Explain Binding Energy per Nucleon and the Binding Energy Curve

Binding Energy per Nucleon and the Binding Energy Curve

Defination: Binding Energy per Nucleon

The binding energy per nucleon is defined as the total binding energy of a nucleus divided by its mass number .

$E_{bn}=\frac{E_b}{A}$

where:

 $E_{b}$= Total binding energy of the nucleus

 A= Mass number of the nucleus

The binding energy per nucleon is a measure of the stability of a nucleus. A larger value of indicates a more stable nucleus.

Binding Energy per Nucleon Curve

The variation of binding energy per nucleon with mass number shows the following features:

1. The binding energy per nucleon increases rapidly for light nuclei.

2. It remains nearly constant at about 8 MeV per nucleon for nuclei with

30 < A < 170

3. The maximum value of binding energy per nucleon is about 8.75 MeV for

A = 56

4. For very heavy nuclei, the binding energy per nucleon decreases gradually and becomes about 7.6 MeV for

A = 238

5. Thus, light nuclei and heavy nuclei are less tightly bound than medium-mass nuclei.

Conclusions from the Binding Energy Curve

1. Strong Attractive Nuclear Force

The binding energy per nucleon is several MeV. This shows that the nuclear force is a strong attractive force that binds nucleons together inside the nucleus.

2. Saturation Property of Nuclear Force

The binding energy per nucleon is nearly constant for medium-mass nuclei. This indicates that nuclear forces have the saturation property. Since nuclear forces are short-ranged, a nucleon interacts mainly with its nearest neighbours.

3. Nuclear Fission

A very heavy nucleus with

$A \approx 240$

has lower binding energy per nucleon than a nucleus with

$A \approx 120$

Therefore, if a nucleus of mass number splits into two nuclei of mass number , the binding energy per nucleon increases and energy is released.

This process is called nuclear fission.

4. Nuclear Fusion

Two very light nuclei with

$A \le 10$

can combine to form a heavier nucleus. The resulting nucleus has a higher binding energy per nucleon than the original nuclei.

Therefore, energy is released during the process.

This process is called nuclear fusion and is the source of the Sun's energy.

Important Points

Binding energy per nucleon:

$E_{bn}=\frac{E_b}{A}$

Nearly constant binding energy per nucleon:

30 < A < 170

Average value:

$E_{bn}\approx 8\ \text{MeV}$

Maximum binding energy per nucleon:

$E_{bn}\approx 8.75\ \text{MeV}$

for

A = 56

Binding energy per nucleon for uranium region:

$E_{bn}\approx 7.6\ \text{MeV}$

for

A = 238

Fission: Heavy nucleus splits into two medium-mass nuclei and releases energy.

Fusion: Two very light nuclei combine to form a heavier nucleus and release energy.

Summary

The binding energy per nucleon curve shows that nuclei around are the most stable. Medium-mass nuclei have nearly constant binding energy per nucleon of about MeV. The curve explains the release of energy in both nuclear fission of heavy nuclei and nuclear fusion of light nuclei.