Nuclei-Solutions

CBSE Class –XII Physics

NCERT Solutions
Chapter - 13
Nuclei

1: (a) Two stable isotopes of lithium and 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, and . 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 and .

Ans: (a) Mass of lithium isotope , = 6.01512 u

Mass of lithium isotope , = 7.01600 u

Abundance of , = 7.5%

Abundance of , = 92.5%

The atomic mass of lithium atom is given as:

$= \frac{(6.0512 \times 7.5) + (7.01600 \times 92.5)}{92.5 + 7.5}$

(b) Mass of boron isotope , = 10.01294 u

Mass of boron isotope , = 11.00931 u

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%.

2: The three stable isotopes of neon: and 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.

Ans: Atomic mass of , = 19.99 u

Abundance of,

Atomic mass of,

Abundance of,

Atomic mass of,

Abundance of,

The average atomic mass of neon is given as:

= $\frac{(19.99 \times 90.51) + (20.99 \times 0.27) + (21.99 \times 9.22)}{90.51 + 0.27 + 9.22}$

3: Obtain the binding energy (in MeV) of a nitrogen nucleus , given =14.00307 u

Ans: 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,

Where,

Mass of a proton, = 1.007825 u

Mass of a neutron, = 1.008665 u

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

= 7.054775 + 7.06055 - 14.00307

= 0.11236 u

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

Where,

c= Speed of light

∴Eb = 0.11236 × 931.5

= 104.66334 MeV

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

4: Obtain the binding energy of the nuclei and in units of MeV from the following data: = 55.934939 u = 208.980388 u

Ans: Atomicmass of = 55.934939 u

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

Hence, the mass defect of the nucleus,

Where,

∴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:

Eb1= Δmc2

Where,

c= Speed of light

∴Eb1= 0.528461 × 931.5

= 492.26 MeV

Average binding energy per nucleon

Atomic mass of , = 208.980388 u

nucleus has 83 protons and (209 - 83) = 126 neutrons.

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

Δm' Where,

Mass of a proton,

Mass of a neutron,

∴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:

= 1.760877 × 931.5

= 1640.26 MeV

Average binding energy per nucleon =

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 atoms (of mass 62.92960 u).

Ans: 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,

NA= Avogadro's number Mass number = 63 g

nucleus has 29 protons and (63 - 29) = 34 neutrons

∴Mass defect of this nucleus,

Where,

Mass of a proton, mH= 1.007825 u

Mass of a neutron, mn= 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=

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

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

6: Write nuclear reaction equations for

(i) α-decay of

(ii) α-decay of

(iii)

(iv)

(v)

(vi)

(vii) Electron capture of

Ans: α particle is a nucleus of helium and particle is an electron In every α-decay, there is a loss of 2 protons and 4 neutrons. In every decay, there is a loss of 1 proton and a neutrino is emitted from the nucleus. In every -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:

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?

Ans: Half-life of the radioactive isotope = T years

Original amount of the radioactive isotope =

(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,

= Decay constant

t= Time

Since $λ = \frac{0.693}{T}$

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, = 0.693/T

years

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

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 present with the stable carbon isotope . 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, and the measured activity, the age of the specimen can be approximately estimated. This is the principle of 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.

Ans: 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 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, and time, t as:

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

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

Ans: The strength of the radioactive source is given as:

Where,

N= Required number of atoms

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

For : (Avogadro's number) atoms = 60 g

∴Mass of atoms $= \frac{60 \times 7.133 \times 10^{16}}{6.023 \times 10^{23}}$ = 7.106 × 10- 6g.

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

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

Ans: Half life of , = 28 years

= 28 × 365 × 24 × 60 × 60

Mass of the isotope, m= 15 mg

90 g of atom contains (Avogadro's number) atoms.

Therefore, 15 mg of contains:

number of atoms

Rate of disintegration,

Where,

= Decay constant

Hence, the disintegration rate of 15 mg of the given isotope is

11: Obtain approximately the ratio of the nuclear radii of the gold isotope and the silver isotope.

Ans: Nuclear radius of the gold isotope

Nuclear radius of the silver isotope

Mass number of gold, = 197

Mass number of silver, = 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.

12: Find the Q-value and the kinetic energy of the emitted α-particle in the α-decay of (a) and (b) . Given = 226.02540 u, = 222.01750 u,

= 220.01137 u, = 216.00189 u.

Ans: (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)

Where,

c= Speed of light

It is given that:

Q-value = [226.02540 - (222.01750 + 4.002603
=

But 1 u = 931.5 MeV/c

∴Q= 0.005297 × 931.5 = 4.94MeV

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 =

$\approx 6.41 M e V$

Kinetic energy of the α-particle

= 6.29 MeV

13: The radionuclide 11C decays according to

The maximum energy of the emitted positron is 0.960 MeV.

Given the mass values: and calculate Q and compare it with the maximum energy of the positron emitted

Ans: The given nuclear reaction is:

Half life of nuclei,

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:

……..(1)

Where,

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 in the case of and 5 m in the case of .

Hence, equation (1) reduces to:

Here, and are the atomic masses.

∴ΔQ= [11.011434 - 11.009305 - 2 ×0.000548]

But 1 u = 931.5 Mev/

∴ΔQ= 0.001033 ×931.5 0.962 MeV

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

14: The nucleus decays byemission. Write down the decay equation and determine the maximum kinetic energy of the electrons emitted. Given that:

= 22.994466 u

= 22.989770 u.

Ans: 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, = 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.

But 1 u = 931.5 $\frac{M e V}{c^{2}}$

Q = $0.004696 \times 931.5$ = 4.374 MeV

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.

15: The Q value of a nuclear reaction is defined by

Q = 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)

(ii)

Atomic masses are given to be

Ans: (i) The given nuclear reaction is:

It is given that:

Atomic mass

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

But 1 u = 931.5 $\frac{M e V}{c^{2}}$

The negative Q-value of the reaction shows that the reaction is endothermic.

(ii) The given nuclear reaction is:

It is given that:

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.

16: Suppose, we think of fission of a nucleus into two equal fragments,. Is the fission energetically possible? Argue by working out Q of the process. Given and .

Ans: 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:

CORRECTION

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.

17: The fission properties of are very similar to those of .

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 undergo fission?

Ans: Average energy released per fission of ,

Amount of pure ,m = 1 kg = 1000 g

NA= Avogadro number =

Mass number of = 239 g

1 mole of contains NAatoms.

∴mg of contains

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

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

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

Ans: Half life of the fuel of the fission reactor,

= 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 .

∴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 U= 2 ×1538 = 3076 kg

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

Ans: The given fusion reaction is:

Amount of deuterium, m= 2 kg

1 mole, i.e., 2 g of deuterium contains

∴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:

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.)

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

Radius of 1st deuteron + Radius of 2nd deuteron

Radius of a deuteron nucleus = 2 fm =

Charge on a deuteron nucleus = Charge on an electron = e=

Potential energy of the two-deuteron system:

Where,

= Permittivity of free space

CORRECTION

Hence, the height of the potential barrier of the two-deuteron system is 360 keV.

21: From the relation , where R0is 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).

Ans: We have the expression for nuclear radius as:

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.

22: For the (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 emission is energetically allowed, electron capture is necessarily allowed but not vice - versa. (LEFT UNCHECKED)

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

….(1)

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

…..(2)

Nuclear mass of

= Nuclear mass of

= Atomic mass of

= Mass of an electron

c = Speed of light

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

………………(3)

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

…………………(4)

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 if emission 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.

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 (23.98504u), (24.98584u) and (25.98259u). The natural abundance of is 78.99% by mass. Calculate the abundances of other two isotopes.

Ans: Average atomic mass of magnesium, m= 24.312 u

Mass of magnesium isotope = 23.98504 u

Mass of magnesium isotope = 24.98584 u

Mass of magnesium isotope , = 25.98259 u

Abundance of , = 78.99%

Abundance of , = x%

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

We have the relation for the average atomic mass as:

CORRECTION

And

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

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 and from the following data:

= 39.962591 u

= 40.962278 u

= 25.986895 u

= 26.981541 u(LEFT UNCHECKED)

Ans: For : Seperation energy=

For : Seperation energy=

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

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:

= 25.986895 + 1.008665 – 26.981541

= 0.014019 u

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

25: A source contains two phosphorous radio nuclides =14.3d) and = 25.3d). Initially, 10% of the decays come from. How long one must wait until 90% do so?

Ans: Half life of , = 14.3 days

Half life of , = 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 tdays, the source has 10% of nucleus and 90% of nucleus.

Initially:

Number of nucleus = N

Number of nucleus = 9 N

Finally:

Number of nucleus=

For nucleus, we can write the number ratio as:

………………(1)

For , we can write the number ratio as:

………………(2)

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

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

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

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

Ans: Take a emission nuclear reaction:

We know that:

Mass of = 223.01850 u

Mass of , = 208.98107 u

Mass of, = 14.00324 u

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

= (223.01850 - 208.98107 - 14.00324)

∴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 , = 223.01850

Mass of , = 219.00948

Mass of , = 4.00260

Q-value of this nuclear reaction is given as:

= (223.01850 - 219.00948 - 4.00260)

= 0.00642 × 931.5 = 5.98 MeV

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

27: Consider the fission of 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 and . Calculate Q for this fission process.

The relevant atomic and particle masses are

m =238.05079 u

m =139.90543 u

m = 98.90594 u

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

It is given that:

Mass of a nucleus = 238.05079 u

Mass of a nucleus = 139.90543 u

Mass of a nucleus , = 98.90594 u

Mass of a neutron = 1.008665 u

Q-value of the above equation,

Where,

m' = Represents the corresponding atomic masses of the nuclei

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

28: Consider the D - T reaction (deuterium - tritium fusion)

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

= 2.014102 u

= 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.)

Ans: (a) Take the D-T nuclear reaction:

It is given that:

Mass of , = 2.014102 u

Mass of , = 3.016049 u

Mass of = 4.002603 u

Mass of , = 1.008665 u

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

= [2.014102 + 3.016049 - 4.002603 - 1.008665]

= [0.018883 ] u

But 1 u = 931.5 MeV/

∴Q= 0.018883 × 931.5 = 17.59 MeV

(b) Radius of deuterium and tritium, fm = Distance between the two nuclei at the moment when they touch each other, d = r + r = 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, or of kinetic energy (KE) is needed to overcome the Coulomb repulsion between the two nuclei.

However,it is given that:

Where, k= Boltzmann constant =

T = Temperature required for triggering the reaction

Hence, the gas must be heated to a temperature of to initiate the reaction.

29: 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 in a fission reactor.

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

1 mole, i.e., 1 g of hydrogen contains atoms.

∴1000 g of contains

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

1mole, i.e., 235 g of contains atoms.

∴1000 g of contains

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.

30: 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 to be about 200MeV.

Ans: Amount of electric power to be generated, P= MW