Physics 001.002.020 Mass Defect

Alignment

Learning Intentions

By the end of the lesson, students will be able to:

• Describe mass defect as the difference between the mass of a nucleus and the total mass of its separate nucleons.
• Describe binding energy as the energy required to completely separate a nucleus into individual protons and neutrons.
• Describe binding energy per nucleon as a measure of nuclear stability.
• Relate mass defect, binding energy and binding energy per nucleon to nuclear fission and fusion.

Success Criteria

By the end of the lesson, students have successfully:

• Identified that the mass of a nucleus is less than the sum of the masses of its separate protons and neutrons.
• Explained that the “missing mass” is associated with energy stored in the nuclear binding of the nucleus.
• Calculated binding energy per nucleon using when given the binding energy and mass number.
• Interpreted higher binding energy per nucleon as a generally more stable nucleus.
• Explained why energy can be released when nuclei change into more tightly bound arrangements.

Syllabus Reference

• Unit 1: Thermal, Nuclear and Electrical Physics
• Topic 2: Ionising Radiation and Nuclear Reactions
• Energy and Mass Defect
• Describe the concepts of mass defect, binding energy and binding energy per nucleon.

Phenomenon

When hydrogen nuclei fuse in the Sun, a huge amount of energy is released. However, the products of the reaction have slightly less mass than the original particles. The “missing” mass has not disappeared. It has been transformed into energy.

A similar idea applies in nuclear fission. When a large nucleus such as uranium-235 splits, the total mass of the products is slightly less than the original mass. This tiny mass difference corresponds to a very large energy release.

Key Idea

A nucleus has less mass than the total mass of its separated protons and neutrons. This difference is called the mass defect. The mass defect is associated with the binding energy of the nucleus. Binding energy per nucleon is useful because it allows us to compare the stability of different nuclei.

Concept

The concept and thought that best describes the cause of the phenomenon is below.

The particles inside a nucleus are held together by the strong nuclear force. Work must be done to separate the nucleus into individual protons and neutrons. The energy required to do this is called the binding energy.

Because mass and energy are equivalent, the energy holding a nucleus together is associated with a measurable difference in mass:

where:

= binding energy in joules,

= mass defect in kilograms,

= speed of light,

The mass defect is:

Binding energy per nucleon is:

where:

= mass number, or total number of nucleons.

A nucleus with a higher binding energy per nucleon is generally more stable because more energy must be supplied, on average, to remove each nucleon.

Convention

The key conventions associated with the concept and in the branch of established knowledge is below.

• Mass number, , is the total number of protons and neutrons.
• Atomic number, , is the number of protons.
• Number of neutrons is calculated using .
• Mass defect is usually positive for a bound nucleus.
• Binding energy is the energy required to separate a nucleus into individual nucleons.
• Binding energy per nucleon is used to compare stability between nuclei of different sizes.
• Binding energy may be expressed in joules, , or electron volts, .
• Nuclear energies are commonly expressed in mega-electron volts, .

Misconceptions

Common misconceptions students have regarding the concept when applying to various situations and solving problems. It could be a conceptual, mathematical or logical misconception.

• The mass defect is not a mistake in measurement. It is a real difference between the mass of a bound nucleus and the mass of its separated nucleons.
• Binding energy is not energy stored like chemical energy in bonds. It is the energy required to separate the nucleus into its nucleons.
• A larger total binding energy does not always mean a more stable nucleus. Binding energy per nucleon is better for comparing stability.
• The missing mass is not destroyed. It is transformed into energy according to mass–energy equivalence.
• A nucleus with high binding energy per nucleon is harder to break apart, but this does not mean it can never undergo nuclear reactions.

Further Reading

• Nuclear binding energy curve
• Mass–energy equivalence
• Nuclear fission
• Nuclear fusion
• Stability of nuclides
• Strong nuclear force

Explicit Instruction

Begin by asking students:

If a nucleus is made of protons and neutrons, should the mass of the nucleus equal the total mass of those protons and neutrons?

Students will likely answer yes. Then introduce the surprising result:

The mass of the nucleus is slightly less than the total mass of its separated protons and neutrons.

This missing mass is called the mass defect.

Use the analogy of building a stable structure. If separate parts come together and form a more stable arrangement, energy is released. To pull the structure apart again, energy must be supplied. In a nucleus, the strong nuclear force holds nucleons together, and the energy required to pull them apart is the binding energy.

Write:

Then explain:

• The separated nucleons have more mass-energy.
• The bound nucleus has less mass-energy.
• The difference has been released as energy when the nucleus formed.
• The same amount of energy would need to be supplied to completely separate the nucleus.

Then define binding energy per nucleon:

This tells us the average energy required to remove one nucleon from the nucleus. It is useful for comparing nuclei because large nuclei naturally have more total binding energy simply because they have more nucleons.

Worked Examples

Worked Example 1

A helium-4 nucleus has 2 protons and 2 neutrons. The mass of the separated nucleons is greater than the mass of the helium-4 nucleus.

Describe what this tells us about the nucleus.

Solution:

The mass of the separated nucleons is greater than the mass of the helium-4 nucleus. Therefore, the helium-4 nucleus has a mass defect.

The missing mass has been transformed into energy. This energy is associated with the binding energy of the nucleus.

The helium-4 nucleus is stable because energy must be supplied to separate it into its individual protons and neutrons.

Worked Example 2

A nucleus has a total binding energy of and contains 8 nucleons.

Calculate the binding energy per nucleon.

Solution:

Therefore, the binding energy per nucleon is per nucleon.

Worked Example 3

Two nuclei have the following binding energy per nucleon values:

Nucleus A: per nucleon

Nucleus B: per nucleon

Which nucleus is generally more stable?

Solution:

Nucleus B is generally more stable because it has the higher binding energy per nucleon.

This means that, on average, more energy is required to remove each nucleon from nucleus B than from nucleus A.

Therefore, nucleus B is more tightly bound.

Check for Understanding

Check 1

A nucleus has less mass than the total mass of its separated protons and neutrons.

What is this mass difference called?

Answer:

It is called the mass defect.

Check 2

What does binding energy represent?

Answer:

Binding energy is the energy required to completely separate a nucleus into its individual protons and neutrons.

Check 3

Why is binding energy per nucleon more useful than total binding energy when comparing nuclear stability?

Answer:

Binding energy per nucleon accounts for the number of nucleons in the nucleus. This makes it useful for comparing nuclei of different sizes.

Investigation (Alternative to Explicit)

Hypothesis

If a nucleus has a higher binding energy per nucleon, then it will generally be more stable because more energy is required, on average, to remove each nucleon.

Data Collection

Students are given a table of selected nuclei with values for:

• mass number,
• total binding energy,
• binding energy per nucleon,

Example data:

Nuclide	Mass number,	Total binding energy,	Binding energy per nucleon
Hydrogen-2	2		per nucleon
Helium-4	4		per nucleon
Carbon-12	12		per nucleon
Iron-56	56		per nucleon
Uranium-235	235		per nucleon

Analysis

Students answer:

1. Which nucleus has the greatest total binding energy?
2. Which nucleus has the greatest binding energy per nucleon?
3. Why is total binding energy not the best measure of stability?
4. Which nucleus in the table is generally the most stable?
5. What does the data suggest about why energy may be released in fusion of small nuclei?
6. What does the data suggest about why energy may be released in fission of very large nuclei?

Expected analysis:

• Uranium-235 has the greatest total binding energy because it has many nucleons.
• Iron-56 has the greatest binding energy per nucleon in the table.
• Binding energy per nucleon is better for comparing stability.
• Small nuclei can release energy by fusing into more tightly bound nuclei.
• Large nuclei can release energy by splitting into smaller, more tightly bound nuclei.

Evaluation

Students discuss:

• The values in the table have been rounded.
• Binding energy per nucleon is a useful general indicator of stability, but nuclear stability also depends on proton-neutron ratio and nuclear structure.
• The binding energy curve provides stronger evidence than isolated data points.
• The model explains why both fusion and fission can release energy, even though they are opposite processes.

Problems

The following problems are designed to check conceptual understanding and basic calculation of binding energy per nucleon.

1. Define mass defect.

2. Define binding energy.

3. Define binding energy per nucleon.

4. A nucleus has a total binding energy of and a mass number of 15. Calculate the binding energy per nucleon.

5. A nucleus has a total binding energy of and contains 30 nucleons. Calculate the binding energy per nucleon.

6. Nucleus X has a binding energy per nucleon of per nucleon. Nucleus Y has a binding energy per nucleon of per nucleon. Which is generally more stable? Explain your answer.

7. Explain why a uranium nucleus can have a very large total binding energy but still be less stable than iron-56.

8. Complete the sentence: The mass of a bound nucleus is less than the mass of its separated nucleons because some mass has been transformed into __________.

9. Explain why energy is released when separate nucleons form a stable nucleus.

10. Explain why binding energy per nucleon is important when comparing fission and fusion reactions.

Followup

Self-check

Students should be able to answer the following:

• Can I define mass defect in words?
• Can I explain why the mass of a nucleus is less than the mass of its separated nucleons?
• Can I define binding energy?
• Can I calculate binding energy per nucleon using ?
• Can I explain why higher binding energy per nucleon usually means greater nuclear stability?
• Can I describe why nuclear reactions can release energy when products are more tightly bound than reactants?

Next Topic

Describe the mass–energy equivalence relationship.

This lesson leads into the relationship:

In the next lesson, students will use this equation to calculate the energy released or absorbed when mass is transformed into energy in nuclear reactions.