# Mass Defect & Binding Energy(OCR A Level Physics)

Author

Katie M

Expertise

Physics

## Mass Defect & Binding Energy

#### Mass Defect & Binding Energy

• Experiments into nuclear structure have found that the total mass of a nucleus is less than the sum of the masses of its constituent nucleons
• This difference in mass is known as the mass defect or mass deficit
• Mass defect is defined as:

The difference between the measured mass of a nucleus and the sum total of the masses of its constituents

A system of separated nucleons has a greater mass than a system of bound nucleons

• Due to the equivalence of mass and energy, this decrease in mass implies that energy is released in the process
• Since nuclei are made up of neutrons and protons, there are forces of repulsion between the positive protons
• Therefore, it takes energy, ie. the binding energy, to hold nucleons together as a nucleus

• Binding energy is defined as:

The energy required to break a nucleus into its constituent protons and neutrons

• Energy and mass are proportional, so, the total energy of a nucleus is less than the sum of the energies of its constituent nucleons
• The formation of a nucleus from a system of isolated protons and neutrons is therefore an exothermic reaction - meaning that it releases energy

#### Exam Tip

Avoid describing the binding energy as the energy stored in the nucleus – this is not correct – it is energy that must be put into the nucleus to pull it apart.

## Binding Energy per Nucleon Graph

• In order to compare nuclear stability, it is more useful to look at the binding energy per nucleon
• The binding energy per nucleon is defined as:

The binding energy of a nucleus divided by the number of nucleons in the nucleus

• A higher binding energy per nucleon indicates a higher stability
• In other words, it requires more energy to pull the nucleus apart

• Iron (A = 56) has the highest binding energy per nucleon, which makes it the most stable of all the elements

By plotting a graph of binding energy per nucleon against nucleon number, the stability of elements can be inferred

#### Key Features of the Graph

• At low values of A:
• Nuclei tend to have a lower binding energy per nucleon, hence, they are generally less stable
• This means the lightest elements have weaker electrostatic forces and are the most likely to undergo fusion

• Helium (4He), carbon (12C) and oxygen (16O) do not fit the trend
• Helium-4 is a particularly stable nucleus hence it has a high binding energy per nucleon
• Carbon-12 and oxygen-16 can be considered to be three and four helium nuclei, respectively, bound together

• At high values of A:
• The general binding energy per nucleon is high and gradually decreases with A
• This means the heaviest elements are the most unstable and likely to undergo fission

#### Worked example

The following equation represents one possible decay of the induced fission of a nucleus of uranium-235.

The graph shows the binding energy per nucleon plotted against nucleon number A.

Calculate the energy released by the fission of uranium-235.

Step 1: Use the graph to identify each isotope’s binding energy per nucleon

• Binding energy per nucleon (U-235) = 7.5 MeV
• Binding energy per nucleon (Sr-91) = 8.2 MeV
• Binding energy per nucleon (Xe-142) = 8.7 MeV

Step 2: Determine the binding energy of each isotope

Binding energy = Binding Energy per Nucleon × Mass Number

• Binding energy of U-235 nucleus = (235 × 7.5) = 1763 MeV
• Binding energy of Sr-91 = (91 × 8.2) = 746 MeV
• Binding energy of Xe-142 = (142 × 8.7) = 1235 MeV

Step 3: Calculate the energy released

Energy released = Binding energy after (Sr + Xe) – Binding energy before (U)

Energy released = (1235 + 746) – 1763 = 218 MeV

#### Exam Tip

Checklist on what to include (and what not to include) in an exam question asking you to draw a graph of binding energy per nucleon against nucleon number:

• You will be expected to draw the best fit curve AND a cross to show the anomaly that is helium
• Do not begin your curve at A = 0, this is not a nucleus!
• Make sure to correctly label both axes AND units for binding energy per nucleon
• You will be expected to include numbers on the axes, mainly at the peak to show the position of iron (56Fe)

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