What Is Mass Defect?

If we measure the mass of a nucleus, we discover something surprising:

The mass of the nucleus is slightly less than the total mass of the individual protons and neutrons that make it.

The difference between these two masses is called the mass defect.

Mass defect = mass of separate nucleons − mass of nucleus

Using symbols:

Δm = (Zmp + Nmn) − mnucleus

where:

Δm = mass defect

Z = number of protons

N = number of neutrons

mp = mass of one proton

mn = mass of one neutron

mnucleus = measured mass of the nucleus

The mass defect is closely connected to nuclear binding energy.

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Why Is There a Mass Difference?

Imagine starting with separate protons and neutrons.

When these nucleons come together to form a nucleus, the strong nuclear interaction creates a bound system.

As the nucleus forms, energy is released.

Because mass and energy are related, the loss of energy from the system corresponds to a decrease in its mass.

Einstein's equation describes this relationship:

E = mc²

Therefore, the final bound nucleus has less mass-energy than the original separated nucleons.

The difference in mass is the mass defect.

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Has the Mass Disappeared?

No.

The word "defect" can make it sound as though some mass has mysteriously vanished.

Instead, some of the original mass-energy has been released from the system as energy when the nucleus formed.

So:

separate nucleons → nucleus + released energy

The nucleus has a lower total mass-energy than the separated nucleons.

To completely separate the nucleus again, this energy must be supplied back to the system.

That required energy is the nuclear binding energy.

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Mass Defect and Binding Energy

Mass defect and binding energy describe the same change from two different perspectives.

Mass defect → difference in mass

Binding energy → corresponding difference in energy

They are connected by:

Eᵦ = Δmc²

where:

Eᵦ = nuclear binding energy

Δm = mass defect

c = speed of light

A larger mass defect generally corresponds to a larger total binding energy.

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Calculating Mass Defect

To calculate mass defect, we compare:

total mass of separate nucleons

with:

measured mass of the nucleus

The general method is:

Step 1: Determine the number of protons

Protons = Z

Step 2: Determine the number of neutrons

N = A − Z

Step 3: Calculate the mass of the separate nucleons

Mass of nucleons = (Z × mp) + (N × mn)

Step 4: Subtract the nuclear mass

Δm = mass of separate nucleons − mass of nucleus

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Example 1: A Simple Nucleus

Suppose a nucleus contains:

2 protons

2 neutrons

For this simplified example, use:

mass of proton = 1.0073 u

mass of neutron = 1.0087 u

Measured nuclear mass:

4.0015 u

Step 1: Calculate the mass of the protons

2 × 1.0073 = 2.0146 u

Step 2: Calculate the mass of the neutrons

2 × 1.0087 = 2.0174 u

Step 3: Calculate the total mass of the separate nucleons

2.0146 + 2.0174 = 4.0320 u

Step 4: Calculate the mass defect

Δm = 4.0320 − 4.0015

Δm = 0.0305 u

Therefore:

Mass defect = 0.0305 u

The bound nucleus has 0.0305 u less mass than the equivalent separated nucleons.

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Converting Mass Defect to Binding Energy

The atomic mass unit can be converted into energy using:

1 u ≈ 931.5 MeV/c²

Therefore:

Binding energy (MeV) = mass defect (u) × 931.5

For our previous example:

Δm = 0.0305 u

Therefore:

Eᵦ = 0.0305 × 931.5

Eᵦ ≈ 28.4 MeV

So the nucleus has a binding energy of approximately:

28.4 MeV

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What Does 28.4 MeV Mean?

A binding energy of 28.4 MeV means approximately 28.4 MeV of energy would have to be supplied to completely separate the nucleus into its individual protons and neutrons.

The reverse is also true.

Approximately 28.4 MeV would be released when the separated nucleons form that nucleus.

Therefore:

nucleus + 28.4 MeV → separate nucleons

or:

separate nucleons → nucleus + 28.4 MeV

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Example 2: Carbon-12

Consider a carbon-12 nucleus:

¹²₆C

From the nuclear notation:

A = 12

Z = 6

Therefore:

Protons = 6

Neutrons = 12 − 6 = 6

Suppose we use:

mp = 1.0073 u

mn = 1.0087 u

and the nuclear mass is approximately:

11.9967 u

Mass of separate protons

6 × 1.0073 = 6.0438 u

Mass of separate neutrons

6 × 1.0087 = 6.0522 u

Total mass of separate nucleons

6.0438 + 6.0522 = 12.0960 u

Mass defect

Δm = 12.0960 − 11.9967

Δm = 0.0993 u

The corresponding binding energy is:

Eᵦ = 0.0993 × 931.5

Eᵦ ≈ 92.5 MeV

So the carbon-12 nucleus has a total binding energy of approximately:

92.5 MeV

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Binding Energy per Nucleon

We can go one step further.

Carbon-12 contains 12 nucleons.

Therefore:

Binding energy per nucleon = total binding energy ÷ A

= 92.5 ÷ 12

≈ 7.71 MeV/nucleon

This value tells us the average binding energy associated with each nucleon.

Binding energy per nucleon is particularly useful when comparing the relative stability of different nuclei.

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Example 3: Finding the Missing Nuclear Mass

Mass defect calculations can also be reversed.

Suppose a nucleus contains:

3 protons

4 neutrons

The separate nucleons have a combined mass of:

7.0567 u

The mass defect is:

0.0421 u

Find the mass of the nucleus.

We know:

Δm = mass of separate nucleons − mass of nucleus

Rearrange:

mass of nucleus = mass of separate nucleons − Δm

Therefore:

mass of nucleus = 7.0567 − 0.0421

mass of nucleus = 7.0146 u

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Example 4: Finding Binding Energy from Mass Defect

Suppose:

Δm = 0.0850 u

Calculate the binding energy.

Use:

Eᵦ = Δm × 931.5

Therefore:

Eᵦ = 0.0850 × 931.5

Eᵦ ≈ 79.2 MeV

So:

Binding energy ≈ 79.2 MeV

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Using Kilograms Instead of Atomic Mass Units

Mass defect can also be expressed in kilograms.

In that case, use Einstein's equation directly:

E = Δmc²

where:

c = 3.00 × 10⁸ m/s

For example, suppose:

Δm = 5.00 × 10⁻²⁹ kg

Then:

E = (5.00 × 10⁻²⁹)(3.00 × 10⁸)²

First:

(3.00 × 10⁸)² = 9.00 × 10¹⁶

Therefore:

E = (5.00 × 10⁻²⁹)(9.00 × 10¹⁶)

E = 4.50 × 10⁻¹² J

Even an extremely small mass difference can therefore correspond to a measurable amount of nuclear energy.

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Why Is c² So Important?

The speed of light is:

c ≈ 3.00 × 10⁸ m/s

Therefore:

c² ≈ 9.00 × 10¹⁶ m²/s²

This is an enormous number.

As a result:

tiny mass change × enormous c² = significant energy change

This is why very small changes in nuclear mass can correspond to large energy releases.

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Atomic Mass or Nuclear Mass?

There is an important detail when solving mass-defect problems.

Tables may provide either:

• nuclear masses, or
• atomic masses

An atomic mass includes the electrons surrounding the nucleus.

A nuclear mass does not.

Therefore, you must use masses consistently.

If a problem provides the mass of the nucleus, compare it with the masses of separate protons and neutrons.

If a problem provides neutral atomic masses, calculations are often arranged using the mass of a hydrogen atom instead of a bare proton so that electron masses cancel correctly.

For introductory problems, always follow the mass values provided in the question.

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A Reliable Calculation Method

For a nuclide:

ᴬZX

use this sequence:

1. Protons = Z

2. Neutrons = A − Z

3. Calculate mass of separate nucleons

4. Subtract measured nuclear mass

5. The result is Δm

6. Convert Δm to energy if required

If Δm is in atomic mass units:

Eᵦ(MeV) = Δm(u) × 931.5

If Δm is in kilograms:

Eᵦ(J) = Δm(kg)c²

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Interpreting a Mass Defect Calculation

Suppose two nuclei have these mass defects:

Nucleus B has the larger total mass defect.

Therefore, nucleus B also has the larger total binding energy, because:

Eᵦ = Δmc²

However, we cannot automatically conclude that nucleus B is more stable.

Why?

Because nucleus B may contain many more nucleons.

To compare how tightly different nuclei are bound, we usually calculate:

binding energy per nucleon

This distinction is important:

Mass defect → connected to total binding energy

Binding energy per nucleon → better for comparing how tightly nuclei of different sizes are bound

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Connecting Mass Defect to Nuclear Reactions

Mass defect is also central to understanding why nuclear reactions can release energy.

Suppose a nuclear reaction begins with particles having a total mass:

minitial

and ends with products having total mass:

mfinal

If:

minitial > mfinal

then the difference in mass has been released as energy.

ΔE = Δmc²

This principle explains the enormous energy available from processes such as:

• nuclear fusion
• nuclear fission
• radioactive decay

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Mass Defect and Fusion

When light nuclei undergo fusion, the resulting nucleus can be more tightly bound.

The final system can have less mass than the original nuclei.

The mass difference appears as released energy.

This is one of the reasons stars can produce enormous amounts of energy.

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Mass Defect and Fission

During nuclear fission, a heavy nucleus splits into smaller nuclei.

The resulting nuclei can have a greater binding energy per nucleon.

Again, the total mass of the products is slightly less than the original mass.

That mass difference is converted into released energy.

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Did You Know?

The mass defect of a nucleus is usually only a small fraction of its total mass.

However, because of the enormous conversion factor in:

E = mc²

that tiny mass difference can represent a very large amount of energy.

This is why nuclear reactions can release far more energy per reaction than ordinary chemical reactions.

Chemical reactions mainly rearrange electrons.

Nuclear reactions change the structure and energy of the nucleus itself.

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Key Terms

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

Nuclear binding energy – The energy required to completely separate a nucleus into its individual nucleons.

Nucleon – A proton or neutron.

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

MeV – Mega-electronvolt, a unit of energy commonly used in nuclear physics.

Mass-energy equivalence – The relationship between mass and energy described by E = mc².

Binding energy per nucleon – The average binding energy associated with each nucleon in a nucleus.

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Key Takeaways

• The mass of a bound nucleus is less than the combined masses of its separate protons and neutrons.
• This difference is called the mass defect.
• Mass is not simply lost; the difference corresponds to energy released when the nucleus forms.
• Mass defect can be calculated using Δm = mass of separate nucleons − mass of nucleus.
• Mass defect and binding energy are connected by Eᵦ = Δmc².
• When mass defect is measured in atomic mass units, 1 u corresponds to approximately 931.5 MeV/c².
• A larger mass defect means a larger total binding energy.
• Binding energy per nucleon is more useful than total mass defect when comparing nuclei of different sizes.
• Mass defect helps explain the energy released during fusion, fission, and other nuclear processes.
• Mass and energy are conserved together as mass-energy.