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Page 1
1(f). Binding Energy
When nuclear masses are measured, it is found that they are less than the sum of the
masses of the neutrons and protons of which they are composed. This is in agreement
with Einstein’s theory of relativity, according to which the mass of a system bound by
energy B is less than the mass of its constituents by
2
/ B c (where c is the velocity of
light).
The Binding energy B of a nucleus is defined as the difference between the energy of the
constituent particles and of the whole nucleus. For a nucleus of atom
A
Z
X ,
2 2
– [ – ( )]
A A
p N Z H N Z
B ZM NM M c ZM NM M X c ? ? ? ? ? ?
? ?
If mass is expressed in atomic mass unit
– 931.5 [ – ( )] 931.5
A A
p N Z H N Z
B ZM NM M MeV ZM NM M X MeV ? ? ? ? ? ? ? ?
? ?
:
p
M Mass of free proton, :
N
M M
N
: Mass of free neutron,
:
H
M mass of hydrogen atom
A
Z
M : mass of the nucleus,
: Z Number of proton, : N Number of neutron,
? ?
A
Z
M X : mass of atom.
9.1.5.1 Binding Energy per Nucleon
The binding energy per nucleon for a given nucleus is found by dividing its total
binding energy by the number of nucleon it contains. Thus binding energy per nucleon is
2 2
– [ – ( )]
A A
p N Z H N Z
B c c
ZM NM M ZM NM M X
A A A
? ? ? ? ? ?
? ?
The binding energy per nucleon for
2
1
H is
2.224
1.112 MeV / nucleon
2
? and for
209
63
Bi it
is
1640 MeV
7.8 MeV / nucleon
209
? .
Page 2
1(f). Binding Energy
When nuclear masses are measured, it is found that they are less than the sum of the
masses of the neutrons and protons of which they are composed. This is in agreement
with Einstein’s theory of relativity, according to which the mass of a system bound by
energy B is less than the mass of its constituents by
2
/ B c (where c is the velocity of
light).
The Binding energy B of a nucleus is defined as the difference between the energy of the
constituent particles and of the whole nucleus. For a nucleus of atom
A
Z
X ,
2 2
– [ – ( )]
A A
p N Z H N Z
B ZM NM M c ZM NM M X c ? ? ? ? ? ?
? ?
If mass is expressed in atomic mass unit
– 931.5 [ – ( )] 931.5
A A
p N Z H N Z
B ZM NM M MeV ZM NM M X MeV ? ? ? ? ? ? ? ?
? ?
:
p
M Mass of free proton, :
N
M M
N
: Mass of free neutron,
:
H
M mass of hydrogen atom
A
Z
M : mass of the nucleus,
: Z Number of proton, : N Number of neutron,
? ?
A
Z
M X : mass of atom.
9.1.5.1 Binding Energy per Nucleon
The binding energy per nucleon for a given nucleus is found by dividing its total
binding energy by the number of nucleon it contains. Thus binding energy per nucleon is
2 2
– [ – ( )]
A A
p N Z H N Z
B c c
ZM NM M ZM NM M X
A A A
? ? ? ? ? ?
? ?
The binding energy per nucleon for
2
1
H is
2.224
1.112 MeV / nucleon
2
? and for
209
63
Bi it
is
1640 MeV
7.8 MeV / nucleon
209
? .
Figure below shows the binding energy per nucleon against the number of nucleons in
various atomic nuclei.
The greater the binding energy per nucleon, the more stable the nucleus is. The graph has
the maximum of 8.8 MeV / nucleon when the number of nucleons is 56. The nucleus that
has 56 protons and neutrons is
56
26
Fe an iron isotope. This is the most stable nucleus of
them all, since the most energy is needed to pull a nucleon away from it.
Two remarkable conclusions can be drawn from the above graph.
(i) If we can somehow split a heavy nucleus into two medium sized ones, each of the new
nuclei will have more binding energy per nucleon than the original nucleus did. The extra
energy will be given off, and it can be a lot. For instance, if the uranium nucleus
235
92
U is
broken into two smaller nuclei, the binding energy difference per nucleon is about
0.8 MeV. The total energy given off is therefore
? ?
MeV
0.8 235 nucleon 188 MeV
nucleon
? ?
?
? ?
? ?
This process is called as nuclear fission.
Mass number, A
Fe
56
26
Binding energy per nucleon, MeV
? ? ? E
Fusion
? ? ? E
Fission
Figure: Binding energy per nucleon as function of mass number.
Page 3
1(f). Binding Energy
When nuclear masses are measured, it is found that they are less than the sum of the
masses of the neutrons and protons of which they are composed. This is in agreement
with Einstein’s theory of relativity, according to which the mass of a system bound by
energy B is less than the mass of its constituents by
2
/ B c (where c is the velocity of
light).
The Binding energy B of a nucleus is defined as the difference between the energy of the
constituent particles and of the whole nucleus. For a nucleus of atom
A
Z
X ,
2 2
– [ – ( )]
A A
p N Z H N Z
B ZM NM M c ZM NM M X c ? ? ? ? ? ?
? ?
If mass is expressed in atomic mass unit
– 931.5 [ – ( )] 931.5
A A
p N Z H N Z
B ZM NM M MeV ZM NM M X MeV ? ? ? ? ? ? ? ?
? ?
:
p
M Mass of free proton, :
N
M M
N
: Mass of free neutron,
:
H
M mass of hydrogen atom
A
Z
M : mass of the nucleus,
: Z Number of proton, : N Number of neutron,
? ?
A
Z
M X : mass of atom.
9.1.5.1 Binding Energy per Nucleon
The binding energy per nucleon for a given nucleus is found by dividing its total
binding energy by the number of nucleon it contains. Thus binding energy per nucleon is
2 2
– [ – ( )]
A A
p N Z H N Z
B c c
ZM NM M ZM NM M X
A A A
? ? ? ? ? ?
? ?
The binding energy per nucleon for
2
1
H is
2.224
1.112 MeV / nucleon
2
? and for
209
63
Bi it
is
1640 MeV
7.8 MeV / nucleon
209
? .
Figure below shows the binding energy per nucleon against the number of nucleons in
various atomic nuclei.
The greater the binding energy per nucleon, the more stable the nucleus is. The graph has
the maximum of 8.8 MeV / nucleon when the number of nucleons is 56. The nucleus that
has 56 protons and neutrons is
56
26
Fe an iron isotope. This is the most stable nucleus of
them all, since the most energy is needed to pull a nucleon away from it.
Two remarkable conclusions can be drawn from the above graph.
(i) If we can somehow split a heavy nucleus into two medium sized ones, each of the new
nuclei will have more binding energy per nucleon than the original nucleus did. The extra
energy will be given off, and it can be a lot. For instance, if the uranium nucleus
235
92
U is
broken into two smaller nuclei, the binding energy difference per nucleon is about
0.8 MeV. The total energy given off is therefore
? ?
MeV
0.8 235 nucleon 188 MeV
nucleon
? ?
?
? ?
? ?
This process is called as nuclear fission.
Mass number, A
Fe
56
26
Binding energy per nucleon, MeV
? ? ? E
Fusion
? ? ? E
Fission
Figure: Binding energy per nucleon as function of mass number.
(ii) If we can somehow join two light nuclei together to give a single nucleus of medium
size also means more binding energy per nucleon in the new nucleus. For instance, if two
2
1
H deuterium nuclei combine to form a
4
2
He
helium nucleus, over 23 MeV is released.
Such a process, called nuclear fusion, is also very effective way to obtain energy. In fact,
nuclear fusion is the main energy source of the sun and other stars.
Example: The measured mass of deuteron atom
? ?
2
1
H , Hydrogen atom
? ?
1
1
H , proton and
neutron is 2.01649 u ,1.00782 u , 1.00727 u and 1.00866 u . Find the binding energy of
the deuteron nucleus (unit / MeV nucleon ).
Solution: Here 2, 1, 1 A Z N ? ? ?
2
1
. . [ – ( )] 931.5
H N
B E ZM NM M H MeV ? ? ?
[1 1.00782 1 1.00866 2.01649] 931.5 MeV ? ? ? ? ? ?
[0.00238] 931.5 2.224 MeV MeV ? ? ?
Example: The binding energy of the neon isotope
20
10
Ne is 160.647 MeV. Find its atomic
mass.
Solution: Here 10, 10, 10 A Z N ? ? ?
? ?
A
Z H N
B
M ( X) ZM NM -
931.5 MeV / u
? ?
? ? ? ?
20
10
160.647
M ( Ne) 10 1.00782 10 1.00866 19.992u
931.5 MeV / u
? ? ? ? ? ?
? ?
Example:
(a) Find the energy needed to remove a neutron from the nucleus of the calcium
isotope
42
20
Ca
.
(b) Find the energy needed to remove a proton from this nucleus.
(c) Why are these energies different?
Given: atomic masses of
42
20
41.958622 Ca u ?
,
41
20
40.962278 Ca u ?
,
41
19
40.961825 K u ?
,
and mass of
1
0
1.008665 n u ?
,
1
1
1.007276 p u ?
.
Page 4
1(f). Binding Energy
When nuclear masses are measured, it is found that they are less than the sum of the
masses of the neutrons and protons of which they are composed. This is in agreement
with Einstein’s theory of relativity, according to which the mass of a system bound by
energy B is less than the mass of its constituents by
2
/ B c (where c is the velocity of
light).
The Binding energy B of a nucleus is defined as the difference between the energy of the
constituent particles and of the whole nucleus. For a nucleus of atom
A
Z
X ,
2 2
– [ – ( )]
A A
p N Z H N Z
B ZM NM M c ZM NM M X c ? ? ? ? ? ?
? ?
If mass is expressed in atomic mass unit
– 931.5 [ – ( )] 931.5
A A
p N Z H N Z
B ZM NM M MeV ZM NM M X MeV ? ? ? ? ? ? ? ?
? ?
:
p
M Mass of free proton, :
N
M M
N
: Mass of free neutron,
:
H
M mass of hydrogen atom
A
Z
M : mass of the nucleus,
: Z Number of proton, : N Number of neutron,
? ?
A
Z
M X : mass of atom.
9.1.5.1 Binding Energy per Nucleon
The binding energy per nucleon for a given nucleus is found by dividing its total
binding energy by the number of nucleon it contains. Thus binding energy per nucleon is
2 2
– [ – ( )]
A A
p N Z H N Z
B c c
ZM NM M ZM NM M X
A A A
? ? ? ? ? ?
? ?
The binding energy per nucleon for
2
1
H is
2.224
1.112 MeV / nucleon
2
? and for
209
63
Bi it
is
1640 MeV
7.8 MeV / nucleon
209
? .
Figure below shows the binding energy per nucleon against the number of nucleons in
various atomic nuclei.
The greater the binding energy per nucleon, the more stable the nucleus is. The graph has
the maximum of 8.8 MeV / nucleon when the number of nucleons is 56. The nucleus that
has 56 protons and neutrons is
56
26
Fe an iron isotope. This is the most stable nucleus of
them all, since the most energy is needed to pull a nucleon away from it.
Two remarkable conclusions can be drawn from the above graph.
(i) If we can somehow split a heavy nucleus into two medium sized ones, each of the new
nuclei will have more binding energy per nucleon than the original nucleus did. The extra
energy will be given off, and it can be a lot. For instance, if the uranium nucleus
235
92
U is
broken into two smaller nuclei, the binding energy difference per nucleon is about
0.8 MeV. The total energy given off is therefore
? ?
MeV
0.8 235 nucleon 188 MeV
nucleon
? ?
?
? ?
? ?
This process is called as nuclear fission.
Mass number, A
Fe
56
26
Binding energy per nucleon, MeV
? ? ? E
Fusion
? ? ? E
Fission
Figure: Binding energy per nucleon as function of mass number.
(ii) If we can somehow join two light nuclei together to give a single nucleus of medium
size also means more binding energy per nucleon in the new nucleus. For instance, if two
2
1
H deuterium nuclei combine to form a
4
2
He
helium nucleus, over 23 MeV is released.
Such a process, called nuclear fusion, is also very effective way to obtain energy. In fact,
nuclear fusion is the main energy source of the sun and other stars.
Example: The measured mass of deuteron atom
? ?
2
1
H , Hydrogen atom
? ?
1
1
H , proton and
neutron is 2.01649 u ,1.00782 u , 1.00727 u and 1.00866 u . Find the binding energy of
the deuteron nucleus (unit / MeV nucleon ).
Solution: Here 2, 1, 1 A Z N ? ? ?
2
1
. . [ – ( )] 931.5
H N
B E ZM NM M H MeV ? ? ?
[1 1.00782 1 1.00866 2.01649] 931.5 MeV ? ? ? ? ? ?
[0.00238] 931.5 2.224 MeV MeV ? ? ?
Example: The binding energy of the neon isotope
20
10
Ne is 160.647 MeV. Find its atomic
mass.
Solution: Here 10, 10, 10 A Z N ? ? ?
? ?
A
Z H N
B
M ( X) ZM NM -
931.5 MeV / u
? ?
? ? ? ?
20
10
160.647
M ( Ne) 10 1.00782 10 1.00866 19.992u
931.5 MeV / u
? ? ? ? ? ?
? ?
Example:
(a) Find the energy needed to remove a neutron from the nucleus of the calcium
isotope
42
20
Ca
.
(b) Find the energy needed to remove a proton from this nucleus.
(c) Why are these energies different?
Given: atomic masses of
42
20
41.958622 Ca u ?
,
41
20
40.962278 Ca u ?
,
41
19
40.961825 K u ?
,
and mass of
1
0
1.008665 n u ?
,
1
1
1.007276 p u ?
.
Solution:
(a)
42 41 1
20 20 0
Ca Ca n ? ?
;
Total mass of the
41 1
20 0
41.970943 Ca and n u ?
Mass defect 41.970943 41.958622 0.012321 m u ? ? ? ?
So, B.E. of missing neutron 931.5 11.48 m MeV ? ? ? ?
(b)
42 41 1
20 19 1
Ca K p ? ? ;
Total mass of the
41 1
19 1
41.919101 K and p u ?
Mass defect 41.919101 41.958622 0.010479 m u ? ? ? ?
So, B.E. of missing protron 931.5 10.27 m MeV ? ? ? ?
(c) The neutron was acted upon only by attractive nuclear forces whereas the proton was
also acted upon by repulsive electric forces that decrease its binding energy.
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FAQs on Binding Energy - Nuclear and Particle Physics for GATE - GATE Physics
| 1. What is binding energy in the context of physics? |  |
Ans. Binding energy refers to the energy required to disassemble a nucleus into its constituent parts (protons and neutrons) or to separate electrons from an atom. It is the energy that holds particles together within a nucleus or an atom.
| 2. How is binding energy calculated? | |
Ans. The binding energy of a nucleus can be calculated using the Einstein's mass-energy equivalence principle (E = mc²), where E is the binding energy, m is the mass defect (difference between the sum of individual masses of nucleons and the actual mass of the nucleus), and c is the speed of light in a vacuum (3 × 10^8 m/s).
| 3. What is the significance of binding energy in nuclear reactions? | |
Ans. Binding energy plays a crucial role in nuclear reactions, particularly in nuclear fission and fusion. In nuclear fission, the nucleus of a heavy atom splits into two smaller nuclei, releasing a significant amount of binding energy. In nuclear fusion, two light nuclei combine to form a heavier nucleus, and the excess binding energy is released.
| 4. How does binding energy affect the stability of an atom or nucleus? | |
Ans. The binding energy per nucleon determines the stability of a nucleus. Nuclei with higher binding energy per nucleon are more stable. If a nucleus has a lower binding energy per nucleon, it can release energy by undergoing nuclear reactions such as fission or fusion to achieve a more stable configuration.
| 5. Can binding energy be converted into other forms of energy? | |
Ans. Yes, binding energy can be converted into other forms of energy, such as thermal energy or electromagnetic radiation. In nuclear reactions, the release of binding energy can be harnessed to generate electricity in nuclear power plants or to create destructive explosions in nuclear weapons.
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