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# Nuclear Binding Energy

## Key Concepts

• Mass and energy are equivalent: E = mc2

• The mass defect for an atom represents the amount of mass that has been "converted" into energy.

• Nuclear Binding Energy (binding energy) is the term used to refer to this energy.

• Nuclear binding energy is the energy required to separate the nucleus of an atom into its parts.

• Protons and neutrons in the nucleus of an atom are referred to collectively as nucleons.

• Nuclear binding energy per nucleon tells us how tightly bound the nucleons are.

• The greater the nuclear binding energy per nucleon the more stable the nucleus is:

 nuclear binding energy per nucleon nuclear stability low → → → → → high low → → → → → high

## Theory

Using Mass Spectroscopy we can measure the mass of individual isotopes of an element, the observed mass of an atom.

We can use the rest mass of protons, neutrons and electrons given in the table on the right to calculate what the mass of an atom should be, the predicted mass of an atom:
predicted mass = mass of all protons + mass of all neutrons + mass of all electrons

We find that the observed mass of an atom is less than the mass we predict:

predicted mass > observed mass
Subatomic
Particle
Rest Mass
atomic mass units
Rest Mass
kg
proton 1.007276 1.673 x 10-27
neutron 1.008665 1.675 x 10-27
electron 0.000549 9.109 x 10-31

This difference in mass is known as the mass defect (Δm):

mass defect = predicted mass - observed mass

Mass cannot just disappear, this would be contrary to the Law of Mass Conservation, but it can be converted into energy.

We can use Einstein's famous equation for mass-energy equivalence, E = mc2,

E = energy (J)
m = mass (kg)
c = speed of light = 3 × 108 m s-1 (vacuum)

to convert the mass defect, Δm in kilograms (kg), into an energy term in Joules (J):

E = Δm × c2
= Δm × 3 × 108
= Δm × 9 × 1016

The binding energy must be related to the energy released when the subatomic particles come together to form an atom:

We could think of this like a chemical equation:

protons + neutrons → nucleus + energy

The binding energy of a nucleus is in reality defined the other way around, that is, as the energy required to completely separate the individual components of the nucleus:
If we were to write a chemical equation for this it would look like the one below:

nucleus + binding energy → protons + neutrons

We would expect that the binding energy of a nucleus would increase as the number of nucleons (numbers of protons and neutrons) in the nucleus increases, that is, because there are more particles in the nucleus (nucleons) it will take more energy to separate them completely.
So a better way of comparing the binding energy of different nuclei is to compare the binding energy per nucleon:

binding energy per nucleon = binding energy for the atom ÷ number of nucleons
binding energy per nucleon = binding energy for the atom ÷ (no. protons + no. neutrons)

Since the number of protons and neutrons in the nucleus of an atom is given by its mass number (A), we carn write:
binding energy per nucleon = binding energy for the atom ÷ mass number
binding energy per nucleon = binding energy for the atom ÷ A

You will sometimes see the binding energy given the symbol B (or B.E.) in which case we can write:
binding energy per nucleon = B ÷ A

A nucleus that is very stable will require a lot of energy to break it up, so it will have a large value for the binding energy per nucleon.
A nucleus that is not very stable will require much less energy to break it up, so it will have a smaller value for the binding energy per nucleon.

## Binding Energy Calculations in Joules

E = Δm × c2
E = binding energy in joules (J)
Δm = mass defect in kilograms (kg)
c = speed of light in metres per second (m s-1) = 3 × 108 m s-1

Steps to solve binding energy problems in SI units:

Step 1: Calculate the mass defect in kilograms (kg)

Step 2: Convert this mass defect into energy in joules (J)

### Example: Calculate the binding energy for helium-4 using SI units

• Step 1: Calculate the mass defect in SI units of kilograms (kg)

We can predict the mass of an atom of helium-4 using the SI units for the rest mass of protons, neutrons and electrons given above.
No. protons = atomic number = Z = 2 (from Periodic Table)
No. electrons = no. protons = 2 (for a neutral atom number of protons = number of electrons)
No. neutrons = mass number (A) - atomic number (Z) = 4 - 2 = 2 (mass number given in name)

 predicted mass = (no. protons × mass proton) + (no. neutrons × mass neutron) + (no. electrons × mass electron) = (2 × 1.673 × 10-27) + (2 × 1.675 × 10-27) + (2 × 9.109 × 10-31) = (3.346 × 10-27) + (3.35 × 10-27) + (1.8218 × 10-30) = 6.6978 × 10-27 kg

Using Mass Spectroscopy, the mass of an atom of helium-4 is found to be 6.646 × 10-27 kg

 Mass defect = predicted mass - observed mass Δm = 6.6978 × 10-27 - 6.646 × 10-27 = 5.18 × 10-29 kg

• Step 2: Convert this mass defect (kg) into energy (J)

 E = Δm × c2 = 5.18 × 10-29 × (3 × 108)2 = 5.18 × 10-29 × 9 × 1016 = 4.662 × 10-12 J

### Calcuating the binding energy for helium-4 using atomic mass units (u)

• Step 1: Calculate the mass defect in kilograms (kg)

We can predict the mass of an atom of helium-4 using the non-SI units of "atomic mass units" (u) for the rest mass of protons, neutrons and electrons given above.
No. protons = atomic number = Z = 2 (from Periodic Table)
No. electrons = no. protons = 2 (for a neutral atom number of protons = number of electrons)
No. neutrons = mass number (A) - atomic number (Z) = 4 - 2 = 2 (mass number given in name)

 predicted mass = (no. protons × mass proton) + (no. neutrons × mass neutron) + (no. electrons × mass electron) = (2 × 1.007276) + (2 × 1.008665) + (2 × 0.000549) = 2.014552 + 2.01733 + 0.001098 = 4.03298 u

Using Mass Spectroscopy, the mass of an atom of helium-4 is found to be 4.003 u

 Mass defect = predicted mass - observed mass Δm = 4.03298 - 4.003 = 0.02998 u

Convert this mass in atomic mass units (u) to a mass in kilograms (kg):

1 atomic mass unit = 1/12 × mass carbon-12 atom in kg
1 atomic mass unit = 1/12 × 1.993 × 10-26 = 1.6608 × 10-27 kg
0.02998 u = 0.02998 × 1.6608 × 10-27 = 4.980 × 10-29 kg

(Note there is a difference between the mass defect in kg calculate here and the one above. This is due to rounding errors in the rest masses of particles given in kg in the table)

• Step 2: Convert this mass defect (kg) into energy (J)

 E = Δm × c2 = 4.980 × 10-29 × (3 × 108)2 = 4.980 × 10-29 × 9 × 1016 = 4.482 × 10-12 J

## Energy Conversions: J and MeV

Physicists often use the electron-volt (eV) as a unit of energy.
1 electron-volt (1 eV) is the energy gained by an electron accelerated through a potential difference of 1 volt (1 V).
1 eV = 1.602 × 10-19 J

1 megaelectron-volt (1 MeV) = 1,000,000 eV
1 MeV = 1,000,000 × 1.602 × 10-19 = 1.602 × 10-13 J

In order to convert the binding energy in joules (J) to megaelectron-volts (Mev), we need to divide the energy in J by 1.602 × 10-13 J/MeV:

energy (MeV) = energy (J) ÷ 1.602 × 10-13 (J/MeV)

### Convert binding energy of helium-4 in J to MeV

binding energy of helium-4 = 4.482 × 10-12 J (calculated above)

binding energy of helium-4 in MeV = (4.482 × 10-12) ÷ (1.602 × 10-13) = 27.98 MeV

## Binding Energy per Nucleon

Binding energy per nucleon can be calculated using the binding energy of the nucleus (as above) and the number of protons and neutrons in the nucleus of the atom (number of nucleons):
binding energy per nucleon = binding energy for the nucleus ÷ number of nucleons
binding energy per nucleon = binding energy for the nucleus ÷ (number of protons + number of neutrons)
binding energy per nucleon = binding energy for the nucleus ÷ mass number

### Calculating Binding Energy per Nucleon for Helium-4

Binding energy per nucleon tells us how much energy per nucleon we need to apply in order to completely separate the components of a nucleus.
The binding energy for helium-4 was calculate above to be 4.482 × 10-12 J (or 27.98 MeV).
The number of nucleons = the number of protons + the number of neutrons = mass number = A = 4
binding energy per nucleon (J/nucleon) = 4.482 × 10-12 J ÷ 4 = 1.1205 × 10-12 J
binding energy per nucleon (MeV/nucleon) = 27.98 MeV ÷ 4 = 6.995 MeV

## Binding Energy per Nucleon and Isotope Stability

Carbon-12 is known to be a stable isotope of carbon, whereas carbon-14 (used for radiocarbon dating) is an unstable isotope of carbon.

The mass of the carbon-12 atom is defined as being exactly 12.000 u, whereas the mass of the carbon-14 atom has been found to be 14.003 u.
Let us calculate, and compare, the binding energy per nucleon for each of these isotopes of carbon.

 carbon-12 carbon-14 no. protons = Z = 6 6 no. neutrons = A - Z = 12 -6 = 6 14 - 6 = 8 no. electrons = no. protons = 6 6 predicted mass (u) = (6×1.007276)+(6×1.008665)+(6×0.000549) (6×1.007276)+(8×1.008665)+(6×0.000549) = 12.09894 u = 14.11627 u observed mass (u) = 12.000 14.003 mass defect (u) = 12.09894 - 12.000 14.11627 - 14.003 = 0.09894 u = 0.11327 u mass defect (kg) 0.09894 × 1.6608 × 10-27 0.11327 × 1.6608 × 10-27 = 1.643 × 10-28 kg = 1.881 × 10-28 kg binding energy (J) 1.643 × 10-28 × (3 × 108)2 1.881 × 10-28 × (3 × 108)2 = 1.4787 × 10-11 J = 1.6929 × 10-11 J binding energy per nucleon (J) 1.4787 × 10-11 J ÷ 12 1.6929 × 10-11 J ÷ 14 = 1.23225 × 10-12 J = 1.20921 × 10-12 J binding energy per nucleon (MeV) 1.23225 × 10-12 J ÷ 1.602 × 10-13 J/MeV 1.20921 × 10-12 J ÷ 1.602 × 10-13 J/MeV = 7.692 MeV = 7.548 MeV

We can see that the more stable carbon isotope, carbon-12, has a higher binding energy per nucleon (7.692 MeV) than the unstable (radioactive) carbon-14 isotope (7.548 MeV).

## Drill

Quick Questions: 1

Calculate the nuclear binding energy per nucleon in MeV for oxygen-17 which has an observed mass of 16.999 u
mass of a proton = 1.007276 u
mass of a neutron = 1.008665 u
mass of an electron = 0.000549 u

B/A = MeV   Check

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