CHAPTER 13

NUCLEI
Composition of Nucleus
o Nucleus consists of protons and neutrons.

o Protons are positively charged particles which are present inside

the nucleus and neutrons are neutral as they don’t have any
charge.
o Atomic number: -

o Atomic number constitutes the total number of protons which

are present in the nucleus of that atom.
o It is denoted by ‘Z’.

o Atomic mass:-

o Atomic mass is the total number of neutrons and protons

which are present inside the nucleus.
o Mass of electrons is not considered while calculating the mass
of the atom and only the mass of neutrons and protons are
considered;
o Since the electrons are the lightest particles their mass is not
considered.
o It is also known as Mass Number.

o It is denoted by ‘A’.

o Nucleons --> Protons + Neutrons

o General representation of the element: -(AZX) where A = mass
number and Z = atomic number.
o For example:- Hydrogen 11H where atomic number=1 and
mass number =1
o Oxygen168O where atomic number=8 and mass number =16(8
protons and 8 neutrons).
Measurement of Atomic mass unit
o Mass of atom is very small as compared to the measurable
masses which we see around us.
o Atomic Mass Unit (a.m.u) is used to measure mass of an atom

o It is denoted by u.

o Atomic mass unit is defined as (1/12th) of the mass of the carbon.

o 1 a.m.u =(1/12) x 1.992647 x10-26 kg
 Where 1.992647 x10-26 = mass of 1 carbon atom.
o 1 a.m.u =1.67 x10-27 kg
o To get the exact measurement of the atomic mass, an
instrument known as Mass spectrometer is used.

Different types of nuclei

1. Isotopes: - Two nuclei with the same atomic number and different
mass number are isotopes of each other.
For example: - There are 3 isotopes of carbon(C) having same
atomic number 6 but their mass numbers are different .
(126C), (136C), (146C).

2. Isobars: -The nuclei which have different atomic number but same
mass number are known as isobars.
For example: - Nitrogen (147N) and Carbon (146C) are both isobars as
their mass numbers are same 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.
For example: -Boron (125B) and Carbon (136C).
Boron: - Atomic number = 5 and mass number =12.
Carbon: - Atomic number = 6and mass number =13.
But the number of neutrons in Boron = (12-5 =7) and number of
neutrons in
carbon = (13-6 = 7) are same.
4. Nuclide: - Nuclides are collection of nuclei with same atomic
number having same number of neutrons.

Charge on a Nucleus
o 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.
o Charge on the nucleus is such that the entire atom is electrically
neutral as a whole.
o Atom constitutes of electrons,protons and neutrons.

o Consider an element ZX
 o Where Z= atomic number, (the number of protons =Z and
the number of electrons =Z).
o 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
o Rutherford performed an experiment which proved that the size
of the nucleus is extremely small.
o In Rutherford scattering experiment a beam of alpha particles
were made to pass through a small thin gold foil.
o Very few alpha particles were deflected.

o 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.
o Very small number of alpha particles got deflected proving That
nucleus is very small in size.
o It was found that the radius of a nucleus(R) of mass number A is
given as :-
o R=R0A1/3 where A = mass number and R0=constant.

o Volume of a nucleus is ∝to the mass number.

o V =(4/3)πR3 , Also R ∝(A)1/3

o => (R)3∝A
o Therefore V ∝ (R) 3∝

Einstein’s Mass-Energy equivalence

o According to Einstein mass is also a form of energy.

o Mass – energy can also be converted into other forms of energy.

o Einstein gave mass-energy equivalence relation as: -

o E=mc2.
o Any object which has got mass ‘m’ has mass energy associated
with it and it is given as mc2.
o This relation helps in understanding nuclear masses and
interaction of nuclei with each other.
 Nuclear binding energy
o Nuclear binding energy is the energy required to hold an atom’s
protons and neutrons together in the nucleus.
o Energy required holding neutrons and protons together therefore
keeps the nucleus intact.
o It can also be defined as the energy needed to separate the
nucleons from each other.

Importance of nuclear binding energy

It 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.
o If nuclear binding energy is high -> high amount of energy is
needed to separate the nucleons this means nucleus is very
stable.
o 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:-
o Mass defect is the difference in the mass of nucleus and its
constituents neutrons and protons).
o It is denoted by ΔM.

o Mathematically :- ΔM = [Z mp+ (A-Z) mn]- M

o Where mp=mass of 1 proton, Z=number of protons,(A-Z)=
mass of neutrons, mN = mass of 1 neutron and M =nuclear
mass of the atom.
o For example: -(168O) à Oxygen atom has 8 protons and 8
neutrons.
o Mass of 8 protons à (8x1.00866) u and Mass of 8 neutrons
à(8x1.00727) u.
o Therefore Oxygen nucleus à(8p+8n) à8(1.00866 + 1.00727) =
16.12744u.
o From spectroscopy ->Atomic mass of (168O) =15.9949u.
 o Mass of 8 electrons =(8x0.00055) u.
o Therefore Nuclear mass of (168O) = (15.9949 – (8x0.00055))
=15.99053u.
o Nuclear mass is less than sum of the masses of its constituents.
o This difference in mass is known as mass defect.
o It is also known as excess mass.
Relation between Mass defect and Nuclear binding energy:-
o Nuclear binding energy is denoted by Eb.
o Eb= ΔMc2
o Where Eb = nuclear binding energy, ΔM=mass defect.

o As there is difference in the mass so there is energy associated

with it. This energy is known as nuclear binding energy.
binding energy per nucleon
It is the average energy required to extract one nucleon from the nucleus. It
is obtained by dividing the binding energy of the nucleus by the number of
nucleons it contains (mass number). The higher value of binding energy /
nucleon indicates comparatively greater stability of the nucleus.
Binding energy curve. It is the graph which shows the relation between the
binding energy per nucleon, Ebn and the mass number A.
Main features of the graph.
• The BE per nucleon Ebn is practically constant that is practically
independent of the atomic number for nuclei of middle mass
number (30 < A < 170)
• Initially there is a rapid increase in the value of binding energy
per nucleon and narrow spikes in the curve which shows extra
stability.
• Between mass numbers 4 and 20, the curve shows cyclic
recurrence of peaks corresponding to 2He4,4Be7,6C12,8O16 and
20
10Ne . This shows that BE/N of these elements is greater than
those of their immediate neighbours.
• After A=20,there is a gradual increase in BE per nucleon. The
maximum value of 8.8Mev is reached at A=56.Therefore iron
nucleus is the most stable nucleus.
• Mass numbers ranging from 40 to 120 are close to the
maximum value so these elements are highly stable and
radioactive.
• Beyond A=120 the value decreases and falls to 7.6Mev for
uranium(A=238).
• Beyond A=128 there is a rapid decrease in the value which
make elements beyond uranium quite unstable and
radioactive.
• Ebn is lower for both lighter nuclei and heavier nuclei.

Note: Lighter nuclei like 1H2,1H3etc have low Ebn. So it combine to

form a heavier nuclei of high Ebn releasing energy. i.e. it undergoes
nuclear fission. Heavier nuclei like 238U have low Ebn. So it split up
into nuclei of high Ebn releasing energy. i.e. it undergoes fission.

• A very heavy nucleus, say A = 240, has lower binding

energy per nucleon compared to that of a nucleus with A
= 120. Thus if a nucleus A = 240 breaks into two A = 120
 nuclei, nucleons get more tightly bound. This implies
energy would be released in the process.

• Consider two very light nuclei (A ≤ 10) joining to form a

heavier nucleus. The binding energy per nucleon of the
fused heavier nuclei is more than the binding energy per
nucleon of the lighter nuclei. This means that the final
system is more tightly bound than the initial system.
Again energy would be released in such a process of
fusion. This is the energy source of the sun.

Nuclear force
o The force with which the nucleons are bound together is known
as nuclear force.
o It is the strong attractive force that binds the nucleons together.

o 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.
o As protons are positively charged they repel each other. This
force of repulsion is given by Coulomb’s force of repulsion.
o This nuclear force is stronger than the coulomb’s force so it
overcomes the force of repulsion.
o This is the reason neutrons and protons are held together inside
the nucleus.
o It is independent of electric charge. Magnitude of nuclear force is
same between proton-proton, proton-neutron or neutron-neutron.
o Nuclear force cannot be given mathematically.
Variation of potential energy with distance of separation
between nucleons.
The PE of a pair of nucleons as a function of their separations is
shown in fig:-.At a particular distance r0 PE is minimum. The force is
attractive when r > r0 and is repulsive when r < r0. The value of r0 is
about 0.8 x 10 - 15m.

o .