Physics 001.002.022 Mass-Energy Equivalence Problems
Alignment
Learning Intentions
By the end of the lesson, students will be able to:
- Identify when mass has been converted into energy in a nuclear reaction.
- Use
to calculate energy released or absorbed from a change in mass. - Convert mass units correctly before using
. - Rearrange
to solve for , or . - Interpret answers using appropriate SI units and scientific notation.
Success Criteria
By the end of the lesson, students have successfully:
- Defined
, and in the relationship . - Substituted mass in kilograms and speed in
. - Calculated energy changes in joules.
- Converted between atomic mass units and kilograms where required.
- Explained why very small mass changes can produce very large energy changes.
Syllabus Reference
- Unit 1: Thermal, Nuclear and Electrical Physics
- Topic 2: Ionising Radiation and Nuclear Reactions
- Energy and Mass Defect
- Solve problems involving the mass–energy equivalence relationship using
.
Phenomenon
A tiny amount of matter can release an enormous amount of energy. In nuclear fission, nuclear fusion and radioactive processes, the total mass of the products is often slightly less than the total mass of the reactants. This missing mass has not disappeared; it has been transformed into energy.
For example, the energy released by the Sun comes from nuclear fusion. Each fusion reaction converts a small fraction of mass into energy, but the Sun contains such a large number of particles that the total energy output is enormous.
Key Idea
Concept
The mass–energy equivalence relationship states that mass and energy are equivalent forms of the same physical quantity. A change in mass corresponds to a change in energy according to:
where:
is the energy released or absorbed, measured in joules is the change in mass, measured in kilograms is the speed of light in a vacuum,
Because
Convention
The key conventions associated with this concept are:
- Use
unless another value is given. - Mass must be converted into kilograms before substituting into
. - Energy calculated using SI units will be in joules.
- If mass is lost by a nuclear system, energy is released.
- In many school-level problems, use
so that released energy gives a positive value. - If using atomic mass units, convert using
. - If converting joules to electron volts, use
. - If converting joules to megaelectron volts, use
.
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.
- Thinking that mass is “destroyed” rather than transformed into energy.
- Forgetting to convert mass from grams or atomic mass units into kilograms.
- Using
instead of . - Assuming a small mass change means a small energy change.
- Confusing
with the total mass of the nucleus rather than the change in mass.
Further Reading
- Mass defect and binding energy
- Nuclear fission and nuclear fusion
- Binding energy per nucleon
- Energy units: joules, electron volts and megaelectron volts
- Conservation of mass-energy in nuclear reactions
Explicit Instruction
The mass–energy equivalence relationship is:
This means that the energy change is equal to the mass change multiplied by the square of the speed of light.
Since:
then:
Therefore:
This explains why nuclear reactions release so much energy. A mass change of only
This is a very large amount of energy from only one milligram of mass.
Problem-solving method:
- Identify what the question gives.
- Identify what the question asks for.
- Convert mass into kilograms if needed.
- Write the formula
. - Substitute values with units.
- Calculate using scientific notation.
- Check that the answer is reasonable and has the correct unit.
Worked Examples
Worked Example 1
A nuclear reaction converts
Known values:
Use:
Solution:
Answer:
The energy released is
Worked Example 2
A nuclear reaction releases
Known values:
Use:
Rearrange:
Solution:
Answer:
The mass converted into energy is
Worked Example 3
In a fusion reaction, the mass before the reaction is
Known values:
Step 1: Calculate the mass defect in atomic mass units.
Step 2: Convert into kilograms.
Step 3: Use
Answer:
The energy released is
Check for Understanding
Check 1
A student writes:
What mistake has the student made?
Expected answer:
The student used
Check 2
A nuclear process converts
Expected answer:
Convert grams into kilograms:
Check 3
The mass before a nuclear reaction is greater than the mass after the reaction. What does this suggest?
Expected answer:
The missing mass has been converted into energy. The reaction has released energy.
Investigation (Alternative to Explicit)
Hypothesis
If a small amount of mass is converted into energy, then the energy released will be very large because the mass change is multiplied by
Data Collection
Students are given a table of hypothetical mass changes:
| Reaction | Mass change |
|---|---|
| A | |
| B | |
| C | |
| D | |
| E |
Students calculate
Analysis
Students plot
Expected relationship:
- The graph should be linear.
- The gradient should be equal to
. - Since
, the gradient should be very large.
Students answer:
- What happens to
when doubles? - What does the gradient represent?
- Why is the graph linear?
- Why does a small change in mass produce a large energy change?
Evaluation
Students evaluate:
- Whether their calculated values use correct units.
- Whether their graph supports the relationship
. - Whether any errors came from incorrect scientific notation or unit conversion.
- Whether using rounded values of
affects the precision of the final answer.
Problems
The following problems are designed to develop fluency with
-
A reaction converts
of mass into energy. Calculate the energy released. -
A nuclear process converts
of mass into energy. Calculate . -
A reaction releases
of energy. Calculate the mass converted into energy. -
A nuclear power process releases
of energy. What mass was converted into energy? -
A reaction has a mass before of
and a mass after of . Calculate: - the mass defect in
- the mass defect in
- the energy released in
- the mass defect in
-
A student calculates that
of mass is converted into energy in a small laboratory nuclear reaction. Explain why this answer is likely unreasonable. -
A nuclear reaction releases
of energy. Calculate the mass defect in kilograms. -
A mass defect of
occurs in a nuclear reaction. Calculate the energy released in joules. -
Compare the energy released when
of mass is converted into energy with the energy released when of mass is converted into energy. -
Challenge: A nuclear reaction releases
per reaction. How many reactions are needed to release of energy?
Suggested answers:
-
-
-
-
-
is a very large mass change for a small nuclear reaction. Since is very large, this would release an enormous amount of energy. -
-
- For
, - For
, releases times more energy than
- For
-
reactions
Followup
Self-check
Students should be able to answer:
- Can I explain what
represents? - Can I explain why mass must be in kilograms?
- Can I correctly square
? - Can I rearrange
? - Can I decide whether an answer is reasonable?
- Can I explain why nuclear reactions release large amounts of energy?
Exit ticket:
A nuclear reaction has a mass defect of
- Calculate the energy released.
- Explain why this energy is large compared with the very small mass defect.
Expected answer:
This is large for a single nuclear-scale event because the mass defect is multiplied by
Next Topic
Explain that more energy is released per nucleon in nuclear fusion than in nuclear fission because a greater percentage of the mass is transformed into energy.