Mass and Energy Are Connected

For a long time, mass and energy were treated as fundamentally different quantities.

Mass described how much matter an object contained, while energy described its ability to produce change.

Special Relativity revealed a deeper relationship:

mass itself corresponds to energy.

Einstein expressed this relationship using one of the most famous equations in physics:

E = mc²

This equation tells us that even a small amount of mass corresponds to an enormous amount of:

energy.

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What Does E = mc² Mean?

The equation is:

E = mc²

where:

• E = rest energy, measured in joules (J)
• m = rest mass, measured in kilograms (kg)
• c = speed of light in vacuum
• c ≈ 3.00 × 10⁸ m/s

More precisely, E = mc² describes the rest energy of an object.

This is the energy associated with an object's mass even when the object is:

at rest.

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The Speed of Light Squared

The speed of light is:

c = 3.00 × 10⁸ m/s.

Therefore:

c² = (3.00 × 10⁸)²

c² = 9.00 × 10¹⁶ m²/s².

So the equation can be written approximately as:

E = m(9.00 × 10¹⁶).

That enormous multiplier explains why a small amount of mass corresponds to:

a very large amount of energy.

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Checking the Units

The equation also works dimensionally.

Mass is measured in:

kg.

c² has units:

m²/s².

Therefore:

mc² = kg·m²/s².

But:

1 J = 1 kg·m²/s².

Therefore:

mc² has units of joules.

So:

E = mc²

has the correct units for energy.

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Rest Energy

An object does not need to be moving to possess energy.

An object with mass has:

rest energy.

The rest energy is:

E₀ = mc².

The subscript zero is sometimes used to emphasize that this is the energy associated with the object's:

rest mass.

For example, a stationary particle still possesses:

rest energy.

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Worked Example 1: Energy Equivalent of 1 kg

Suppose:

m = 1.00 kg.

Use:

E = mc².

Substitute:

E = (1.00)(3.00 × 10⁸)².

Therefore:

E = 9.00 × 10¹⁶ J.

So 1 kg of mass has a rest-energy equivalent of:

9.00 × 10¹⁶ J.

That is an extraordinarily large amount of energy.

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Worked Example 2: Energy Equivalent of 1 Gram

Suppose:

m = 1.00 g.

First convert to kilograms:

1.00 g = 0.00100 kg

or:

1.00 × 10⁻³ kg.

Now:

E = mc²

E = (1.00 × 10⁻³)(3.00 × 10⁸)²

E = (1.00 × 10⁻³)(9.00 × 10¹⁶)

E = 9.00 × 10¹³ J.

Even one gram of mass corresponds to:

9.00 × 10¹³ J.

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Why Such a Small Mass Represents So Much Energy

The key is:

c².

Since:

c² = 9.00 × 10¹⁶ m²/s²,

every kilogram of mass corresponds to approximately:

9 × 10¹⁶ joules of rest energy.

This enormous conversion factor is why tiny changes in mass can correspond to substantial:

energy changes.

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Calculating Energy from Mass

To calculate the energy equivalent of a mass:

Step 1

Write:

E = mc²

Step 2

Convert the mass to:

kilograms.

Step 3

Use:

c = 3.00 × 10⁸ m/s.

Step 4

Square c:

c² = 9.00 × 10¹⁶.

Step 5

Multiply:

m × c².

Step 6

Give the answer in:

joules.

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Worked Example 3

A process results in a mass decrease of:

2.0 × 10⁻⁶ kg.

What energy corresponds to this mass change?

Use:

E = mc².

E = (2.0 × 10⁻⁶)(9.00 × 10¹⁶)

Combine the numbers:

2.0 × 9.00 = 18

Combine the powers:

10⁻⁶ × 10¹⁶ = 10¹⁰

Therefore:

E = 18 × 10¹⁰ J

or:

E = 1.8 × 10¹¹ J.

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Mass Equivalent of Energy

The equation can also be rearranged.

Starting with:

E = mc²

divide both sides by c²:

m = E/c².

This allows us to calculate the mass equivalent of a quantity of:

energy.

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Worked Example 4: Energy to Mass

Suppose:

E = 4.5 × 10¹⁴ J.

Find the mass equivalent.

Use:

m = E/c².

Substitute:

m = (4.5 × 10¹⁴)/(9.00 × 10¹⁶).

Therefore:

m = 5.0 × 10⁻³ kg.

So:

m = 0.0050 kg

or:

5.0 g.

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Worked Example 5

An energy change of:

1.8 × 10¹² J

corresponds to what change in mass?

Use:

m = E/c².

m = (1.8 × 10¹²)/(9.00 × 10¹⁶)

m = 2.0 × 10⁻⁵ kg.

Therefore:

m = 0.000020 kg.

Converting to grams:

m = 0.020 g.

A very small mass change can therefore correspond to a large:

energy change.

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Mass-Energy Equivalence

The phrase mass-energy equivalence means that mass and energy are two related properties of a physical system.

Mass contributes to the system's:

total energy.

Energy stored within a system also contributes to the system's:

mass.

This is deeper than simply saying:

"mass can turn into energy."

A better statement is:

mass and energy are related components of the physical description of a system.

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Does Mass Simply Disappear?

Not exactly.

In an isolated system, total:

energy and momentum are conserved.

During a nuclear reaction, for example, the total rest mass of the products may be lower than that of the initial particles.

The difference appears as other forms of energy, such as:

• kinetic energy
• electromagnetic radiation
• particle energy

So the complete system still obeys:

conservation laws.

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Mass Defect

In nuclear physics, the difference between the mass of a bound nucleus and the total mass of its separated components is related to:

binding energy.

This difference is often called the:

mass defect.

If the mass difference is:

Δm,

the corresponding energy is:

ΔE = Δmc².

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Binding Energy

The binding energy of a nucleus is the energy required to completely separate the nucleus into its individual:

protons and neutrons.

A stable bound nucleus has less mass than the total mass of the same nucleons when separated.

The difference corresponds to:

nuclear binding energy.

Therefore:

ΔE = Δmc².

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Nuclear Fission

Nuclear fission occurs when a heavy nucleus splits into smaller nuclei.

During fission:

1. a heavy nucleus undergoes splitting
2. smaller nuclei are produced
3. additional particles may be released
4. the products have slightly less total rest mass than the initial system
5. the difference appears primarily as kinetic energy and radiation

The released energy corresponds to the change in mass:

ΔE = Δmc².

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Why Nuclear Reactions Release So Much Energy

Chemical reactions mainly involve changes in:

electron arrangements and chemical bonds.

Nuclear reactions involve changes in:

atomic nuclei.

The mass differences involved in nuclear reactions are much larger relative to the system than those in ordinary chemical reactions.

Therefore, nuclear reactions can release far more energy per kilogram of material than:

chemical reactions.

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Nuclear Fusion

Nuclear fusion occurs when lighter nuclei combine to form heavier nuclei.

One of the most important examples occurs in:

stars.

In the Sun, nuclear reactions ultimately convert hydrogen into helium.

The resulting products have slightly less mass than the original particles.

The difference appears as:

released energy.

This energy eventually leaves the Sun as:

• electromagnetic radiation
• kinetic and thermal energy
• neutrinos

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Why the Sun Shines

The Sun's energy ultimately comes from:

nuclear fusion.

A simplified overall description is:

hydrogen nuclei → helium nucleus + energy

The mass of the final products is slightly lower than the mass of the initial system.

That mass difference corresponds to energy according to:

ΔE = Δmc².

Mass-energy equivalence therefore helps explain the energy output of:

stars.

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Matter and Antimatter

Another dramatic example occurs when:

matter and antimatter interact.

A particle and its antiparticle can annihilate, producing other particles such as:

photons.

For example:

electron + positron → photons

The initial rest energy of the electron and positron appears in the energy of the:

products.

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Pair Production

The reverse process can also occur.

Sufficient energy can produce:

massive particles.

For example, a sufficiently energetic photon interacting appropriately can contribute to producing:

a particle-antiparticle pair.

This is called:

pair production.

This demonstrates that the mass-energy relationship works:

in both directions.

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Energy Can Contribute to Mass

Suppose you heat a sealed container.

The particles inside gain:

internal energy.

Because the total energy of the container increases, its total mass also increases by:

Δm = ΔE/c².

The increase is extraordinarily small for everyday amounts of energy, but it is:

real.

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Example: Heating an Object

Suppose a sealed object absorbs:

9.0 × 10⁶ J

of energy.

Its mass increases by:

Δm = ΔE/c².

Therefore:

Δm = (9.0 × 10⁶)/(9.00 × 10¹⁶)

Δm = 1.0 × 10⁻¹⁰ kg.

This is only:

0.0000000001 kg.

The mass change is far too small for ordinary scales to detect easily.

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Chemical Reactions and Mass-Energy

Mass-energy equivalence also applies to:

chemical reactions.

When a chemical reaction releases energy, the products have very slightly less total mass than the reactants if the released energy leaves the system.

When energy is stored in a chemical system, the system has very slightly:

more mass.

However, the changes are extremely small because chemical energies are tiny compared with:

mc².

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Batteries and Mass

Consider a rechargeable battery.

A fully charged battery contains more stored energy than:

the same battery when discharged.

Therefore, the charged battery also has slightly:

greater mass.

The difference is extraordinarily small, but according to:

E = mc²

it must exist.

This is an excellent example showing that mass-energy equivalence applies beyond:

nuclear physics.

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Energy Stored in a Spring

The same idea applies to a compressed:

spring.

A compressed spring stores:

elastic potential energy.

Therefore, a system containing the compressed spring has slightly more mass than the same system with the spring:

relaxed.

Again, the difference is extremely small.

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A Hot Object Has Slightly More Mass

A hotter object contains more:

internal energy.

Therefore, all else being equal, a hot object has slightly more mass than the same object when:

cooler.

This may seem surprising, but it follows directly from:

mass-energy equivalence.

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Relativistic Energy

The equation E = mc² is specifically the rest-energy equation.

For a moving particle, the complete energy relationship is:

E² = (pc)² + (mc²)²

where:

• E = total energy
• p = relativistic momentum
• m = rest mass
• c = speed of light

If the particle is at rest:

p = 0.

Therefore:

E² = (mc²)²

and:

E = mc².

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Total Energy of a Moving Particle

For a massive particle moving at speed v:

E = γmc²

where:

γ = 1/√(1 − v²/c²).

When:

v = 0,

then:

γ = 1.

Therefore:

E = mc².

This confirms that mc² is the:

rest energy.

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Kinetic Energy in Relativity

The relativistic kinetic energy is:

KE = (γ − 1)mc².

At speeds much lower than c, this approaches the familiar classical equation:

KE ≈ ½mv².

At speeds close to c, however, the classical equation becomes inaccurate and the:

relativistic equation

must be used.

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Photons and E = mc²

A common misconception is that photons must have mass because they possess energy.

Photons have:

zero rest mass.

Yet photons have:

energy and momentum.

For a photon:

m = 0.

Using:

E² = (pc)² + (mc²)²

gives:

E = pc.

Therefore, E = mc² should not be interpreted as saying that every form of energy must belong to a particle with nonzero rest mass.

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Mass-Energy in Particle Physics

Particle accelerators provide striking demonstrations of:

mass-energy equivalence.

Particles are accelerated to very high energies and collided.

The energy available in the collision can produce:

new massive particles,

provided the conservation laws and required energy conditions are satisfied.

This is one reason high-energy particle accelerators are powerful tools for studying:

fundamental particles.

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Mass-Energy and Conservation Laws

Modern physics usually emphasizes conservation of:

total energy

rather than treating mass and energy as completely separate conserved quantities.

During a process:

• rest energy may change
• kinetic energy may change
• radiation may be produced
• particles may be created or destroyed

But the total energy of an isolated system remains:

conserved.

Momentum and other applicable conserved quantities must also be:

conserved.

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Worked Example 6: Nuclear Mass Change

A nuclear reaction decreases the total rest mass by:

5.0 × 10⁻⁴ kg.

Calculate the corresponding released energy.

Use:

ΔE = Δmc².

ΔE = (5.0 × 10⁻⁴)(9.00 × 10¹⁶)

Therefore:

ΔE = 4.5 × 10¹³ J.

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Worked Example 7: Finding the Mass Change

A process releases:

2.7 × 10¹⁵ J.

Calculate the corresponding decrease in rest mass.

Use:

Δm = ΔE/c².

Δm = (2.7 × 10¹⁵)/(9.00 × 10¹⁶)

Δm = 3.0 × 10⁻² kg.

Therefore:

Δm = 0.030 kg

or:

30 g.

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Worked Example 8: A Tiny Mass Change

Suppose:

Δm = 4.0 × 10⁻⁹ kg.

Calculate the corresponding energy.

ΔE = Δmc²

ΔE = (4.0 × 10⁻⁹)(9.00 × 10¹⁶)

ΔE = 3.6 × 10⁸ J.

Even a mass change of only a few billionths of a kilogram corresponds to:

hundreds of millions of joules.

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A Useful Calculation Shortcut

Since:

c² ≈ 9.00 × 10¹⁶,

for introductory calculations you can often use:

E ≈ m(9.00 × 10¹⁶).

Or, when finding mass:

m ≈ E/(9.00 × 10¹⁶).

Always make sure mass is measured in:

kilograms.

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Common Mistake: Forgetting to Square c

The equation is:

E = mc²

not:

E = mc.

Remember:

c² = 9.00 × 10¹⁶,

not:

3.00 × 10⁸.

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Common Mistake: Using Grams

SI calculations require mass in:

kilograms.

For example:

5 g = 0.005 kg

or:

5 × 10⁻³ kg.

Do the conversion before using:

E = mc².

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Common Misconception: E = mc² Is the Total Energy Formula in Every Situation

For an object at rest:

E₀ = mc².

For a moving particle:

E = γmc².

More generally:

E² = p²c² + m²c⁴.

Therefore, the famous equation is best understood as the relationship between:

rest mass and rest energy.

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Common Misconception: Nuclear Reactions Convert All Mass into Energy

Usually they do not.

In fission and fusion, only a relatively small difference between the initial and final rest masses corresponds to the:

released energy.

Most of the mass remains associated with the:

reaction products.

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Common Misconception: Only Nuclear Reactions Involve E = mc²

Mass-energy equivalence applies to:

all physical systems.

It applies to:

• nuclear reactions
• chemical reactions
• heated objects
• batteries
• compressed springs
• particle collisions
• matter-antimatter processes

The reason nuclear reactions are emphasized is that their mass changes are much larger and easier to observe than those of:

ordinary processes.

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Common Misconception: Mass Is Simply "Destroyed"

Modern physics does not require mass to be independently conserved in every process.

Instead, the complete system obeys conservation of:

energy and momentum.

Changes in rest mass correspond to changes in other forms of:

energy.

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Why E = mc² Changed Physics

Mass-energy equivalence transformed our understanding of:

matter and energy.

It helped explain:

• nuclear binding energy
• stellar energy production
• nuclear fission
• nuclear fusion
• particle-antiparticle annihilation
• particle creation
• high-energy particle collisions

It revealed that mass is deeply connected to the:

energy content of physical systems.

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From Stars to Subatomic Particles

The same fundamental principle helps explain processes across an extraordinary range of scales.

Inside stars:

mass differences correspond to released energy.

Inside atomic nuclei:

binding energy contributes to nuclear mass.

In particle accelerators:

collision energy can produce new particles.

In matter-antimatter interactions:

rest energy can appear in other particles and radiation.

The same relationship connects them:

E = mc².

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Check Your Understanding

1. What does E = mc² describe?

2. What does each symbol in the equation represent?

3. What is the approximate value of c?

4. Calculate c².

5. Why does a small mass correspond to a large amount of energy?

6. Calculate the rest energy of 2.0 kg of matter.

7. Calculate the rest energy of 0.50 kg of matter.

8. Calculate the energy equivalent of 2.0 g of mass.

9. Rearrange E = mc² to make m the subject.

10. Calculate the mass equivalent of 9.0 × 10¹³ J.

11. What is meant by rest energy?

12. Explain the meaning of mass-energy equivalence.

13. What is a mass defect?

14. How is mass defect related to nuclear binding energy?

15. Explain how mass-energy equivalence applies to nuclear fission.

16. Explain how it applies to nuclear fusion in stars.

17. Why does a charged battery have slightly more mass than the same discharged battery?

18. Why can particle collisions produce new massive particles?

19. Why is E = mc² not the complete energy equation for a moving particle?

20. Explain why mass-energy equivalence is important to modern physics.

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Key Terms

• Mass-energy equivalence: Principle that mass contributes to the energy of a physical system.
• Rest mass: Invariant mass of an object measured in its rest frame.
• Rest energy: Energy associated with rest mass, given by E₀ = mc².
• Speed of light (c): Fundamental invariant speed, approximately 3.00 × 10⁸ m/s.
• Mass defect: Difference between masses associated with separated components and the corresponding bound system.
• Binding energy: Energy required to separate a bound system into its components.
• Nuclear fission: Process in which a heavy nucleus splits into smaller nuclei.
• Nuclear fusion: Process in which lighter nuclei combine to form heavier nuclei.
• Annihilation: Process in which a particle and antiparticle transform into other particles while conserving energy and momentum.
• Pair production: Production of a particle-antiparticle pair from sufficient available energy under appropriate conditions.
• Relativistic energy: Energy described using the equations of Special Relativity.
• Relativistic momentum: Momentum of a particle calculated using relativistic mechanics.
• Photon: Quantum of electromagnetic radiation with zero rest mass.
• Joule: SI unit of energy.
• Conservation of energy: Principle that total energy of an isolated system remains constant.

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Key Takeaways

• Einstein's equation E = mc² expresses the relationship between rest mass and rest energy.
• E represents energy, m represents rest mass, and c represents the speed of light.
• The speed of light is approximately 3.00 × 10⁸ m/s.
• Therefore, c² ≈ 9.00 × 10¹⁶ m²/s².
• Because c² is enormous, even a small amount of mass corresponds to a very large amount of energy.
• To calculate rest energy, use E = mc².
• To calculate mass equivalent, use m = E/c².
• Mass should normally be converted to kilograms before using the equation in SI units.
• E = mc² specifically describes the rest energy associated with mass.
• For a moving particle, the more general relationship is E² = p²c² + m²c⁴.
• Nuclear binding energy is associated with measurable differences in rest mass.
• Nuclear fission releases energy because the final products have lower total rest energy than the initial system.
• Nuclear fusion in stars also releases energy through differences in binding energy and rest mass.
• The Sun's energy ultimately comes from nuclear fusion.
• Matter-antimatter annihilation provides another striking example of mass-energy relationships.
• Energy can also contribute to the creation of massive particles when conservation laws and energy requirements are satisfied.
• A system with more stored energy has slightly greater mass.
• A charged battery therefore has slightly more mass than the same battery when discharged.
• A heated object has slightly more mass than the same object when cooler.
• Mass-energy equivalence applies to chemical processes as well as nuclear processes, although chemical mass changes are extremely small.
• Modern physics emphasizes conservation of total energy and momentum, rather than treating mass as a separately conserved quantity in all processes.
• Mass-energy equivalence connects Special Relativity to nuclear physics, astrophysics, and particle physics.
• E = mc² is important not merely because it predicts large energies, but because it reveals a fundamental relationship between the mass and energy content of physical systems.