constant, λand time, t as:

Hence, the approximate age of the Indus-Valley civilisation is 4223.5 years.

Answer

The strength of the radioactive source is given as:

Where,

N = Required number of atoms

Half-life of,  = 5.3 years

= 5.3 × 365 × 24 × 60 × 60

= 1.67 × 10⁸ s

For decay constant λ, we have the rate of decay as:

Where, λ 

For:

Mass of 6.023 × 10²³ (Avogadro’s number) atoms = 60 g

∴Mass of atoms 

Hence, the amount of  necessary for the purpose is 7.106 × 10⁻⁶ g.

Answer

Half life of , = 28 years

= 28 × 365 × 24 × 60 × 60

= 8.83 × 10⁸ s

Mass of the isotope, m = 15 mg

90 g of atom contains 6.023 × 10²³ (Avogadro’s number) atoms.

Therefore, 15 mg of  contains:

Rate of disintegration, 

Where,

λ = Decay constant 

Hence, the disintegration rate of 15 mg of the given isotope is
7.878 × 10¹⁰ atoms/s.

Answer

Nuclear radius of the gold isotope = R_Au

Nuclear radius of the silver isotope = R_Ag

Mass number of gold, A_Au = 197

Mass number of silver, A_Ag = 107

The ratio of the radii of the two nuclei is related with their mass numbers as:

Hence, the ratio of the nuclear radii of the gold and silver isotopes is about 1.23.

Answer

(a) Alpha particle decay of emits a helium nucleus. As a result, its mass number reduces to (226 − 4) 222 and its atomic number reduces to (88 − 2) 86. This is shown in the following nuclear reaction.

Q-value of

emitted α-particle = (Sum of initial mass − Sum of final mass) c²

Where,

c = Speed of light

It is given that:

Q-value = [226.02540 − (222.01750 + 4.002603)] u c²
= 0.005297 u c²

But 1 u = 931.5 MeV/c²

∴Q = 0.005297 × 931.5 ≈ 4.94 MeV

Kinetic energy of the α-particle 

(b) Alpha particle decay of  is shown by the following nuclear reaction.

It is given that:

Mass of = 220.01137 u

Mass of = 216.00189 u

∴Q-value = 

≈ 641 MeV

Kinetic energy of the α-particle 

= 6.29 MeV

Answer

The given nuclear reaction is:

Atomic mass of = 11.011434 u

Atomic mass of 

Maximum energy possessed by the emitted positron = 0.960 MeV

The change in the Q-value (ΔQ) of the nuclear masses of the  nucleus is given as:

Where,

m_e = Mass of an electron or positron = 0.000548 u

c = Speed of light

m’ = Respective nuclear masses

If atomic masses are used instead of nuclear masses, then we have to add 6 m_e in the case ofand 5 m_e in the case of.

Hence, equation (1) reduces to:

∴ΔQ = [11.011434 − 11.009305 − 2 × 0.000548] c²

= (0.001033 c²) u

But 1 u = 931.5 Mev/c²

∴ΔQ = 0.001033 × 931.5 ≈ 0.962 MeV

The value of Q is almost comparable to the maximum energy of the emitted positron.

Answer

In  emission, the number of protons increases by 1, and one electron and an antineutrino are emitted from the parent nucleus.

 emission of the nucleus  is given as:

It is given that:

Atomic mass of = 22.994466 u

Atomic mass of = 22.989770 u

Mass of an electron, m_e = 0.000548 u

Q-value of the given reaction is given as:

There are 10 electrons in  and 11 electrons in. Hence, the mass of the electron is cancelled in the Q-value equation.

The daughter nucleus is too heavy as compared to and . Hence, it carries negligible energy. The kinetic energy of the antineutrino is nearly zero. Hence, the maximum kinetic energy of the emitted electrons is almost equal to the Q-value, i.e., 4.374 MeV.

Answer

(i) The given nuclear reaction is:

It is given that:

Atomic mass 

Atomic mass 

Atomic mass 

According to the question, the Q-value of the reaction can be written as:

The negativeQ-value of the reaction shows that the reaction is endothermic.

(ii) The given nuclear reaction is:

It is given that:

Atomic mass of 

Atomic mass of 

Atomic mass of 

The Q-value of this reaction is given as:

The positive Q-value of the reaction shows that the reaction is exothermic.

Answer

The fission of can be given as:

It is given that:

Atomic mass of  = 55.93494 u

Atomic mass of 

The Q-value of this nuclear reaction is given as:

The Q-value of the fission is negative. Therefore, the fission is not possible energetically. For an energetically-possible fission reaction, the Q-value must be positive.

Answer

Average energy released per fission of, 

Amount of pure, m = 1 kg = 1000 g

N_A= Avogadro number = 6.023 × 10²³

Mass number of= 239 g

1 mole of contains N_A atoms.

∴m g of contains

∴Total energy released during the fission of 1 kg ofis calculated as:

Hence,  is released if all the atoms in 1 kg of pure undergo fission.

Answer

Half life of the fuel of the fission reactor, years

= 5 × 365 × 24 × 60 × 60 s

We know that in the fission of 1 g of nucleus, the energy released is equal to 200 MeV.

1 mole, i.e., 235 g of contains 6.023 × 10²³ atoms.

∴1 g  contains

The total energy generated per gram ofis calculated as:

The reactor operates only 80% of the time.

Hence, the amount of consumed in 5 years by the 1000 MW fission reactor is calculated as:

∴Initial amount of = 2 × 1538 = 3076 kg

Answer

The given fusion reaction is:

Amount of deuterium, m = 2 kg

1 mole, i.e., 2 g of deuterium contains 6.023 × 10²³ atoms.

∴2.0 kg of deuterium contains 

It can be inferred from the given reaction that when two atoms of deuterium fuse, 3.27 MeV energy is released.

∴Total energy per nucleus released in the fusion reaction:

Power of the electric lamp, P = 100 W = 100 J/s

Hence, the energy consumed by the lamp per second = 100 J

The total time for which the electric lamp will glow is calculated as:

Answer

When two deuterons collide head-on, the distance between their centres, d is given as:

Radius of 1^st deuteron + Radius of 2^nd deuteron

Radius of a deuteron nucleus = 2 fm = 2 × 10⁻¹⁵ m

∴d = 2 × 10⁻¹⁵ + 2 × 10⁻¹⁵ = 4 × 10⁻¹⁵ m

Charge on a deuteron nucleus = Charge on an electron = e = 1.6 × 10⁻¹⁹ C

Potential energy of the two-deuteron system:

Where,

 = Permittivity of free space

Hence, the height of the potential barrier of the two-deuteron system is

360 keV.

Answer

We have the expression for nuclear radius as:

R = R₀A¹/³

Where,

R₀ = Constant.

A = Mass number of the nucleus

Nuclear matter density, 

Let m be the average mass of the nucleus.

Hence, mass of the nucleus = mA

Hence, the nuclear matter density is independent of A. It is nearly constant.

Answer

Let the amount of energy released during the electron capture process be Q₁. The nuclear reaction can be written as:

Let the amount of energy released during the positron capture process be Q₂. The nuclear reaction can be written as:

= Nuclear mass of 

= Nuclear mass of 

= Atomic mass of 

= Atomic mass of 

m_e = Mass of an electron

c = Speed of light

Q-value of the electron capture reaction is given as:

Q-value of the positron capture reaction is given as:

It can be inferred that if Q₂ > 0, then Q₁ > 0; Also, if Q₁> 0, it does not necessarily mean that Q₂ > 0.

In other words, this means that ifemission is energetically allowed, then the electron capture process is necessarily allowed, but not vice-versa. This is because the Q-value must be positive for an energetically-allowed nuclear reaction.

Answer

Average atomic mass of magnesium, m = 24.312 u

Mass of magnesium isotope, m₁ = 23.98504 u

Mass of magnesium isotope, m₂ = 24.98584 u

Mass of magnesium isotope, m₃ = 25.98259 u

Abundance of, η₁= 78.99%

Abundance of, η₂ = x%

Hence, abundance of, η₃ = 100 − x − 78.99% = (21.01 − x)%

We have the relation for the average atomic mass as:

Hence, the abundance of is 9.3% and that of is 11.71%.

Answer

For 

For

A neutron is removed from a nucleus. The corresponding nuclear reaction can be written as:

It is given that:

Mass = 39.962591 u

Mass) = 40.962278 u

Mass = 1.008665 u

The mass defect of this reaction is given as:

Δm = 

∴Δm = 0.008978 × 931.5 MeV/c²

Hence, the energy required for neutron removal is calculated as:

For, the neutron removal reaction can be written as:

It is given that:

Mass = 26.981541 u

Mass = 25.986895 u

The mass defect of this reaction is given as:

Hence, the energy required for neutron removal is calculated as:

Answer

Half life of, T_{1/2 =} 14.3 days

Half life of, T’_1/2 = 25.3 days

 nucleus decay is 10% of the total amount of decay.

The source has initially 10% of nucleus and 90% of nucleus.

Suppose after t days, the source has 10% of nucleus and 90% of  nucleus.

Initially:

Number of nucleus = N

Number of nucleus = 9 N

Finally:

Number of 

Number of 

For nucleus, we can write the number ratio as:

For, we can write the number ratio as:

On dividing equation (1) by equation (2), we get:

Hence, it will take about 208.5 days for 90% decay of .

Answer

Take a  emission nuclear reaction:

We know that:

Mass of m₁ = 223.01850 u

Mass of m₂ = 208.98107 u

Mass of, m₃ = 14.00324 u

Hence, the Q-value of the reaction is given as:

Q = (m₁ − m₂ − m₃) c²

= (223.01850 − 208.98107 − 14.00324) c²

= (0.03419 c²) u

But 1 u = 931.5 MeV/c²

∴Q = 0.03419 × 931.5

= 31.848 MeV

Hence, the Q-value of the nuclear reaction is 31.848 MeV. Since the value is positive, the reaction is energetically allowed.

Now take a emission nuclear reaction:

We know that:

Mass of m₁ = 223.01850

Mass of m₂ = 219.00948

Mass of, m₃ = 4.00260

Q-value of this nuclear reaction is given as:

Q = (m₁ − m₂ − m₃) c²

= (223.01850 − 219.00948 − 4.00260) C²

= (0.00642 c²) u

= 0.00642 × 931.5 = 5.98 MeV

Hence, the Q value of the second nuclear reaction is 5.98 MeV. Since the value is positive, the reaction is energetically allowed.

Answer

In the fission of, 10 β⁻ particles decay from the parent nucleus. The nuclear reaction can be written as:

It is given that:

Mass of a nucleusm₁ = 238.05079 u

Mass of a nucleus m₂ = 139.90543 u

Mass of a nucleus, m₃ = 98.90594 u

Mass of a neutronm₄ = 1.008665 u

Q-value of the above equation,

Where,

m’ = Represents the corresponding atomic masses of the nuclei

= m₁ − 92m_e

= m₂ − 58m_e

= m₃ − 44m_e

= m₄

Hence, the Q-value of the fission process is 231.007 MeV.

Answer

(a) Take the D-T nuclear reaction: 

It is given that:

Mass of, m₁= 2.014102 u

Mass of, m₂ = 3.016049 u

Mass of m₃ = 4.002603 u

Mass of, m₄ = 1.008665 u

Q-value of the given D-T reaction is:

Q = [m₁ + m₂− m₃ − m₄] c²

= [2.014102 + 3.016049 − 4.002603 − 1.008665] c²

= [0.018883 c²] u

But 1 u = 931.5 MeV/c²

∴Q = 0.018883 × 931.5 = 17.59 MeV

(b) Radius of deuterium and tritium, r ≈ 2.0 fm = 2 × 10⁻¹⁵ m

Distance between the two nuclei at the moment when they touch each other, d = r + r = 4 × 10⁻¹⁵ m

Charge on the deuterium nucleus = e

Charge on the tritium nucleus = e

Hence, the repulsive potential energy between the two nuclei is given as:

Where,

∈₀ = Permittivity of free space

Hence, 5.76 × 10⁻¹⁴ J or of kinetic energy (KE) is needed to overcome the Coulomb repulsion between the two nuclei.

However, it is given that:

KE

Where,

k = Boltzmann constant = 1.38 × 10⁻²³ m² kg s⁻² K⁻¹

T = Temperature required for triggering the reaction

Hence, the gas must be heated to a temperature of 1.39 × 10⁹ K to initiate the reaction.

Answer

It can be observed from the given γ-decay diagram that γ₁ decays from the 1.088 MeV energy level to the 0 MeV energy level.

Hence, the energy corresponding to γ₁-decay is given as:

E₁ = 1.088 − 0 = 1.088 MeV

hν₁= 1.088 × 1.6 × 10⁻¹⁹ × 10⁶ J

Where,

h = Planck’s constant = 6.6 × 10⁻³⁴ Js

ν₁ = Frequency of radiation radiated by γ₁-decay

It can be observed from the given γ-decay diagram that γ₂ decays from the 0.412 MeV energy level to the 0 MeV energy level.

Hence, the energy corresponding to γ₂-decay is given as:

E₂ = 0.412 − 0 = 0.412 MeV

hν₂= 0.412 × 1.6 × 10⁻¹⁹ × 10⁶ J

Where,

ν₂ = Frequency of radiation radiated by γ₂-decay

It can be observed from the given γ-decay diagram that γ₃ decays from the 1.088 MeV energy level to the 0.412 MeV energy level.

Hence, the energy corresponding to γ₃-decay is given as:

E₃ = 1.088 − 0.412 = 0.676 MeV

hν₃= 0.676 × 10⁻¹⁹ × 10⁶ J

Where,

ν₃ = Frequency of radiation radiated by γ₃-decay

Mass of = 197.968233 u

Mass of = 197.966760 u

1 u = 931.5 MeV/c²

Energy of the highest level is given as:

β₁ decays from the 1.3720995 MeV level to the 1.088 MeV level

∴Maximum kinetic energy of the β₁ particle = 1.3720995 − 1.088

= 0.2840995 MeV

β₂ decays from the 1.3720995 MeV level to the 0.412 MeV level

∴Maximum kinetic energy of the β₂ particle = 1.3720995 − 0.412

= 0.9600995 MeV

Answer

(a) Amount of hydrogen, m = 1 kg = 1000 g

1 mole, i.e., 1 g of hydrogen () contains 6.023 × 10²³ atoms.

∴1000 g of  contains 6.023 × 10²³ × 1000 atoms.

Within the sun, four  nuclei combine and form one  nucleus. In this process 26 MeV of energy is released.

Hence, the energy released from the fusion of 1 kg is:

(b) Amount of  = 1 kg = 1000 g

1 mole, i.e., 235 g of  contains 6.023 × 10²³ atoms.

∴1000 g ofcontains 

It is known that the amount of energy released in the fission of one atom of is 200 MeV.

Hence, energy released from the fission of 1 kg of  is:

∴ 

Therefore, the energy released in the fusion of 1 kg of hydrogen is nearly 8 times the energy released in the fission of 1 kg of uranium.

Answer

Amount of electric power to be generated, P = 2 × 10⁵ MW

10% of this amount has to be obtained from nuclear power plants.

∴Amount of nuclear power, 

= 2 × 10⁴ MW

= 2 × 10⁴ × 10⁶ J/s

= 2 × 10¹⁰ × 60 × 60 × 24 × 365 J/y

Heat energy released per fission of a ²³⁵U nucleus, E = 200 MeV

Efficiency of a reactor = 25%

Hence, the amount of energy converted into the electrical energy per fission is calculated as:

Number of atoms required for fission per year:

1 mole, i.e., 235 g of U²³⁵ contains 6.023 × 10²³ atoms.

∴Mass of 6.023 × 10²³ atoms of U²³⁵ = 235 g = 235 × 10⁻³ kg

∴Mass of 78840 × 10²⁴ atoms of U²³⁵

Hence, the mass of uranium needed per year is 3.076 × 10⁴ kg.