4. Mass-Energy Equivalence

Mass and Energy Are Connected

Before the development of modern physics, mass and energy were often treated as completely separate quantities.

Albert Einstein showed that they are fundamentally connected.

His famous mass-energy relationship is:

E = mc²

where:

E = energy, measured in joules (J)

m = mass, measured in kilograms (kg)

c = speed of light = 3.00 × 10⁸ m/s

This equation tells us that mass is a form of energy.

Because the speed of light squared is an enormous number, even a very small amount of mass corresponds to a very large amount of energy.

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Understanding E = mc²

The speed of light is approximately:

c = 3.00 × 10⁸ m/s

Squaring this gives:

c² = 9.00 × 10¹⁶ m²/s²

Therefore:

E = m × 9.00 × 10¹⁶

This enormous conversion factor means that a tiny change in mass can correspond to a substantial amount of energy.

For example, a mass of only:

0.001 kg

has a mass-energy equivalent of:

E = (0.001)(3.00 × 10⁸)²

E = 9.00 × 10¹³ J

That is:

90 trillion joules

This does not mean that ordinary objects spontaneously release all their rest energy. It shows the enormous amount of energy associated with mass.

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Mass Changes in Nuclear Physics

In nuclear physics, we are usually interested in a change in mass, rather than converting the entire mass of an object.

For this reason, the equation is often written:

ΔE = Δmc²

where:

ΔE = change in energy

Δm = change in mass

If the products of a nuclear process have slightly less mass than the starting particles, the difference in mass appears as other forms of energy.

Therefore:

mass decrease → energy released

The energy may appear as:

• kinetic energy of particles
• electromagnetic radiation such as gamma rays
• energy carried by other emitted particles

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Is Mass Really Destroyed?

It is more accurate to say that mass-energy is conserved.

Consider a nuclear reaction:

Initial particles → final particles + released energy

If the final particles have less rest mass than the initial particles, the difference has appeared as other forms of energy.

So we can think of:

rest mass-energy → kinetic energy + radiation + other energy

The total mass-energy of the complete isolated system remains conserved.

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Example 1: Energy from a Mass Change

Suppose a nuclear reaction results in a mass decrease of:

2.00 × 10⁻²⁹ kg

Calculate the energy released.

Use:

ΔE = Δmc²

Substitute:

ΔE = (2.00 × 10⁻²⁹)(3.00 × 10⁸)²

First calculate:

(3.00 × 10⁸)² = 9.00 × 10¹⁶

Therefore:

ΔE = (2.00 × 10⁻²⁹)(9.00 × 10¹⁶)

ΔE = 1.80 × 10⁻¹² J

So:

Energy released = 1.80 × 10⁻¹² J

This may appear small, but remember that this is the energy from a change involving only a tiny number of particles. A macroscopic sample contains an enormous number of nuclei.

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Example 2: A Larger Mass Change

Suppose:

Δm = 5.00 × 10⁻⁶ kg

Calculate the equivalent energy.

ΔE = Δmc²

ΔE = (5.00 × 10⁻⁶)(3.00 × 10⁸)²

ΔE = (5.00 × 10⁻⁶)(9.00 × 10¹⁶)

ΔE = 4.50 × 10¹¹ J

Even a mass change of only a few millionths of a kilogram corresponds to an enormous amount of energy.

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Rearranging the Equation

Sometimes the energy is known and the mass change must be calculated.

Starting with:

ΔE = Δmc²

divide both sides by c²:

Δm = ΔE / c²

This allows us to determine how much mass corresponds to a particular amount of energy.

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Example 3: Finding the Mass Change

Suppose a process releases:

1.80 × 10¹⁴ J

Calculate the equivalent mass change.

Use:

Δm = ΔE / c²

Substitute:

Δm = (1.80 × 10¹⁴) / (3.00 × 10⁸)²

Δm = (1.80 × 10¹⁴) / (9.00 × 10¹⁶)

Δm = 2.00 × 10⁻³ kg

Therefore:

Δm = 0.00200 kg

or:

2.00 g

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Using Atomic Mass Units

Working in kilograms is not always convenient when dealing with individual nuclei.

Nuclear masses are commonly measured in atomic mass units (u).

A useful relationship is:

1 u = 1.6605 × 10⁻²⁷ kg

Using E = mc², this corresponds to an energy of approximately:

1 u = 931.5 MeV/c²

Therefore, a mass change of:

1 u

corresponds to an energy change of:

931.5 MeV

For nuclear calculations, we can therefore use:

ΔE (MeV) = Δm (u) × 931.5

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Example 4: Using Atomic Mass Units

Suppose a nuclear reaction has a mass decrease of:

0.0250 u

Calculate the energy released.

Use:

ΔE = Δm × 931.5

ΔE = 0.0250 × 931.5

ΔE ≈ 23.3 MeV

Therefore:

Energy released ≈ 23.3 MeV

This method is much quicker than first converting atomic mass units into kilograms.

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Joules and Electronvolts

Nuclear energies can be expressed in either joules or electronvolts.

An electronvolt (eV) is a very small unit of energy.

Useful units include:

1 keV = 10³ eV

1 MeV = 10⁶ eV

1 GeV = 10⁹ eV

Nuclear reaction energies are commonly measured in MeV.

A useful conversion is:

1 eV ≈ 1.602 × 10⁻¹⁹ J

Therefore:

1 MeV ≈ 1.602 × 10⁻¹³ J

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Mass-Energy and Nuclear Binding

Mass-energy equivalence explains the connection between mass defect and nuclear binding energy.

When separate protons and neutrons combine to form a nucleus:

nucleons → bound nucleus + energy

The bound nucleus has slightly less mass than the separated nucleons.

The difference is the mass defect:

Δm = mass of separate nucleons − mass of nucleus

The corresponding energy is the nuclear binding energy:

Eᵦ = Δmc²

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Mass-Energy in Nuclear Reactions

Mass-energy equivalence is essential for understanding nuclear reactions.

For any nuclear reaction, we can compare the total rest mass before and after the reaction.

Mass before > mass after

means that energy can be released.

The mass difference corresponds to:

ΔE = Δmc²

The released energy often appears primarily as kinetic energy of the reaction products and radiation.

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Nuclear Fusion

In nuclear fusion, light nuclei combine to form heavier nuclei.

For example, stars ultimately convert hydrogen into helium through a sequence of nuclear reactions.

The total rest mass of the final products is slightly less than the total rest mass of the original particles.

That mass difference is released as energy.

The relationship is:

mass of initial particles > mass of final products

Therefore:

mass difference → released energy

This is the fundamental connection between Einstein's equation and the energy produced by stars.

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Nuclear Fission

Mass-energy equivalence also explains the energy released during nuclear fission.

In fission, a heavy nucleus splits into smaller nuclei.

For example:

heavy nucleus → smaller nuclei + neutrons + energy

The combined rest mass of the products can be slightly smaller than the initial rest mass.

The mass difference is released as energy.

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Why Can Both Fusion and Fission Release Energy?

This can seem confusing.

Fusion joins nuclei together.

Fission splits nuclei apart.

How can both release energy?

The answer comes from nuclear binding energy.

Light nuclei can become more tightly bound by fusion.

Very heavy nuclei can become more tightly bound by fission.

In both cases, the products can have a lower total rest mass-energy than the starting nuclei.

The difference is released as energy.

This connects three major ideas:

Mass defect ↔ Binding energy ↔ E = mc²

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The Q-Value of a Nuclear Reaction

The energy released or absorbed in a nuclear reaction is often called the Q-value.

A simple expression is:

Q = (minitial − mfinal)c²

If:

Q > 0

energy is released.

If:

Q < 0

energy must be supplied.

This gives physicists a quick way to determine whether a particular nuclear process releases or requires energy.

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Example 5: Energy Released in a Nuclear Reaction

Suppose the total mass before a reaction is:

4.0350 u

and the total mass after the reaction is:

4.0080 u

Step 1: Calculate the mass difference

Δm = 4.0350 − 4.0080

Δm = 0.0270 u

Step 2: Convert mass to energy

ΔE = 0.0270 × 931.5

ΔE ≈ 25.2 MeV

Therefore:

Energy released ≈ 25.2 MeV

Because the final mass is smaller than the initial mass, the reaction releases energy.

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A Reliable Problem-Solving Method

For mass-energy problems, use this sequence.

Step 1: Identify what is given.

Is the mass in:

kg

or:

u?

Step 2: Choose the correct relationship.

For kilograms:

ΔE = Δmc²

For atomic mass units:

ΔE(MeV) = Δm(u) × 931.5

Step 3: Substitute carefully.

Pay particular attention to powers of ten.

Step 4: Include units.

Energy should normally be expressed in:

J or MeV

Step 5: Interpret your answer.

Ask whether the process:

releases energy

or:

requires energy.

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Common Mistakes

Mistake 1: Forgetting to square c

The equation is:

E = mc²

not:

E = mc

Remember:

(3.00 × 10⁸)² = 9.00 × 10¹⁶

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Mistake 2: Using grams instead of kilograms

If you are calculating energy in joules using SI units, mass must be in:

kilograms

For example:

2 g = 0.002 kg

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Mistake 3: Using the total mass instead of the mass change

In most nuclear reaction calculations, we need:

Δm

not the entire mass of the nucleus.

Calculate:

Δm = initial mass − final mass

and then use:

ΔE = Δmc²

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Mistake 4: Confusing MeV with MeV/c²

Mass can be expressed as:

MeV/c²

Energy is expressed as:

MeV

The conversion:

1 u ≈ 931.5 MeV/c²

means that a mass difference of 1 u corresponds to an energy difference of approximately:

931.5 MeV

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Did You Know?

The Sun converts roughly 4 million tonnes of mass into other forms of energy every second.

That sounds enormous, but the Sun itself has an enormous mass, so this represents only a tiny fraction of its total mass.

Einstein's mass-energy relationship explains how such a relatively small mass change can provide the tremendous energy radiated by the Sun.

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Connecting the Ideas

Mass-energy equivalence connects several major concepts in nuclear physics:

Nuclear composition

↓

Nucleons bind together

↓

Mass defect occurs

↓

Mass corresponds to binding energy

↓

E = Δmc²

↓

Nuclear reactions rearrange nuclear binding

↓

Differences in mass-energy can be released as kinetic energy and radiation

This relationship is fundamental to understanding radioactivity, nuclear fusion, nuclear fission, stars, and nuclear energy.

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Key Terms

Mass-energy equivalence – The principle that mass and energy are different forms of the same physical quantity.

Rest energy – The energy associated with the rest mass of an object, given by E = mc².

Mass defect – The difference between the mass of separated nucleons and the mass of the bound nucleus.

Binding energy – The energy required to completely separate a nucleus into its nucleons.

Atomic mass unit (u) – A unit commonly used for atomic and nuclear masses.

Electronvolt (eV) – A unit of energy commonly used in atomic and particle physics.

Mega-electronvolt (MeV) – One million electronvolts.

Q-value – The net energy released or absorbed during a nuclear reaction.

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Key Takeaways

• Einstein's mass-energy relationship is E = mc².
• Mass and energy are fundamentally related.
• In nuclear physics, we often use ΔE = Δmc² to calculate energy associated with a mass change.
• Because c² is extremely large, a tiny mass change can correspond to substantial energy.
• Mass is not simply destroyed; total mass-energy is conserved.
• Nuclear binding energy is related to mass defect through Eᵦ = Δmc².
• When mass is measured in atomic mass units, 1 u corresponds to approximately 931.5 MeV of energy.
• Nuclear fusion releases energy when the products have lower total rest mass-energy than the starting particles.
• Nuclear fission can release energy for the same reason.
• The energy of a nuclear reaction can be calculated using its Q-value.
• Mass-energy equivalence provides the connection between mass defect, binding energy, fusion, fission, and nuclear energy.