Physics 001.002.021 Mass-Energy Equivalence

Alignment

Learning Intentions

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

• Describe the mass–energy equivalence relationship.
• Recognise that mass can be transformed into energy during nuclear reactions.
• Identify that a small change in mass can correspond to a very large change in energy because is extremely large.
• Connect mass–energy equivalence to nuclear fission, fusion, binding energy and mass defect.

Success Criteria

By the end of the lesson, students have successfully:

• Stated the relationship and .
• Described the meaning of , , , and .
• Explained why nuclear reactions can release large amounts of energy.
• Distinguished between total mass–energy and change in mass–energy.
• Used correct SI units for mass, energy and the speed of light.

Syllabus Reference

• Unit 1: Thermal, Nuclear and Electrical Physics
• Topic 2: Ionising Radiation and Nuclear Reactions
• Energy and Mass Defect
• Describe the mass–energy equivalence relationship.
• Solve problems involving the mass–energy equivalence relationship using .

Phenomenon

The Sun releases enormous amounts of energy every second. This energy does not come from burning fuel like a fire. Instead, it comes from nuclear fusion, where a small amount of mass is transformed into energy.

A similar idea explains why nuclear fission in a reactor can release far more energy per kilogram of fuel than chemical reactions such as burning coal. In nuclear reactions, the total mass of the products can be slightly less than the total mass of the reactants. The “missing” mass has been transformed into energy.

Key Idea

Mass and energy are equivalent. Mass can be thought of as a concentrated form of energy.

The relationship is:

For nuclear reactions, we are usually interested in the change in mass:

where:

• is the energy released or absorbed in joules
• is the change in mass in kilograms
• is the speed of light,

Because is very large, even a tiny change in mass can produce a very large amount of energy.

Concept

The mass–energy equivalence relationship describes the idea that mass and energy are two forms of the same physical quantity.

In nuclear reactions, the mass of the products is often different from the mass of the reactants. If the products have less mass than the reactants, the difference in mass has been released as energy.

This mass difference is called the mass defect when discussing nuclei and binding energy.

The energy equivalent of this mass difference is calculated using:

This relationship explains why nuclear reactions release much more energy than chemical reactions. Chemical reactions involve rearranging electrons. Nuclear reactions involve changes in the nucleus, where mass differences are much more significant.

Convention

The key conventions associated with mass–energy equivalence are:

• Use kilograms for mass when calculating energy in joules.
• Use .
• Use when calculating the energy released or absorbed due to a change in mass.
• A loss of mass corresponds to energy being released.
• A gain in mass corresponds to energy being absorbed.
• The symbol means the change in mass, not the total mass.
• The symbol represents the speed of light in a vacuum.

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.

• Mass is not “destroyed”; it is transformed into energy.
• does not mean every object is constantly releasing all of its mass as energy.
• In nuclear reaction problems, students should usually use , not the total mass of the object.
• The mass change must be in kilograms if the answer is required in joules.
• A very small mass defect can still produce a very large energy release because is extremely large.

Further Reading

• Einstein’s theory of special relativity
• Nuclear fission and nuclear fusion
• Mass defect and binding energy
• Binding energy per nucleon
• Energy production in stars

Explicit Instruction

Mass–energy equivalence is summarised by:

This means that a mass has an equivalent amount of energy .

In nuclear physics, we are usually interested in the energy released or absorbed when mass changes. Therefore, we use:

The speed of light is:

So:

Therefore:

This means that every kilogram of mass transformed into energy would release:

In real nuclear reactions, the change in mass is usually much smaller than , but the energy released can still be very large.

Worked Examples

Worked Example 1

A nuclear reaction has a mass defect of . Calculate the energy released.

Therefore, the energy released is .

Worked Example 2

A nuclear reaction releases of energy. Calculate the mass transformed into energy.

Therefore, the mass transformed into energy is .

Worked Example 3

In a fusion reaction, the reactants have a total mass of and the products have a total mass of . Calculate the energy released.

First calculate the mass defect:

Now calculate the energy released:

Therefore, the energy released is .

This is a tiny amount of energy for one reaction, but stars contain enormous numbers of nuclei undergoing fusion.

Check for Understanding

Check 1

What does the equation mean?

Expected response:

Mass and energy are equivalent. A quantity of mass has an equivalent amount of energy.

Check 2

Why can a very small mass defect release a large amount of energy?

Expected response:

Because the mass change is multiplied by , and is extremely large.

Check 3

A reaction has a mass defect of . Does this seem too small to matter?

Expected response:

No. Even a very small mass defect can release a large amount of energy because .

Investigation (Alternative to Explicit)

Hypothesis

If mass can be transformed into energy, then a small decrease in mass during a nuclear reaction should correspond to a large amount of released energy.

Data Collection

Students are provided with a table of hypothetical nuclear reactions showing:

• mass of reactants
• mass of products
• calculated mass defect
• energy released

Students calculate and for each reaction.

Example data:

Reaction	Mass of reactants	Mass of products
A
B
C

Analysis

Students calculate:

Students compare the size of the mass defect with the size of the energy released.

Evaluation

Students discuss:

• Why the mass changes are difficult to measure directly.
• Why nuclear reactions release more energy than chemical reactions.
• Why the mass defect must be calculated carefully using consistent units.
• Whether the data supports the relationship .

Problems

The following problems are designed to develop understanding of the mass–energy equivalence relationship.

1. State the mass–energy equivalence relationship.

2. Define each symbol in .

3. Explain why is more useful than in nuclear reaction calculations.

4. A reaction has a mass defect of . Calculate the energy released.

5. A nuclear reaction releases of energy. Calculate the mass transformed into energy.

6. The reactants in a nuclear reaction have a total mass of . The products have a total mass of . Calculate the mass defect and energy released.

7. Explain why chemical reactions do not usually involve noticeable changes in mass.

8. Explain how mass–energy equivalence helps account for energy production in stars.

9. A student says, “The missing mass disappears.” Explain why this statement is incorrect.

10. A nuclear reaction produces products with more mass than the reactants. Would energy be released or absorbed? Explain your answer.

Followup

Self-check

Students should be able to answer the following questions:

• Can I state the equation ?
• Can I explain what mass–energy equivalence means?
• Can I describe why a small mass change can release a large amount of energy?
• Can I identify when to use ?
• Can I explain the connection between mass defect and energy release?

Next Topic

Solve problems involving the mass–energy equivalence relationship using .