Mass Defect and Binding Energy

Overview

Mass Defect and Binding Energy explains why the mass of a nucleus is less than the total mass of its separate nucleons, and how this missing mass corresponds to energy.

This topic is central to:

• nuclear stability
• energy released in nuclear reactions
• understanding Nuclear Fission
• understanding Nuclear Fusion

Main hub: Nuclear Physics

Definition

Mass defect is the difference between the total rest mass of the separate nucleons and the actual mass of the bound nucleus. The corresponding energy is the binding energy.

Why It Matters

This idea explains:

• why bound nuclei have lower mass than separate nucleons
• why energy is released in nuclear formation and some nuclear reactions
• why binding energy per nucleon is a good measure of stability

Key Representations

Rest Mass and Nuclear Mass

A nucleus contains:

• $Z$ protons
• $N$ neutrons

If all nucleons were separated and at rest, the total rest mass would be:

$$m_{\text{separate}}=Z{m}_p+N{m}_n$$

where:

• $m_p$ = proton mass
• $m_n$ = neutron mass

However, the actual nucleus has measured mass:

$$m_{\text{nucleus}}$$

and:

$$m_{\text{nucleus}}<m_{\text{separate}}$$

Important note:

• if nuclear masses are used, compare bare nucleon masses with nuclear mass
• if atomic masses of neutral atoms are used, use atomic masses consistently so electron masses cancel appropriately
• in that atomic-mass method, hydrogen-1 atomic mass is typically used instead of bare proton mass

Mass Defect

Mass defect is the difference between the total separated-nucleon mass and the nuclear mass:

$$\Delta m=(Z{m}_p+N{m}_n)-m_{\text{nucleus}}$$

Meaning:

• the missing mass has been converted into energy when the nucleus formed

Binding Energy

Binding energy is the minimum energy required to separate a nucleus completely into free nucleons.

Using Einstein’s relation:

$$E=\Delta m{c}^2$$

So:

$$\text{binding energy}=\Delta m{c}^2$$

Physical meaning:

• larger binding energy means the nucleus is harder to break apart
• strongly bound nuclei are generally more stable

Why Nuclear Mass Is Smaller

When nucleons bind together:

• energy is released to the surroundings
• total system energy decreases
• the mass equivalent of that released energy is lost

Hence:

$$m_{\text{nucleus}}<Z{m}_p+N{m}_n$$

Binding Energy per Nucleon

$$\text{binding energy per nucleon}=\frac{\text{binding energy}}{A}$$

where:

$$A=Z+N$$

Why it matters:

• it compares how tightly each nucleon is bound on average

Important distinction:

• total binding energy = whole nucleus
• binding energy per nucleon = average per nucleon

For stability comparisons, use binding energy per nucleon.

Units and Conversion Ideas

Atomic mass unit:

$$1u=1.66\times 10^{-27}\text{kg}$$

Nuclear energies are often measured in electronvolts:

• $1\text{eV}$
• $1\text{MeV}=10^6\text{eV}$

Useful conversion:

$$1u\approx 931\text{MeV}/c^2$$

Therefore, if mass defect is in $u$:

$$E(\text{MeV})=\Delta m\times 931$$

approximately.

Stability Curve Interpretation

Plotting binding energy per nucleon against nucleon number $A$ gives:

• rapid rise for light nuclei
• peak near the iron / nickel region
• slow decline for very heavy nuclei

Meaning:

• nuclei near the peak are the most stable
• light nuclei can release energy by fusion
• heavy nuclei can release energy by fission

Why Higher Binding Energy per Nucleon Means Greater Stability

If binding energy per nucleon is high:

• more energy is needed to remove each nucleon
• nucleons are strongly held together
• the nucleus resists breakup better

Thus it is more stable.

Relation to Nuclear Fission

Very heavy nuclei have lower binding energy per nucleon than medium nuclei.

When a heavy nucleus splits:

• products have higher binding energy per nucleon
• total binding energy increases
• energy is released

See Nuclear Fission.

Relation to Nuclear Fusion

Very light nuclei have lower binding energy per nucleon.

When light nuclei combine:

• the product has higher binding energy per nucleon
• total binding energy increases
• energy is released

See Nuclear Fusion.

Worked Example 1: Mass Defect in Symbolic Form

For helium-4:

$${}_2^4\text{He}$$

It contains:

• $Z=2$ protons
• $N=2$ neutrons

Mass defect:

$$\Delta m=2m_p+2m_n-m_{\text{He nucleus}}$$

Worked Example 2: Convert to Energy

If:

$$\Delta m=0.030u$$

Then:

$$E=0.030\times 931=27.9\text{MeV}$$

Worked Example 3: Binding Energy per Nucleon

If total binding energy is $56\text{MeV}$ for a nucleus with $A=8$:

$$\frac{56}{8}=7.0\text{MeV per nucleon}$$

Summary

• nuclear mass is less than the sum of nucleon masses
• the difference is mass defect
• missing mass corresponds to binding energy
• greater binding energy per nucleon means greater stability
• fission and fusion release energy because products move toward higher binding energy per nucleon

Links

• Nuclear Physics
• Nuclear Fission
• Nuclear Fusion
• Nuclear Physics Common Exam Traps