# Episode 525: Binding energy

**Summary**

- Discussion: Introducing mass defect and atomic mass units (10 minutes)
- Discussion: Mass defect and binding energy (10 minutes)
- Worked example: Calculating binding energy (10 minutes)
- Student questions: Calculations (20 minutes)
- Student activity: Spreadsheet calculations (20 minutes)
- Student activity: Spreadsheet calculations of binding energy per nucleon (20 minutes)
- Discussion: Fission and fusion linked to binding energy graph (10 minutes)

**Discussion: Introducing mass defect and atomic mass units**

Ask your students to consider whether the following data is self-consistent:

proton mass, m_{p} = 1.673 ´ 10^{-27} kg

neutron mass, m_{n} = 1.675 ´ 10^{-27} kg

mass of a

nucleus = 6.643 ´ 10^{-27} kg

The mass of a

nucleus is **less** than the sum of the masses of its parts; this is true for **all** nuclides. So much for conservation of mass.

Introduce the atomic mass unit (amu, or u) as a convenient unit of nuclear mass. 1 amu or 1 u = 1/12 the mass of a neutral ^{12}C atom (i.e. including its six electrons) = 1.66056 ´ 10^{-27} kg. Thus:

m_{p} = 1.0073 u

m_{n} = 1.0087 u

m_{e} = 0.00055 u

mass of a neutral

atom = 4.0026 u

**Discussion: Mass defect and binding energy**

What has happened to the missing mass – or mass defect – between the whole and the sum of the parts? To separate the particles, they must be pulled apart against the attractive strong force. They thus have potential energy when they are separated.

When the particles come together to form a nucleus, their potential energy decreases.

So energy must be put in to separate the nucleons of a nucleus. This energy is known as the binding energy, a rather confusing term because students often think that this means that energy is required to bind nucleons together. As with chemical bonds, this is the opposite of the truth. Energy is needed to break bonds.

Einstein’s Special Theory of Relativity (1905) relates mass and energy via the equation E = mc^{2} (where c is the speed of light in a vacuum). In this case, we have:

binding energy = mass defect ´ c^{2} or DE = Dm ´ c^{2}

(It is not advisable to talk about mass being ‘converted to energy’ or similar expressions. It is better to say that, in measuring an object’s mass, we are determining its energy. A helium nucleus has less mass than its constituent nucleons; in pulling them apart, we do work and so give them energy; hence their mass is greater.)

**Worked example: Calculating binding energy**

Calculate the mass defect and binding energy for

(Mass defect = 0.053 ´ 10^{-27} kg; binding energy = 1.59 ´ 10^{-12} J = 9.94 MeV

**Student questions**

Episode 525-1: Change in energy: Change in mass (Word, 189 KB)

Episode 525-2: Finding binding energy (Word, 52 KB)

Episode 525-3: Fusion in a kettle? (Word, 35 KB)

**Student activity: A data analysis exercise using Excel**

This uses a spreadsheet to calculate binding energy for a number of nuclides.

Episode 525-4: A binding energy calculator (Word, 130 KB)

**Student activity**

Another spreadsheet activity, this time looking at the binding energy per nucleon. Note that it is desirable to plot this graph with a negative energy axis; this means that the lowest values are for the most stable nuclides.

Episode 525-5: Binding energy of nuclei (Word, 79 KB)

**Discussion**

Briefly discuss fission and fusion in terms of the graph. Although the fission ‘jump’ looks quite small compared to a typical fusion jump, the graph is plotting BE **per nucleon**. Many more nucleons are involved in the fission of heavy atoms than in the fusion of lighter ones. (This topic can be developed further when discussing nuclear power.)

Episode 525-6: Binding energy per nucleon (Word, 45 KB)

**Download this episode**

Episode 525: Binding energy (Word, 495 KB)