## Class 12 Physics Nuclei Revision Notes & Important Question for Exam

• September 27, 2020

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## Class 12th Physics Chapter 13 Nuclei Notes

Introduction

• We will discuss about the nucleus which is the core of the atom.
• Nucleus is located in the centre of the atom and it is positively charged whereas electrons negatively charged and revolve around the nucleus in the orbits.
• We will also look at the composition of the structure of a nucleus and itsdifferent properties.
• How the present day energy crisis can be solved by using the nuclear energy.
• The interaction between different nuclei.

Structure of an atom

Composition of Nucleus

• Nucleus consists of protons and neutrons.
• Protons are positively charged particles which are present inside the nucleus and neutrons are neutral as they don’t have any charge.
• Atomic number: –
• Atomic number constitutes the total number of protons which are present in the nucleus of that atom.
• It is denoted by ‘Z’.
• Atomic mass:-
• Atomic mass is the total number of neutrons and protons which are present inside the nucleus.
• Mass of electrons is not considered while calculating the mass of the atom and only the mass of neutrons and protons are considered;
• Since the electrons are the lightest particles their mass is not considered.
• It is also known as Mass Number.
• It is denoted by ‘A’.
• Nucleons  –> Protons + Neutrons
• General representation of the element: -(AZX) where A = atomicmass and Z = atomic number.
• For example:- Hydrogen 11H  where atomic number=1 and mass number =1
• Oxygen168O where atomic number=8 and mass number =16(8 protons and 8 neutrons).

Measurement of Atomic mass unit

• Mass of atom is very small as compared to the measurable masses which we see around us.
• Atomic Mass Unit (a.m.u) is used to measure mass of an atom
• It is denoted by u.
• Atomic mass unit is defined as (1/12th) of the mass of the carbon.
• 1 a.m.u =(1/12) x 1.992647 x10-26 kg
• Where 1.992647 x10-26 = mass of 1 carbon atom.
• 1 a.m.u =1.67 x10-27
• To get the exact measurement of the atomic mass, an instrument known as Mass spectrometer is used.

Mass Spectrometer

Problem: –The three stable isotopes of neon: 1020Ne,1021Ne and 1022Ne have respective abundances of90.51%, 0.27% and 9.22%. The atomic masses of the three isotopes are

19.99 u, 20.99u and 21.99 u, respectively. Obtain the average atomic mass of neon.

Atomic mass of1020Ne, m1= 19.99 u

Abundance of 1020Ne, η1 = 90.51%

Atomic mass of1021Ne, m2 = 20.99 u

Abundance of 1021Ne, η2 = 0.27%

Atomic mass of1022Ne, m3 = 21.99 u

Abundance of1022Ne, η3 = 9.22%

The average atomic mass of neon is given as:

m= (m1 η1+ m2 η2+ m3 η3)/ (η1+ η2+ η3)

= (19.99×90.51+20.99×0.27+21.99×9.22)/ (90.51+0.27+9.22)

=20.1771u

Nuclei types

1. Isotopes: – Two nuclei with the same atomic number and different mass number are isotopes of each other.
1. For example: – There are 3 isotopes of carbon(C) having same atomic number 6 but their mass numbersare differenti.e. 12,13 and 14.(126C), (136C), (146C).
2. Isobars: -The nuclei which have different atomic number but same mass number are known as isobars.
1. For example: – Nitrogen (147N) and Carbon (146C)areboth isobars as their mass numbers aresame which is 14 but their atomic numbers are 7 and 6 respectively.
3. Isotones: -Isotones are those nuclei which have different atomic number but same number of neutrons.
1. For example: -Boron (125B) and Carbon (136C).
2. Boron: – Atomic number = 5 and mass number =12.
3. Carbon: – Atomic number = 6and mass number =13.
4. But the number of neutronsin Boron = (12-5 =7) and number of neutrons in

carbon = (13-6 = 7) are same.

1. Nuclide: – Nuclides are collection of nuclei with same atomic number having same number of neutrons.

Charge on a Nucleus

• Nucleus is positively charged and consists of protons which are (+ively) charged and neutrons are neutral.As a whole nucleus has to be positively charged.
• Charge on the nucleus is such that the entireatom is electrically neutral as a whole.
• Atom constitutes of electrons,protons and neutrons.
• Consider an element ZX
• Where Z= atomic number, (the number of protons =Z and the number of electrons =Z).
• Total charge on all the electrons e= -(Ze).

Total charge on the nucleus has to be equal and opposite of the charge on electron i.e. it should be =+ (Ze)for the atom to be electrically neutral.

Size of Nucleus

• Rutherfordperformed an experiment which proved that the size of the nucleus is extremely small.
• In Rutherford scattering experiment a beam of alpha particles were made to pass through a small thin gold foil.
• Very few alpha particles were deflected.
• The alpha particles got deflected because of repulsion with the nucleus as alpha particles are positively charged.They get repelled because they are both positively charged.
• Very small number of alpha particles got deflected provingthat nucleus is very small in size.
• It was found that the radius of a nucleus(R) of mass number A is given as :-
• R=R0A1/3 where A = mass number and R0=constant.
• Volume of a nucleus is ∝to the mass number.
• V =(4/3)πR3 , Also R ∝(A)1/3
• => (R)3∝A
• Therefore  (R) 3
• Density of nucleus is independent of mass number.

Problem:- Given the mass of iron nucleus as 55.85u and A=56,find the nuclear density?

Answer:- Given: – mFe = 55.85, u = 9.27 × 10–26 kg

Nuclear density =mass/volume

= (9.27 × 10–26)/((4 π /3)(1.2x 10-15)3)x (1/56)

= 2.29 × 1017 kg m–3

The density of matter in neutron stars (an astrophysical object) iscomparable to this density. This shows that matter in these objectshas been compressed to such an extent that they resemble a bignucleus.

Rutherford’s Gold Foil Experiment

Einstein’s Mass-Energy equivalence

• According to Einstein mass is also a form of energy.
• Mass – energy can also be converted into other forms of energy.
• Einstein gave mass-energy equivalence relation as: – E=mc2.
• Any object which has got mass ‘m’ has mass energy associated with it and it is given as mc2.
• This relation helps in understanding nuclear masses and interaction of nuclei with each other.

Problem:- Calculate the energy equivalent of 1 g of substance.

Answer:- Energy, E = 10–3 × (3 × 108)2 J

E = 10–3 × 9 × 1016 = 9 × 1013 J

Thus, if one gram of matter is converted to energy, there is a release of enormous amount of energy.

Nuclear binding energy

• Nuclear binding energy is the energy required to hold an atom’s protons and neutrons together in the nucleus.
• Energy required holding neutrons and protons together therefore keeps the nucleus intact.
• It can also be defined as the energyneeded to separate the nucleons from each other.
• Importance of nuclear binding energy describes how strongly nucleons are bound to each other. By determining its value we will come to know whether the neutrons and protons are tightly or loosely bound to each other.
• If nuclear binding energy is high -> high amount of energy is needed to separate the nucleons this means nucleus is very stable.
• If nuclear binding energy is low -> low amount of energy is needed to separate the nucleons this means nucleus is not very stable.
• Mass defect:-
• Mass defect is the difference in the mass of nucleus and its constituents(neutrons and protons).
• It is denoted by ΔM.
• Mathematically :- ΔM = [Z mp+ (A-Z) mn]- M
• Where mp=mass of 1 proton, Z=number of protons,(A-Z)= mass of neutrons, mN = mass of 1 neutronand M =nuclearmass of the atom.
• For example: -(168O)àOxygen atom has 8 protons and 8 neutrons.
• Mass of 8 protons à (8×1.00866) u and Mass of 8 neutrons à(8×1.00727) u.
• Therefore Oxygen nucleus à(8p+8n) à8(1.00866 + 1.00727) = 16.12744u.
• From spectroscopy ->Atomic mass of (168O) =15.9949u.
• Mass of 8 electrons =(8×0.00055) u.
• Therefore Nuclear mass of (168O) = (15.9949 – (8×0.00055)) =15.99053u.
• Nuclear mass is less than sum of the masses of its constituents.
• This difference in mass is known as mass defect.
• It is also known as excess mass.
• Relation between Mass defect and Nuclear binding energy:-
• Nuclear binding energy is denoted by Eb.
• Eb= ΔMc2
• Where Eb = nuclear binding energy, ΔM=mass defect.
• As there is difference in the mass so there is energy associated with it. This energy is known as nuclear binding energy.
• Nuclear binding energy is a measure of how well a nucleus is held together.

Problem:- Find the energy equivalent of one atomic mass unit,first in Joules and then in MeV. Using this, express the mass defectof (816O) in MeV/c2.

Answer:-  1u = 1.6605 × 10–27 kg

To convert it into energy units, we multiply it by c2 and find thatenergy equivalent

= 1.6605 × 10–27 × (2.9979 × 108)2 kg m2/s2

= 1.4924 × 10–10 J

= (1.4924 × 10–10)/ (1.602×10-19) eV

= 0.9315 × 109 eV

= 931.5 MeV

Or, 1u = 931.5 MeV/c2

For (816O), ΔM = 0.13691 u = 0.13691×931.5 MeV/c2

= 127.5 MeV/c2

The energy needed to separate (816O) into its constituents is thus127.5 MeV/c2.

Problem:- Find the energy equivalent of one atomic mass unit,first in Joules and then in MeV. Using this, express the mass defectof 168O in MeV/c2.

Answer:- 1u = 1.6605 × 10–27 kg

To convert it into energy units, we multiply it by c2 and find thatenergy equivalent

= 1.6605 × 10–27 × (2.9979 × 108)2 kg m2/s2

= 1.4924 × 10–10 J

= (1.4924 × 10–10 J)/ (1.602×10-19) eV

= 0.9315 × 109 eV

= 931.5 MeV

Or, 1u = 931.5 MeV/c2

For 168O, ΔM = 0.13691 u = 0.13691×931.5 MeV/c2

= 127.5 MeV/c2

The energy needed to separate 168O into its constituents is thus127.5 MeV/c2.

Nuclear binding energy per nucleon

• Nuclear binding energy per nucleon is defined as the average energy per nucleon needed to separate a nucleus into its individual constituents.
• It is denoted by Ebn.
• Experimentally there was a graph plotted between binding energy per nucleon and the mass number(A).
• Following are the observations from the graph:-
1. Initially the graph was increasing.This implies that Ebn is very less for lesser mass number.
2. In the middle range the Ebn becomes constant.This means Ebn is independent of mass number.
3. In the end Ebnstarts decreasing.This shows Ebn is less when mass number is more.

Problem: – Obtain the binding energy (in MeV) of a nitrogen nucleus (147N), given m (147N) =14.00307u?

Answer:- Atomic mass of (147N ) nitrogen, m = 14.00307 u

A nucleus of (147N) nitrogen contains 7 protons and 7 neutrons.

Hence, the mass defect of this nucleus, ∆m = 7mH + 7mn − m

Where,

Mass of a proton, mH = 1.007825 u

Mass of a neutron, mn= 1.008665 u

Therefore, ∆m = (7 × 1.007825 + 7 × 1.008665 − 14.00307)

= 7.054775 + 7.06055 − 14.00307

= 0.11236 u

But 1 u = 931.5 MeV/c2

∆m = 0.11236 × 931.5 MeV/c2

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

Eb = ∆mc2

Where, c = Speed of light

Eb = 0.11236 × 931.5(MeV/c2)/c2

= 104.66334 MeV

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

Deriving Nuclear force from Ebn

1. Lighter nuclei: –
1. In the initial part of the graph A(mass number) is less therefore Ebn is also less.As a result lesser energy is required to separate the nucleons.
2. This showsnuclei areunstable.
3. The nuclei are unstable and in order to become stable lighter nuclei combine with each other to form heavier nuclei.
4. Let the energy of heavier nuclei formed is E’bn and of lighter nuclei be Ebn. This implies E’bn> Ebn.
5. Energy is released when 2 lighter nuclei combine together to form a heavier nuclei.
6. This process is known as Nuclear Fusion.
2. For heavier nuclei:-
1. Mass number is very high and Ebn is very less.
2. In order to become stable the heavier nuclei will split into 2 lighter nuclei.
3. Energy associated with heavier nuclei =Ebn and energy associated with 2 lighter nuclei =E’bn.
4. This implies E’bn> Ebn. Energy is released in this process by the heavier nuclei in order to attain stability.
5. This process is known as Nuclear Fission.
3. Constancy of Ebnin the mid-range of A:-
1. In this portion the mass number is increasing due to whichnumber of nucleons also increase.
2. The force which is present between the nucleons is of short range.The strength of the force decreases as the distance increases.
3. The nucleons are getting affected by their nearest neighbouring nucleons and not by the nucleons which are far away.
4. As a result Ebn remains constant.
5. But when there are too many nucleons Ebn suddenly starts decreasing.

Problem: – Obtain the binding energy of the nuclei (5626Fe) and (20983Bi) in units of MeV from the following data: m (5626Fe) = 55.934939u and m (20983Bi) = 208.980388 u.

Answer:- Atomic mass of (5626Fe), mFe = 55.934939 u

(5626Fe) nucleus has 26 protons and (56 − 26) = 30 neutrons

Hence, the mass defect of the nucleus, ∆m = ((26 × mH) + (30 × mn)– mFe)

Where,

Mass of a proton, mH = 1.007825 u

Mass of a neutron, mn = 1.008665 u

∆m = ((26 × 1.007825 + 30 × 1.008665) − 55.934939)

= (26.20345 + 30.25995 − 55.934939)

= 0.528461 u

But 1 u = 931.5 MeV/c2

Therefore, ∆m = 0.528461 × 931.5 MeV/c2

The binding energy of this nucleus is given as:

Eb1 = ∆mc2

Where, c =Speed of light

Eb1 = 0.528461 × 931.5(MeV/c2)/c2

= 492.26 MeV

Average binding energy per nucleon= (492.26/56)=8.76 MeV

Atomic mass of (20983Bi), m2 = 208.980388 u

(20983Bi) nucleus has 83 protons and (209 − 83) = 126 neutrons.

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

∆m’ = (83 × mH + 126 × mn) − m2

Where,

Mass of a proton, mH = 1.007825 u

Mass of a neutron, mn = 1.008665 u

∆m’ = (83 × 1.007825 + 126 × 1.008665 – 208.980388)

= 83.649475 + 127.091790 − 208.980388

= 1.760877 u

But 1 u = 931.5 MeV/c2

Therefore, ∆m’ = 1.760877 × 931.5 (MeV/c2) x c2

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

Eb2 = ∆m’c2

= 1.760877 × 931.5(MeV/c2) x c2

= 1640.26 MeV

Average binding energy per nucleon = (1640.26/209) =7.848MeV

Nuclear force

• The force with which the nucleons are bound together is known as nuclear force.
• It is the strong attractive force that binds the nucleons together.
• When the nuclear force is compared to other forces of nature like gravitational or coulomb’s force etc.it is the strongest of all the forces.
• As protons are positively charged they repel each other.This force of repulsion is given by Coulomb’s force of repulsion.
• This nuclear force is stronger than the coulomb’s force so it overcomes the force of repulsion.
• This is the reason neutrons and protons are held together inside the nucleus.
• It is independent of electric charge.Magnitude of nuclear force is same between proton-proton,proton-neutron or neutron-neutron.
• Nuclear force cannot be given mathematically.

Problem: – The neutron separation energy is defined as the energy required removing a neutronfrom 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 um (2613A) = 25.986895 u and m(2713A) = 26.981541 u.

For (4120Ca): Separation energy=8.363007 MeV

(2713Al): Separation energy=13.059 MeV

(01n) is removed from a (4120Ca).

For a neutron nucleus. The corresponding nuclear reaction can be written as:

4120Ca –> 4020Ca + 01n

It is given that:

m (4020Ca) Mass = 39.962591 u,

m (4120Ca) Mass = 40.962278 u

m (01n)Mass = 1.008665 u

The mass defect of this reaction is given as:

Therefore, ∆m= m(4020Ca) + (01n) –m(4120Ca)

= (39.962591+1.008665-40.962278)u =0.008978u

But 1u=931.5 (MeV/c2)

Therefore,∆m = 0.008978 x 931.5 (MeV/c2)

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

E=∆mc2

=0.008978 x931.5 =8.363007MeV

For (2713Al), the neutron removal reaction can be written as:

2713Al –> 2613Al + 01n

It is given that:

m (2713Al)Mass = 26.981541 u

m (2613Al) Mass =25.986895u

The mass defect of this reaction is given as:

∆m =m (2613Al) +m (01n) –m (2713Al)

=25.986895 + 1.008665 -26.981541

=0.0414019u

=0.0414019 x 931.5(MeV/c2)

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

E=∆mc2

=(0.0414019 x 931.5) =13.059MeV

• Radioactivity is a nuclear phenomenon in which an unstable nucleus undergoes decay to form stable nuclei.
• Radioactivity was discovered accidentally by a scientist Henry Becquerel (1896).

Henry Becquerel

• Only those nuclei which are unstable will show the phenomenon of radioactivity.
• Fluorescence is a property shown by fluorescent objects; when visible light falls on a fluorescent object then these objects will also start emitting light.
• Henry Becquerel performed experiments on some fluorescent objectsand he gave a hypothesis that fluorescent objects along with the visible light also emit some kind of radiation as well.
• Experimental set-up:-
• He took a photographic plate and placed inside a box.He placed a fluorescent object on the top of the box and allowed the sun light to fall on it.
• As the box is opaque it won’t allow the rays of light to pass through it.
• When the photographic plate will be developed there should be no traces of image on the plate.
• He repeated the experiment by taking different compounds and observed there was no image formed on the photographic film.
• This showed that sunlight is the only ray which is emitted in the above case and as the plate was covered with opaque object so the sunlight is not able to reach the plate.
• He then took the compound of uranium known as uranium potassium sulphate.
• When the photographic plate was developed there was a black spot on the photographic plate.
• This shows that some radiation was emitted that reached the photographic plate.
• In order to know from where the radiation was emitted, he kept a blade on top of a box and uranium potassium sulphate compound was placed above the blade.
• After performing the experiment, when the photographic plate was developed, the photo of blade was seen on the plate.
• This shows that the radiation was emitted by the uranium potassium sulphate compound.
• Accidentally while he was performing the experiment it was not sunny that day.
• The result which he got was same i.e. when the photographic plate was developed there was the same image of the blade on the plate.

Conclusion:-

• The radiation was emitted by the uranium potassium sulphate compound and sunlight did not play any role in emitting the radiations.
• After series of experiments he proved that it was uranium alone which was responsible for emitting the radiations.
• He concluded that there are certain elements which emit radiation on their own.
• This phenomenon was named as radioactivity.

• Experimentallyit was observedthat the number of neutrons varies with the number of protons as we move higher in the periodic table.
• For the elements which have lesser atomic number (e.g.: O, C) the number of neutrons and the number of protons are equal.
• From the graph we can see the straight line shows the number of neutrons is equal to the number of protons.
• But as the atomic number increases gradually the number of neutrons become more than the number of protons.
• In the graph there is a specific pattern like zigzag is shownbythe stable nuclides and some which don’t show this pattern represents the unstable nuclides.
• The nuclides which are unstable emit some radiations to become;this phenomenon is known as radioactivity and the nuclides are known as radioactive nuclides.
• Nuclides to the left of the graph arestablewhich shows they have excess neutrons.
• Nuclides to the right of the graph have excess protons.Such nuclides are known as radioactive nuclides.

• Radioactive decay is a phenomenon in which unstable nucleus decay to form stable nuclei.
• There are 3 types of radioactive decay by which a unstable nuclei becomes stable nuclei:-
• Alpha decay:-
• In alpha decay α – particle (helium nucleus) is emitted.
• Beta decay:-
• In beta decay electrons or positrons are emitted.
• If electronsare emitted then it is known as β decay; and when positronsare emitted then the decay is known as β+
• Gamma decay:-
• In gamma decay high energy photons(γ rays) are emitted.
• Law of radioactive decay states that the number of nuclei undergoing decay per unit time is ∝to total number of nuclei in the sample.
• Consider a sample of radioactive nuclei (Uranium). Thesample of uranium will have several uranium nuclei and they will undergo radioactive decay to convert into relatively stable lighter nuclei.
• That means the number of uranium nuclei will gradually keep on decreasing with time.
• For example:-
• If in a sample there are 10 uranium nuclei and in another 100 uranium nuclei the rate of decay will be slower where there are 10 uranium nuclei as compared to 100 uranium nuclei.

Generalized expression for law of radioactive decay:-

Suppose there are ‘N’ active nuclei at any instant of time (‘t’).

In small interval of time ‘dt’ =’dN’ (number of nuclei decay).

Rate of decay (dN/dt) ∝N (number of nuclei which are present in the sample).

(dN/dt) = -(λ N)

• Whereλ = decay constant and (-ive) sign means the number of nuclei is gradually decreasing with time.
• => (dN) =-λ dt, integrating both the sides,
• =>N0N∫(dN)/N = -λ0t∫ N dt
• Where N0=number of nuclei at t=0 and N= number of nuclei after time t.
• [ln N]N0N = -λ[t]0t
• (ln N0 – ln N) =-λt
• =>ln(N/N0) = -λt
• => (N/N0)=e– λt
• =>N = N0 e-λt always N0> N.
• Radioactive law is also known as exponential decay law as number of nuclei decreases exponentially which can be seen from the graph if plotted between N and t.
• It is followed only by radioactive nuclei.

1. Decay Rate: –
1. Decay rate is defined as the number of nuclei decaying per unit time.
2. Denoted by ‘R’.
3. Mathematically:- R =-(dN/dt)
• Where dN=change in the number of nuclei with time and (-ive) sign shows the number of nuclei decreasing.
• From Exponential decay law N = N0 e-λt, therefore
• R=(d/dt)( N0 e-λt)
• =- N0 [e-λt(-λ)]
• R= N0λ e-λt ;
• At t=0 , R0= N0λ (equation (1))
• Therefore R=R0 e-λt=>decay constant also changes exponentially.
1. Activity of radioactive sample is defined as the total decay rate R of a sample of one or more radionuclides.
2. Decay rate of whole sample is considered.
3. I. unit: – Becquerel(Bq); 1Bq = (1 decay/sec).
4. Other unit: – Curie(Ci); 1C i=3.7 x1010
1. Half – life of radioactive sample is defined as the time at which number of nuclei reduces to one-half of their initial values.
2. It is denoted by t (1/2).
3. Half-life tells how long radioactive nuclei can last.
4. Mathematical expression:-
• From radioactive law , N=N0e-λt (equation(1))
• Initially number of sample = N0.
• Later number of samples will become=(N0/2).
• Therefore from equation(1)- (N0/2) =N0e-λt(1/2)
• =>e-λt(1/2)= (1/2)
• =>- λt(1/2) = ln(1/2) (taking log on both sides)
• =>- λt(1/2) =ln(1) – ln(2)
• =>λt (1/2) = ln(2) (using ln (1) =0).
• =>t(1/2) = (0.693/λ)
1. Mean life is defined as the average life of a nuclei in the radioactive sample.
2. Any nuclei can decay at any interval of time.
3. It is denoted by tav.
4. Mathematically:-
• Let number of radioactive sample at t=0 =N0.
• From the law of radioactive decay (dN/dt) =-λN
• => dN= -λNdt (these many nuclei will decay in time dt).
• Therefore number of nuclei which decay between t and (t+dt) =λNdt.
• But of the nuclei will decay fast and some will slowly decay.
• Sum of lives of all these nuclei = tλNdt.
• Therefore average life of the sample =tav= (0∫(tλN)dt/N0)
• = (λ/N00∫ (tN) dt.
• From exponential law of decay :- N =N0 e-λt
• tav= (λ/N0) ∫tN0 e-λt dt = (λ/N0)∫t e-λt dt
• tav = λ0∫te-λtdt
• =λ[t∫e-λt dt -∫(∫e-λt.1)dt
• =λ[(te-λt)/(-λ)]0 -λ [(e-λt)/(-λ) dt]0
• =λ[0] –λ[(e-λt)/λ2]0
• tav = (1/ λ)
• Using t(1/2) = (0.693/λ)  and tav = (1/ λ)
• tav = (t(1/2)/0.693)

Problem:- ( Class 12 Physics Nuclei Notes )

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?

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:

(N/N0)= 3.125 %=( 3.125)/ (100) = (1/32)

But (N/N0) =e λt

Where, λ = Decay, constant t = Time

Therefore e– (λt) = (1/32)

– (λt) = ln (1) – ln (32)

– (λt) = 0 -3.4657

t= (3.4657)/ (λ)

Since λ = (0.693)/ (T)

Therefore t= (3.466)/ ((0.693)/ (T)) ≈ 5T years.

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:

(N/N0)=1 %=( 1/100)

But (N/N0) =e λt

Therefore e– (λt) = (1/100)

– (λt) = ln (1) – ln (100)

– (λt) = 0 – 4.6052

t= (4.6052)/ (λ)

Since λ = (0.693)/ (T)

Therefore t= (4.6052)/ ((0.693)/ (T)) =6.645T years.

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

Problem:- Obtain the amount (6027Co) of necessary to provide a radioactive source of 8.0 mCistrength. The half-life of (6027Co) is 5.3 years.

(dN/dt) =8.0 mCi

=8×10-3x 3.7×1010

=29.6×107 decay/s.

Where,

N = Required number of atoms

Half-life of (6027Co), T1/2 = 5.3 years

= 5.3 × 365 × 24 × 60 × 60

= 1.67 × 108 s

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

(dN/dt) = (λN)

Where λ = (0.693/T1/2) = (0.693)/ (1.67×108) s-1

Therefore N = (1/λ) (dN/dt)

= (29.6×107)/ (0.693)/ (1.67×108) =7.133 x1016 atoms

For (6027Co):

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

Therefore, Mass of (7.133 x 1016) atoms= (60×7.1333×1016)/ (6.023×1023)

=7.106×10-6g

Hence, the amount of (6027Co) necessary for the purpose is 7.106 × 10−6 g.

Alpha decay

• In alpha decay α particles are emitted.Daughter nucleus is formed from the parent nucleus.
• The atomic number decreases by 2 and Mass number increases by 4.
• It is a very spontaneous process and happens on its own.
• It occurs only in radionuclides.
• For example:-
• 23892U (unstable) –> 90234Th + 24He
• (Uranium (U) is known as parent nucleus and Thorium (Th) is known as daughter nucleus).
• 24He is α particle.
• General form of alpha decay: –AZX (parent) à(A-4) (Z-2) Y + 24He
• Where Y is daughter nucleus.

Q-value of alpha decay

• Q-value is a parameter or a characteristic of a nuclear reaction which describeswhether the reaction can take place or not.
• Q-value is defined as difference in the initial mass energy and final mass energy of decayed products.
• Consider the general equation:-
• AZX (parent) –>(A-4) (Z-2) Y + 24
• Initial rest mass energy Ui =[m (AZX) – Zme]c2
• Where m (AZX) = rest mass and Zme = mass of electrons.
• Final rest mass energy Uf = [m((A-4) (Z-2) Y – (Z-2) me + m(24He) – 2me ] c2
• Q =Ui – Uf
• =[m (AZX) – Zme – m((A-4) (Z-2) Y + (Z-2) me – m(24He) + 2me]c2
• Q = [m (AZX) – m((A-4) (Z-2) Y- m(24He)]c2
• If Q =(+ive) then equation is energetically allowed.
• => Q>0, (Ui-Uf) >0=> Ui>Uf
• This implies energy of the reactants is more than the energy of the products.
• There is some free energy that is in the form of kinetic energy.
• If Q =(-ive) then equation is not energetically allowed.

Problem: –  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) =226.0250 u, m (22086Rn) = 220.01337u, m (22286Rn) =222.01750u, m (21684Po) =216.00189u.

(a) Alpha particle decay of (22688Ra) 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.

(22688Ra) –>  (22288Ra) + (42He)

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

c2 Where, c = Speed of light It is given that:

m (22688Ra) =226.0250 u, m (22286Rn) =222.01750u, m (42He) =4.002603u

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

= 0.005297 u c2

But 1 u = 931.5 (MeV/c2)

Therefore Q = (0.005297 × 931.5) ≈ 4.94 MeV

Kinetic energy of the α-particle =

(Mass number after decay)/ (Mass number before decay) x Q

= (222)/ (226) x 4.94 = 4.85MeV.

(b) Alpha particle decay of (22086Rn)

(22086Rn) –> (21684Po) + (42He)

It is given that:

Mass of (22086Rn) = 220.01137 u

Mass of (21684Po) = 216.00189 u

Therefore, Q-value = [220.01137 – (216.00189+4.002603)] x931.5

≈ 641 MeV

Kinetic energy of the α-particle = (220-4)/ (220) x 6.41

= 6.29 MeV

Problem: –We are given the following atomic masses:23892U = 238.05079 u

42He = 4.00260 u, 90234Th = 234.04363 u,11H= 1.00783 u, 23791Pa = 237.05121 u

Here the symbol Pa is for the element protactinium (Z = 91).

(a) Calculate the energy released during the alpha decay of 23892U .

(b) Show that 23892U cannot spontaneously emit a proton.

(a) The alpha decay of 23892U is given by equation

AZ X –> (A-4) (Z-2Y + 42He

The energy releasedin this process is given by:

Q = (MU – MTh – MHe) c2

Substituting the atomic masses as given in the data, we find

Q = (238.05079 – 234.04363 – 4.00260) u × c2

= (0.00456 u) c2

= (0.00456 u) (931.5 MeV/u)

= 4.25 MeV.

(b) If 23892U spontaneously emits a proton, the decay process would be

23892U –> 23791Pa + 11H

The Q for this process to happen is

= (MU – MPa – MH) c2

= (238.05079 – 237.05121 – 1.00783) u × c2

= (– 0.00825 u) c2

= – (0.00825 u) (931.5 MeV/u)

= – 7.68 MeV

Thus, the Q of the process is negative and therefore it cannot proceedspontaneously. We will have to supply energy of 7.68 MeV to a23892U nucleus to make it emit a proton.

Beta decay

• In case of beta decay either electron or positron is emitted.
• Mass number remains the same.
• βdecay –> Electron is emitted.
• Atomic number increases by 1.
• For example:- 3215P à1632S + e + ̅ν  where  ̅ν =anti-neutrino
• Q-value for βdecay:-
• AZA (Z+1) Y + e+ ̅ν
• Initial rest mass energy Ui = [m (AZX) – Zme]c2
• Where m (AZX) = rest mass and Zme = mass of electrons
• Final rest mass energy Uf = [m(A (Z+1) Y) – (Z+1) me + me ] c2
• =[m(A (Z+1) Y) -Zme] c2
• Therefore Q =Ui– Uf
• =[m(AZX) -Zme – m(A (Z+1) Y) + Zme] c2
• Q = [m(AZX) – m(A (Z+1) Y)]

Example:-

• β+decay à Positron(same as electron but with (+ ive) charge) is emitted.
• Atomic number decreases by 1.
• For example:- 2311Na –>1022Ne + e++ν where ν = neutrino
• Q-value for beta (+ive) decay:-
• AZX –>A (Z-1) Y + e+ + ν
• Initial rest mass energy Ui = [m (AZX) – Zme]c2
• Where m (AZX) = rest mass and Zme = mass of electrons
• Final rest mass energy Uf = [m(A (Z-1) Y) – (Z-1) me + me ] c2
• Therefore Q =Ui – Uf
• = [m (AZX) – Zme – m(A (Z-1) Y) – (Z-1)me – me] c2
• Q = [m(AZX) – m(A (Z-1) Y) -2 me] c2
• Example to show whether nuclear reaction can take place or not:-
• 11p(proton) à01n(neutron) + e+ + ν. The mass of neutron is greater than the mass of proton.
• The Q-value = (-ive) as a result the above reaction is not possible.This means conversion of stable proton to neutron is not allowed.
• Consider 01n (neutron) à11p (proton) + e + ̅ν
• The Q-value =(+ive) as a result above reaction is possible.This means conversion of neutron to proton is allowed.

Beta-decay: neutrino and anti-neutrino

• Neutrino (ν) and anti-neutrino ( ̅ν) are neutral particles.
• They also have negligible mass.
• They have extremely high penetration power.
• They have extremely weak interaction with matter.

Problem:- 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 (2311Na) = 22.989770 u.

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

β Emission of the nucleus (2310Ne)

(2310Ne) –> (2311Na) + e +   ̅ν + Q

It is given that:

Atomic mass m (2310Ne) of = 22.994466 u

Atomic mass m (2311Na) of = 22.989770 u

Mass of an electron, me = 0.000548 u

Q-value of the given reaction is given as:

Q= [m (2310Ne) + [m (2311Na) +me]] c2

There are 10 electrons in and 11 electrons in (2311Na). Hence, the mass of the electron is cancelled in the Q-value equation.

Therefore Q = [22.994466 – 22.989770] c2

= (0.004696 c2) u.

But 1u = 9.31 (MeV/c2)

Therefore Q= (0.004696 x 931.5) =4.374MeV.

The daughter nucleus is too heavy as compared to e– and   ̅ν. 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.

Problem:- For the β+(positron) emission from a nucleus, there is another competing process knownas electron capture (electron from an inner orbit, say, the K−shell, is captured by thenucleus and a neutrino is emitted).

e+ + AZX –> A(Z-1)Y + ν

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

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

e+ + AZX –> A(Z-1)Y + ν +Q1   (equation(1))

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

AZX –> A (Z-1) Y + e+ + ν +Q2   (equation (2))

mN (AZX) =Nuclear mass of ( AZX)

mN(A (Z-1) Y) = Nuclear mass of (A (Z-1) Y)

m(AZX) = Atomic mass of ( AZX)

m(A (Z-1) Y) = Atomic mass of (A (Z-1) Y)

me = Mass of an electron

c = Speed of light

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

Q1= [mN(AZX) + me -mN(A (Z-1) Y)] c2

= [m (AZX) -Zme+ me –m (A (Z-1) Y) + (Z-1) me] c2

= [m (AZX) -mA (Z-1) Y] c2   (equation (3))

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

Q2=mN(AZX) -mN(A (Z-1) Y) – me] c2

= [m (AZX) -Zme– m (A (Z-1) Y) + (Z-1) me – me] c2

= [m (AZX) -mA (Z-1) Y – 2me] c2   (equation (4))

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

In other words, this means that if β+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.

Gamma decay

• In Gamma decay γ rays are emitted.γ rays are electromagnetic waves with short wavelength.
• Most of the daughter nuclei of alpha decay and beta decay are in excited state.
• As a result they are unstable.When the daughter nuclides try to transit from excited state to ground state they emit radiations.
• These radiations are known as γThis known as gamma decay.

Comparison of Alpha,Beta & Gamma Rays

Nuclear Energy

• Nuclear energy is the energy that holds together the nuclei of atoms.
• Nuclear energy is obtained from nucleusby either:-
• Breaking of heavy nucleus into 2 relatively lighter nuclei known as nuclear fissionor by
• Combining 2 lighter nuclei to form a heavy nucleus known as nuclear fusion.
• Nuclear energy is becominga possible solutionfor the energy crisis in the world.
• Electric energy can be harnessed from nuclear energy.

Nuclear reactor

Types of Nuclear reactions

Problem:- 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 thecoin is entirely made of (2963Cu) atoms (of mass 62.92960 u).

Answer:– Mass of a copper coin, m’ = 3 g

Atomic mass of (2963Cu) atom, m = 62.92960 u

The total number of (2963Cu) atoms in the coin, N = (NA x m’)/ (Mass number)

Where,

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

Mass number = 63 g

Therefore N = (6.023 × 1023 x3)/ (63) =2.868×1022 atoms.

(2963Cu) nucleus has 29 protons and (63-29) = 34 neutrons.

Therefore, Mass defect of this nucleus, ∆m’ = 29 × mH + 34 × mn − m

Where,

Mass of a proton, mH = 1.007825 u

Mass of a neutron, mn = 1.008665 u

Therefore, ∆m’ = 29 × 1.007825 + 34 × 1.008665 − 62.9296

= 0.591935 u

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

=1.69766958×1022u.

But 1 u = 931.5 (MeV/c2)

Therefore, ∆m = 1.69766958 × 1022 × 931.5 (MeV/c2)

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

Eb= ∆mc2

= 1.69766958 × 1022 × 931.5(MeV/c2)/c2

= 1.581 × 1025 MeV

But 1 MeV = 1.6 × 10−13 J

Eb = 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.

Nuclear Fission

• Scientist named Fermi was the one who performedfirst nuclear fission reaction.
• He considered the sample of 92235U and bombarded it with a neutron 10n i.e.
• 10n + 92235U(Target nuclei)à92236U(unstable so broke down into) –>56144Ba + 8936kr +310n
• Where (56144Ba) = Barium,(8936kr)=Krypton and (10n) =neutron.
• This reaction is known as nuclear fission reaction as heavier nuclei is broken down into lighter nuclei,thereby releasing large amounts of energy.

Types of Nuclear Fission reactions:

1. Uncontrolled chain reaction:-
1. In uncontrolled chain reaction, the reaction is uncontrolled and rapid.The reaction keeps increasing and becomes huge.
2. Energy of the order of mega electron volts is produced.
3. This reaction is known as chain reaction as the product formed in first reaction initiates the second reaction and so on.
4. One of the important applications of this reaction is in making atom bombs,hydrogen bombs and nuclear bombs.
5. (Number of neutrons hitting next target)/ (number of neutrons emitted)>=1.
6. Reaction that takes place :-
1. 10n + 92235U –>92236U(unstable so broke down into) –>56144Ba + 8936kr +3 10n
1. Controlled chain reaction:-
1. In controlled chain reaction, the reaction is controlled and steady.
2. Reaction that takes place :-
1. 10n + 92235U –> 92236U(unstable so broke down into) –>56144Ba + 8936kr +3 10n
2. Using some methods 2 neutrons are removed and only 1 neutron is allowed to hit the next target.
3. Energy released is less as compared to the energy released in the nuclear fission reaction.
4. One of the most important applications is in Nuclear Reactor where electricity can be produced.
5. (Number of neutrons hitting next target)/ (number of neutrons emitted)<1.

Nuclear Reactor

• Nuclear reactors are useful in producing electricity.
• A nuclear reactor is an arrangement to generate electricity whichmakes use of nuclear fission.

Requirements for controlled nuclear fission in reactor:

1. Neutrons to be slowed down.
• Neutrons are slowed down by using moderators which are lighter nuclei which slow down fast moving neutrons by elastic collision.
• Commonly used moderators:-

1.Water

1. Heavy Water
2. Graphite
• Consequence of use of moderators:-
1. Multiplication factor of neutrons increases: – When a neutron hits a target nucleus along with daughter nucleus,it produces 3 neutrons.These 3 neutrons are highly energetic but they need to be slowed down, so they can hit the target nucleus.
2. As a result high multiplication factor results in uncontrolled chain reaction.
1. Excess neutrons to be absorbed.
• As uncontrolled chain reaction is wanted therefore to absorb excess neutrons Control Rods are used.
• These control rods are inserted in the core of the nuclear reactor.
• Control rods are capable of initiating(while taking out of the reactor) and stopping(inserting in the nuclear reactor) the nuclear reaction.
• As they absorb all the excess neutronsthere are no neutrons left to start the reaction.
• Control rods are made up of neutron absorbing materials.
• They decrease the multiplication factor of neutrons to a very small value.
• Commonly used material is Cadmium.

Control Rods

Construction of Nuclear Reactor

• The core of the nuclear reactor consists of uranium (235U) in the form of cylindrical rods.These rods are dipped inside a liquid which is the moderator.
• Whenever one neutron strikes this uranium rod nuclear fission reaction starts and 3 fast moving neutrons are produced.
• Because of the moderator these 3 neutrons undergo elastic collision as a result they slow down before they strike the second rod.
• Geometry of the core is such that only one out of 3 neutrons which are emitted strike the next rod making the reaction a controlled one.
• When the control rods are inserted inside they will absorb all the extra neutrons.Since there are no neutrons nuclear fission reaction will stop.
• Large amount of energy is also released in the core.
• In order to extract the energy from the core water at very high pressure is passed through it.
• As hot water passes through it produces steam in the steam generators.
• This steam is used to run the turbines which in turn produce electricity.
• This process will keep on continuing till the uranium on the rods does not get over.Then the rods have to be replaced in the nuclear reactor.

1. Energy released is extremely large.
2. Needs fuel in extremely small quantity.

1. Spent fuel is highly radioactive and extremely hazardous to all life forms.

Problem:- A 1000 MW fission reactor consumes half of its fuel in 5.00 y. How much (23592U) did it containinitially? Assume that the reactor operates 80% of the time that all the energy generatedarises from the fission of (23592U) and that this nuclide is consumed only by the fissionprocess.

Answer:- Half-life of the fuel of the fission reactor, t(1/2)= 5years

= 5 × 365 × 24 × 60 × 60 s

We know that in the fission of 1 g of (23592U) nucleus, the energy released is equal to 200MeV.

1 mole, i.e., 235 g of (23592U) contains (6.023 × 1023) atoms.

1 g (23592U) contains = (6.023×1023)/ (235) atoms.

The total energy generated per gram of (23592U) is calculated as:

E = ((6.023×1023)/ (235)) x200MeV/g

= (200×6.023×1023x1.6×10-19x106)/ (235) =8.20×1010 J/g

The reactor operates only 80% of the time.

Hence, the amount of (23592U) consumed in 5 years by the 1000 MW fission reactor iscalculated as:

= (5x80x60x60x365x24x1000x106) g / (100×8.20×1010)

=1538kg

Therefore, Initial amount of (23592U) = 2 × 1538 = 3076 kg

Problem:- 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, isreleased if all the atoms in 1 kg of pure (23994Pu) undergo fission?

Answer:- Average energy released per fission of (23994Pu), Eav=180MeV

Amount of pure (23994Pu), m = 1 kg = 1000 g

NA= Avogadro number = 6.023 × 1023

Mass number of (23994Pu) = 239 g

1 mole of (23994Pu) contains NA atoms.

Therefore mg of (23994Pu) contains ((NA)/ (Mass number) x m) atoms

= ((6.023 × 1023)/ (239) x1000) =2.52 x1024 atoms.

Therefore Total energy released during the fission of 1 kg of (23994Pu) is calculated as:

E=Eav x 2.52×1024

=180×2.52×1024 = 4.536×1026MeV

Hence, 4.536×1026MeV is released if all the atoms in 1 kg of pure (23994Pu) undergo fission.

Nuclear Fusion

• In a nuclear fusion reaction two lighter nuclei combine to form a relatively heavier nucleus.
• In this process huge amount of energy is also released.
• Temperature at which protons would have enough energy to overcome the coulomb’s barrier is very high.

Thermonuclear fusion

• Increasing the temperature of the material until the particles have enough energy due to their thermal motions alone –to overcome the coulomb barrier.
• For thermonuclear fusion,extreme conditions of temperature and pressure are required.
• Example of Thermonuclear fusion is generation of energy in stars.
• For example: From the sun we get large amount of energy and this energy generated due to the thermonuclear fusion reaction taking place in the sun.

Energy generation in sun

• In sun the energy generation is a multi- step process.There are total of 4 steps involved in the energy generation inside sun.
• Step1:- 11H(proton) + 11H(proton) –>21H(deuteron)+ e+(positron) + ν(neutrino)+0.42MeV
• Step2:- e+(positron) + e(electron) –>γ(Gamma rays) + 1.02MeV
• Step3:- 21H(deuteron)+ 11H(proton) –>32He(helium) + γ(Gamma rays)+5.49MeV
• Step4:- 32H + 32H –>42He + 11H+11H+12.86MeV
• Step 1,2,and 3 occur twice in the sun and the step 4 occurs only once.
• When all the above 4 reactions are combined together then four hydrogen atoms combine to form a42He atom with a release of 26.7MeV of energy.
• Final reaction is:- 411H +2e– –> 42He +6 γ+2 ν+26.7MeV
• It is also known as proton-proton cycle because this process starts with protons.

Problem:- 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+3.27MeV

Answer:- The given fusion reaction is:

21H + 21H –>32He+n+3.27MeV

Amount of deuterium, m = 2 kg

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

Therefore, 2.0 kg of deuterium contains= ((6.023×1023)/ (2)) x (2000) =6.023×1026atoms

It can be inferred from the given reaction that when two atoms of deuterium fuse, 3.27

MeV energy is released.

Therefore, total energy per nucleus released in the fusion reaction:

E= (3.27/2) x 6.023×1026 MeV

= (3.27/2) x 6.023×1026x 1.6×10-19x 106

=1.576×1014 J

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:

= (1.576×1014)/ (100x60x60x24x365)

= (4.9 x104) years.

Problem:-  From the relation R = R0A1/3, where R0 is a constant and A is the mass number of anucleus, show that the nuclear matter density is nearly constant (i.e. independent of A).

R = R0A1/3

Where,

R0 = Constant.

A = Mass number of the nucleus

Nuclear matter density,ρ = (Mass of the nucleus)/ (volume of nucleus)

Let m be the average mass of the nucleus.

Hence, mass of the nucleus = mA

Therefore ρ=(mA)/((4/3)πR3)=(3mA)/(4π)(R0A(1/3))3=(3m)/(4πR03)

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

Chapter-13-Nuclei

## Nucleus : Previous Year’s Questions

Q.1 Two nuclei have mass numbers in the ratio 1 : 2. What is the ratio of their nuclear densities? [2009 D]
Sol. Nuclear density is independent of mass number.

Q.2 Two nuclei have mass numbers in the ratio 1 : 8. What is the ratio of their nuclear radii? [2009]
where R0= constant, A = mass number
Therefore R1R2=(A1A2)1/3
=(18)1/3=12
R1:R2=1:2

Q.3 Assuming the nuclei to be spherical in shape, how does the surface area of a nucleus of mass number A1 compare with that of a nucleus of mass number A2? [2008 C]
where R0= constant
Therefore Surface Area S=4πR2
=4π(R0A1/3)2
=4πR20A2/3
Therefore Ratio of surface areas, S1S2=(A1A2)2/3

Q.4 How is the size of a nucleus is experimentally determined? Write the relation between the radius and mass number of the nucleus. Show that the  density of nucleus is independent of its mass number. [2011C D,2011C]
Sol. The size of the nucleus is experimentally determined using Rutherford’s α – scattering experiment and the distance of closest  approach and impact parameter.
The relation between radius and mass number of nucleus is
R=R0A1/3
where R0=1.2fm
where, A = mass number R = radius of nucleus
Nuclear density
ρ=MassofnucleusVolumeofnucleus
=mA43π(R0A1/3)3
where, m = mass of each nucleon
ρ=mA43πR30A
ρ=m43πR30
From the above formula, it is clear that r does not depend on mass number.

Q.5 Write any two characteristic properties of nuclear force. [2010]
Sol. Two characteristics of nuclear force
(i) These are short range force.
(ii) These are strong force of attractive nature.

Q.6 Draw a plot representing the law of radioactive decay. Define the activity of a sample of a radioactive nucleus. Write its SI unit. [2006 C,2007,2008 F,2010 D]
Sol. The curve representing the law of radioactive decay is shown below.

The rate or activity of a sample is defined as the rate of disintegration taking place in the sample of radioactive substance.
SI unit of activity is Becquerel (Bq).
1 Bq = 1 disintegration/s.

Q.7 State the law of radioactive decay.
Plot a graph showing the number (N) of undecayed nuclei as a function of time (t) for a given radioactive sample having half life T2√.
Depict in the plot, the number of undecayed nuclei at (a) t=3T2√ and (b) t=5T2√. [2011 D]
Sol. Law of radioactive decay : – The rate of disintegration of radioactive sample at any instant is directly proportional to the number of undisintegrated nuclei present  in the sample at that instant i.e.,
dNdt∝N
dNdt=−λN
where , N = number of undisintegrated nuclei present in the sample at any instant t and dNdt is rate of disintegration.
The required plot is shown below.

Q.8 How is the mean life of a radioactive sample related to its half life? [2011 F]
Sol. Mean life tm=1λ, where λ is decay constant.
But half life T12=In2λ
Therefore λ=In2T12
Therefore tm=T12log2=T120.693

Q.9 A nucleus undergoes β – decay. How does its
(a) Mass number                                         (b) Atomic number change?
[2011 C D]
Sol. During b- decay
(a) No change in mass number.                  (b) Atomic number increases by 1.

Q.10 A nucleus 23892U undergoes alpha- decay and transforms to thorium. What is
(a) The mass number and                                              (b) Atomic number of the nucleus produced?                         [2011 C D]
Sol. In α- decay, the mass number of parent nucleus decreases by 4 units and atomic number decreases by 2 units.
Therefore 23892U→42He+23490Th
(a) Mass number of produced nucleus = 234                             (b) Atomic number of produced nucleus = 90

Q.11 What is the basic mechanism for the emission of β− and β+ particles in nuclide? Give an example by writing explicity a decay process for b– emission.
Is
(a) The energy of the emitted β – particles continuous or discrete?
(b)The daughter nucleus obtained through β – decay, an isotope or an isobar of the parent nucleus? [2010 C D]
Sol.During β− -decay from the nucleus, nuclei undergoes a change in such a way that atomic number increases by one and mass number remain same.
In β+- decay, the mass number of present radioactive nuclei remains same whereas atomic number decrease by one.
Example of β− – decay
3215P→3216S+e−+ν¯¯¯
(Half life = 26 days)
In β- decay, an electron and an antineutrino are created in following manner :
n→p+e−+ν¯¯¯
(a) The energy of emitted b-particles is continuous.
(b) As there is no change in mass number during β-decay. So, the daughter nucleus is isobar of the parent nucleus.

Q.12 A radionuclide sample has N0 nuclei at τ = 0. Its number of undecayed nuclei get reduced N0e at t = τ . What does the term t stand for?
Write, in terms of t, the time interval T in which half of the original number of nuclei, of this radionuclide would have got decayed. [2008 C D]
Sol.(i) The term t stands for mean life.
(ii) The required relation is τ = 1.44 T i.e., mean life of radioactive sample.
= 1.44 × half life

Q.13
(a) What is meant by half life of a radioactive element?
(b) The half life of a radioactive substance is 30s. Calculate
(i) The decay constant and
(ii) Time taken for the sample to decay by 3/4th of the initial value. [2006,2009 F]
Sol.(a) Half life : – Half life of a radioactive element is the time taken by the sample to disintegrate to half of its original amount.
Half life period T12=0.693λ
where, λ is decay constant.
(b) T12=30s
(i) λ=0.693T12=0.69330
⇒0.0231s−1

(ii) Since N=N0(12)n
where, n = number of half lives
N = number of undisintegrated nuclei present in the sample.
N0 = original number of undisintegrated atom.
Here,
N=N0−34 N0
N0=14N0
N=N0(12)n
N04=N0(12)n
⇒ (12)n=(12)n
⇒ n = 2
Totaltime=n×T12
Total time taken = 60s = 1 min.

Q.14 The radioactive isotope D decays according to the sequence
D⟶β−particleD1⟶α−particleD2
If the mass number and atomic number of D2 are 176 and 71 respectively, what is
(a) The mass number
(b) Atomic number of D?                                                                                  [2007 D,2009 D,2010 D]
Sol. In β – decay, the mass number remains same and atomic number increases by 1 unit. In a- decay, the mass number decreases by 4 units and atomic number decreases by 2 units.
Let mass and atomic number of D be A and Z respectively.
Therefore AZD⟶β−particleAZ+1D1⟶α−particleA−4DZ+1−2D2
or A−4Z−1D
AZD⟶AZHD1+0−1B⟶AZHD1⟶A−4Z−1D2+42He
According to question, the mass number and atomic number of D2 are 176 and 71 respectively.
Therefore (a) Mass number of D = Z –1 = 71 ⇒ Z = 72
(b) Atomic number of D = A – 4 = 176 ⇒ 176 + 4 = 180

Q.15 In a given sample, two radioisotopes, A and B, are initially present in the ratio of 1 : 4. The half lives of A and B are respectively 100 yr and 50 yr. Find the time after  which the amounts of A and B become equal. [2012 F]
Sol. Let NA be the concentation of A, then 4 NA is the concentration of B.

Q.16 An observer, in a laboratory, starts with N0 nuclei of a radioactive sample and keeps on observing the number (N) of left over nuclei at regular intervals of 10 min each. She prepares the following table on the basis of her observation.

Use this data to plot a graph of loge(N0/N) vs time(t) and calculate the
(a) decay constant and
(b) half life of the given sample. [2009C D]
Sol. The graph between loge(N0N) and time is shown below.

(a) Since loge(N0N)=λt
⇒ Slope of loge(N0N) vs time t graph gives decay constant (λ)
Therefore λ=3.46510=6.93020=0.3465s−1
(b) Since Half life T12=0.693λ
T12=0.6930.3465=2s
Half life T12=2s

Q.17 (a) Derive the law of radioactive decay viz.
N=N0e−λt
(b) Explain, giving necessary reactions, how energy is released during.
(i) Fission and                                                                  (ii) Fusion.   [2011 C]
Sol. Law of radioactive decay: –The rate of disintegration of radioactive sample at any instant is directly proportional to  the number of undisintegrated nuclei present in the sample at that instant i.e.,
dNdt∝N
dNdt=−λN
where,N = number of undisintegrated nuclei present in the sample at any instant t and dNdt is rate of disintegration.
The required plot is shown

Since dNdt=−λN
where λ = decay constant
N = number of undisintegrated nuclei in the sample of radioactive substance.
or dNN=−λdt
Integrating both sides, we get
∫dNN=−λ∫dt
In N=−λt+C
At t = 0, N = N0= number of undisintegrated
Therefore C = log N.
Therefore log N = −λt+logN0
or logNN0=−λt
Therefore N=N0e−λt

(b) (i) Nuclear Fission : – The phenomenon of splitting of heavy nuclei (mass number > 120) into smaller nuclei of nearly equal masses is known as nuclear fission.
In nuclear fission, the sum of the masses of the product is less than the sum of masses of the reactants. This difference of mass gets converted into energy as per E=mc2  and hence ample amount of energy is released in nuclear fission e.g.,
23592U+10n→14156Ba+9236Kr+310n+Q
Thus, energy is liberated in nuclear fission.

(ii) Nuclear Fusion : – The phenomenon of conversion of two lighter nuclei into a single heavy nucleus is called nuclear fusion.
Since, the mass of the heavier product nucleus is less than the sum of masses of reactant nuclear and therefore certain mass defect occurs which converts into energy as per Einstein’s mass energy relation. Thus, energy is released during nuclear fusion e.g.,
1H1+1H1⟶1H2+e++n+0.42MeV

Q.18 Why is it necessary to slow down the neutrons, produced through the fission of 23592U nuclei (by neutrons) to sustain a chain reaction? What type of nuclei are (preferably) needed for slowing down fast neutrons?
[2008 C]

Sol. Average kinetic energy of neutrons produced in nuclear fission of 92U235 is nearly 2 MeV and may escapse the reactor. So, there is need to  slow down the fast neutrons using appropriate substance namely moderator, into slow thermal neutrons.
Nuclei which have comparable mass of that of neutrons should be prefarable be used to slow down fast neutrons. It is due to the fact that the elastic collision between fast neutrons and slow moving protons in paraffin lead to  interchange the velocities.

Q.19 State the law of radioactive decay. Establish a mathematical relation between half life period and disintegration constant of a radioactive nucleus. [2006,2007 F]
Sol. Law of radioactive decay : – The rate of disintegration of radioactive sample at any instant is directly proportional to the number of undisintegrated nuclei present in  the sample at a instant.
i.e., dNdt∝N
dNdt=−λN , where 1 = decay constant.
N=N0e−λt … (1)
where, N = number of undisintegrated radioactive nuclei present in the sample.
N0 = original number of undisintegrated nuclei present in the sample.
For half life period
t=T12,N=N02
N02=N0e−λ/T12
where T12= half life period
⇒ eλT12=2
Taking in of both sides, we get λT12√ in e = In 2
λT12=In2
λT12=In2λ=0.693λ
T12=0.693λ

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