# Mass Defect and 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 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, i.e. the binding energy, to hold nucleons together as a nucleus

Binding energy is 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 ## Binding Energy per Nucleon - 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 ## Binding Energy per Nucleon Graph The graph below shows how binding energy per nucleon varies with mass number (nucleon number) for different elements:

### 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 ($^{4}$He), carbon ($^{12}$C) and oxygen ($^{16}$O) 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**

Iron ($^{56}$Fe) sits at the peak of the binding energy per nucleon curve, making it the most stable nucleus

## Energy Release in Nuclear Reactions - The daughter nuclei produced as a result of both fission and fusion have a higher binding energy per nucleon than the parent nuclei - Therefore, energy is released as a result of the mass difference between the parent nuclei and the daughter nuclei - The binding energy is equal to the amount of energy released in forming the nucleus, and can be calculated using: $$E = (\Delta m)c^2$$ Where: - $E$ = Binding energy released (J) - $\Delta m$ = mass defect (kg) - $c$ = speed of light (m s$^{-1}$)

Avoid describing the binding energy as energy stored in the nucleus – it is energy that must be put into the nucleus to pull it apart

## Worked Example: Energy from Fission Consider the following fission reaction of uranium-235: $$^{235}_{92}\text{U} + ^{1}_{0}\text{n} \rightarrow ^{139}_{54}\text{Xe} + ^{95}_{38}\text{Sr} + 2^{1}_{0}\text{n} + \text{energy}$$ Given binding energy per nucleon values from a graph: - Binding energy per nucleon (U-235) = 7.5 MeV - Binding energy per nucleon (Sr-95) = 8.74 MeV - Binding energy per nucleon (Xe-139) = 8.39 MeV **Step 1: Determine the total binding energy of each isotope** Total binding energy = Binding energy per nucleon × Mass number - Binding energy of $^{235}$U = $235 \times 7.5 = 1762.5$ MeV - Binding energy of $^{95}$Sr = $95 \times 8.74 = 830.3$ MeV - Binding energy of $^{139}$Xe = $139 \times 8.39 = 1166.21$ MeV **Step 2: Calculate the difference in energy between products and reactants** Energy released = Binding energy after (Sr + Xe) – Binding energy before (U) Energy released = $(830.3 + 1166.21) - 1762.5$ Energy released = $1996.51 - 1762.5 = 234.01$ MeV **Step 3: Interpret the result** - Since the products have higher binding energy per nucleon, they are more stable - The more energy released, the further the reaction moves up the binding energy per nucleon curve - Nuclear reactions tend to favour the more stable route