Binding Energy

Binding energy is a fundamental concept in the field of physics, particularly in the study of atomic and nuclear systems. Binding Energy is defined as the energy required to hold together the constituents of a system, such as the particles within an atomic nucleus or the electrons surrounding an atomic nucleus. Understanding binding energy is crucial for comprehending the stability, structure, and behaviour of atoms, molecules, and nuclei.

Binding energy specifically refers to the amount of energy needed to disassemble a nucleus into its individual protons and neutrons. The binding energy of nuclei is a positive value because every nucleus needs net energy to isolate them into neutrons and protons. Binding Energy is applicable to atoms and ions bound together in crystals.

What is Binding Energy?

The binding energy of a nucleus arises from the interactions between many constituent particles. It is primarily governed by the strong nuclear force as a result of the formation of a stable and tightly bounded system. The binding energy represents energy associated with those interactions that act as a measure of the stability and coherence of the nucleus.

By using Formula, a binding energy of a nucleus can be calculated that takes into account the number of protons and neutrons in the nucleus, as well as the mass of these particles. This calculation provides valuable insights into the nucleus’s stability, allowing scientists to predict whether a nucleus is likely to undergo radioactive decay or remain stable.

The binding energy is a concept that explains other systems in physics. In atomic and molecular systems, it explains the energy required to keep electrons inside an atom or hold atoms together in a molecule. It influences chemical reactions, the stability of materials, and the properties of substances.

Binding Energy Definition

Binding Energy is the energy required to separate any system of particles into individual particles. In the context of nuclear physics binding energy is the energy required to separate the nucleus into its constituent nucleons i.e., proton and neutron.

When we talk about nucleons, we see that they are protons and neutrons plus other nuclear particles which make up the nucleus of an atom. 

What is Binding Energy Formula?

The formula for calculating the binding energy (BE) of a nucleus is given by:

BE = [Z × m_p + (A – Z) × m_n – m] × c²

Where,

• BE represents the binding energy of the nucleus
• Z is the number of protons (atomic number) in the nucleus
• m_p is the mass of a proton [1.00728 atomic mass units]
• A is the total number of nucleons (protons + neutrons) in the nucleus
• m_n is the mass of a neutron [1.00867 amu]
• m represents the actual mass of the nucleus
• c is the speed of light [3.00 x 10⁸ m/s]

In the formula, the terms (Z × m_p) and ((A – Z) × m_n) represent the total mass of all the protons and neutrons in the nucleus, respectively. The term m represents the actual mass of the nucleus, which may be slightly different from the sum of the individual masses due to the mass defect.

Other Formula of Binding Energy

Another formula that can be used to calculate the binding energy is:

E = mc²

Where,

• E is the Energy
• m is the Mass
• c is the Speed of Light

E = mc² is one of the most famous equations in physics and is derived from Einstein’s theory of special relativity. In the context of binding energy, we know that the nuclei of any atom are composed of protons and neutrons which are held by strong nuclear forces. The total mass of a nucleus is slightly less than the sum of the masses of its individual nucleons when they are at rest (measured independently outside the nucleus). This missing mass is known as the mass defect (Δm).

Thus the above formula can be written as:

E_b = Δmc²

Where E_b is the binding energy.

For example, the mass defect of an atom of deuterium is 0.0023884, thus its binding energy from the above formula came out to be nearly equal to 2.23 MeV. This implies that the energy required to disintegrate an atom of deuterium is 2.23 MeV.

Note: This equation is also known as mass-energy equivalence, as this shows that mass and energy are interchangeable.

Read more about Mass Energy Equivalence.

Types of Binding Energy

There are various types of binding energy, in the context of atom and nucleus some of the types are discussed as follows:

Electron Binding Energy

Electron binding energy, or ionization energy, is the amount of energy needed to release an electron from an atom or solid material.

Element	Electron Binding Energy (Ionization Energy)
Hydrogen	13.6 eV
Helium	24.6 eV
Carbon	11.3 eV
Oxygen	13.6 eV
Sodium	5.1 eV
Chlorine	12.97 eV

Atomic Binding Energy

In the context of atoms, binding energy refers to the energy required to remove an electron from an atom and release it as a free particle (ionization energy) or the energy released when an electron is added to an atom (electron affinity).

Nuclear Binding Energy

In the context of atomic nuclei, binding energy refers to the energy required to disassemble a nucleus into its individual protons and neutrons or the energy released when protons and neutrons come together to form a nucleus. It is also known as nuclear binding energy or nuclear binding per nucleon.

Learn more about Nuclear Binding Energy.

Bond-Dissociation Energy

Bond-Dissociation Energy is the measure of energy between atoms in a chemical bond i.e., In other words, we can say that the energy required to break any complex molecule to its constituent atoms is known as Bond-Dissociation Energy. It is also called Bond Energy. For example, in the following table, the bond energy for various bonds is given:

Bond Type	Bond Energy (kJ/mol)	Bond Energy (eV)
Hydrogen (H-H)	436	4.52
Oxygen (O=O)	498	5.16
Carbon-Hydrogen (C-H)	413	4.28
Carbon-Carbon (C-C)	348	3.61
Carbon-Oxygen (C-O)	410	4.25
Nitrogen-Nitrogen (N-N)	945	9.80

Binding Energy Per Nucleon

Binding Energy Per Nucleon is the average energy required to remove an individual nucleon from the nucleus, which means the higher the Binding Energy Per Nucleon or BEN more stable the nucleus is, and the lower the BEN value for any nucleus less stable it is.

Binding Energy Per Nucleon Formula

The binding energy per nucleon (BEN), which is defined by, 

BEN = E_b / A

Where,

• E_b is the Binding Enegery
• A is the Atomic Mass of Nucleus

Applications of Binding Energy

There are some applications of Binding Energy that are :

• Nuclear Power: In the production of nuclear power, the binding energy is used to estimate the amount of power generated and which can further help us take precautionary measures against any catastrophe that can happen. As nuclear power is very dangerous we know that from America’s nuclear bombing on Japan at Hiroshima and Nagasaki. Also, understanding the binding energy of different isotopes helps in optimizing reactor designs and fuel choices.
• Nuclear Medicine: The one of use cases of binding energy and nuclear technology in the medical field is positron emission tomography (PET), which involves the use of radioactive isotopes to observe the parameters in the human body such as blood flow, metabolism, neurotransmitters, and radio-labelled drugs.
• Nuclear Weapons: In modern world warfare, nuclear weapons are the epitome of mass destruction and every nation wants to develop such weapons to protect themself against other powerful nations. In designing nuclear weapons, the binding energy is used to find the output of the weapon.
• Binding energy is also applied in determining whether fusion or fission will be favourable.

Solved Example of Binding Energy Formula

Example 1: Calculate the binding energy per nucleon for a nucleus with atomic number (Z) = 92, mass number (A) = 238, and a binding energy (BE) of 1782.5 MeV.

Solution:

BE/A = BE / A

BE/A = 1782.5 MeV / 238

(Calculate the result)

BE/A ≈ 7.49 MeV/nucleon

Example 2: Calculate the binding energy for a carbon-12 nucleus (6 protons and 6 neutrons) given the following values:

• Mass of a proton (m_p) = 1.00728 atomic mass units (amu)
• Mass of a neutron (m_n) = 1.00867 amu
• Measured mass of carbon-12 nucleus (m) = 12.00000 amu
• Speed of light (c) = 3.00 x 10⁸ meters per second (m/s)

Solution:

We will use the formula: BE = (Z×m_p + (A – Z)×m_n – M) × c²

Where BE represents Binding Energy

Calculate the number of protons (Z), number of neutrons (N), and atomic mass number (A):

• Z (number of protons) = 6
• N (number of neutrons) = 6
• A (atomic mass number) = Z + N = 6 + 6 = 12

Substitute the given values into the formula:

BE = (Z×m_p + (A – Z)×m_n – M) × c²

BE = (6 × 1.00728 + (12 – 6) × 1.00867 – 12.00000) × (3.00 × 10⁸)²

BE = (6.04368 + 6.05196 – 12.00000) × (3.00 x 10⁸)²

BE = 0.09564 * (9.00 x 10¹⁶)

BE = 8.610 x 10¹⁵ Joules

Therefore, the binding energy for a carbon-12 nucleus is approximately 8.610 x 10¹⁵ Joules.

Note: Joules is the unit of Energy on the Binding Energy Formula

FAQs of Binding Energy Formula

1. Define Binding Energy.

It is defined as amount of energy required to hold together the constituents of a system, such as the particles within an atomic nucleus or the electrons surrounding an atomic nucleus.

2. How is Binding Energy-Related to Stability of Atomic Nuclei?

The binding energy of a nucleus represents the energy associated with strong nuclear force that holds both protons and neutrons together. High binding energy represents more stable nucleus as it requires more energy to break apart.

3. What is Significance of Mass Defect in Calculation of Binding Energy?

The mass defect represents the difference between sum of the masses of individual nucleons in a nucleus and the actual mass of the nucleus. The mass defect is calculated when multiplied by the square of the speed of light (c²), is converted into binding energy according to Einstein’s mass-energy equivalence equation (E = mc²).

4. How does Binding Energy per Nucleon Affect Stability of a Nucleus?

The binding energy per nucleon provides a measure of the average energy required to remove a nucleon from the nucleus. Higher values indicate stronger binding forces and greater stability.

5. Write Formula for Binding Energy.

The formula for calculating the binding energy (BE) of a nucleus is as follows:

BE = (Z×m_p + (A – Z)×m_n – M) × c²

6. Can Binding Energy Formula be Used for All Elements?

Yes, the binding energy formula can be used for all elements to approximate the binding energy of their nuclei, although more complex models may be required for unstable or heavy nuclei.