The mass of the nucleus is always less than the sum of the masses. Binding energy and mass defect. Binding energy and nuclear energy

Nucleons inside the nucleus are held together by nuclear forces. They are held by a certain energy. It is quite difficult to measure this energy directly, but it can be done indirectly. It is logical to assume that the energy required to break the bond of nucleons in the nucleus will be equal to or greater than the energy that holds the nucleons together.

Binding energy and nuclear energy

This applied energy is now easier to measure. It is clear that this value will very accurately reflect the amount of energy that holds nucleons inside the nucleus. Therefore, the minimum energy required to split a nucleus into individual nucleons is called nuclear binding energy.

Relationship between mass and energy

We know that any energy is related to body mass in direct proportion. Therefore, it is natural that the binding energy of a nucleus will depend on the mass of the particles that make up this nucleus. This relationship was established by Albert Einstein in 1905. It is called the law of the relationship between mass and energy. In accordance with this law, the internal energy of a system of particles or rest energy is directly proportional to the mass of the particles that make up this system:

where E is energy, m is mass,
c is the speed of light in vacuum.

Mass defect effect

Now suppose that we split the nucleus of an atom into its constituent nucleons or took a certain number of nucleons from the nucleus. We spent some energy to overcome nuclear forces, since we did work. In the case of the reverse process - the synthesis of a nucleus, or the addition of nucleons to an already existing nucleus, energy, according to the law of conservation, on the contrary, will be released. When the rest energy of a system of particles changes due to some processes, their mass changes accordingly. Formulas in this case will be as follows:

∆m=(∆E_0)/c^2 or ∆E_0=∆mc^2,

where ∆E_0 is the change in the rest energy of the particle system,
∆m – change in particle mass.

For example, in the case of fusion of nucleons and the formation of a nucleus, we experience a release of energy and a decrease in the total mass of nucleons. Mass and energy are carried away by the emitted photons. This is the mass defect effect. The mass of a nucleus is always less than the sum of the masses of the nucleons that make up this nucleus. Numerically, the mass defect is expressed as follows:

∆m=(Zm_p+Nm_n)-M_я,

where M_i is the mass of the nucleus,
Z is the number of protons in the nucleus,
N is the number of neutrons in the nucleus,
m_p – mass of a free proton,
m_n is the mass of a free neutron.

The value ∆m in the two formulas above is the amount by which the total mass of the particles of the nucleus changes when its energy changes due to rupture or fusion. In the case of synthesis, this quantity will be a mass defect.

In order to break a nucleus into separate (free) nucleons that do not interact with each other, it is necessary to do work to overcome nuclear forces, that is, impart a certain energy to the nucleus. On the contrary, when free nucleons combine into a nucleus, the same energy is released (according to the law of conservation of energy).

• The minimum energy required to split a nucleus into individual nucleons is called the nuclear binding energy

How can one determine the value of the binding energy of a nucleus?

The simplest way to find this energy is based on the application of the law on the relationship between mass and energy, discovered by the German scientist Albert Einstein in 1905.

Albert Einstein (1879-1955)
German theoretical physicist, one of the founders of modern physics. Discovered the law of the relationship between mass and energy, created the special and general theories of relativity

According to this law, there is a direct proportional relationship between the mass m of a particle system and the rest energy, i.e., the internal energy E 0 of this system:

where c is the speed of light in vacuum.

If the rest energy of a system of particles as a result of any processes changes by the value ΔE 0 1, then this will entail a corresponding change in the mass of this system by the value Δm, and the relationship between these quantities will be expressed by the equality:

ΔE 0 = Δmс 2.

Thus, when free nucleons merge into a nucleus, as a result of the release of energy (which is carried away by the photons emitted during this process), the mass of the nucleons should also decrease. In other words, the mass of a nucleus is always less than the sum of the masses of the nucleons of which it consists.

The lack of nuclear mass Δm compared to the total mass of its constituent nucleons can be written as follows:

Δm = (Zm p + Nm n) - M i,

where M i is the mass of the nucleus, Z and N are the number of protons and neutrons in the nucleus, and m p and m n are the masses of the free proton and neutron.

The quantity Δm is called the mass defect. The presence of a mass defect is confirmed by numerous experiments.

Let us calculate, for example, the binding energy ΔE 0 of the nucleus of a deuterium (heavy hydrogen) atom, consisting of one proton and one neutron. In other words, let's calculate the energy required to split a nucleus into a proton and a neutron.

To do this, we first determine the mass defect Δm of this nucleus, taking the approximate values ​​of the masses of nucleons and the mass of the nucleus of the deuterium atom from the corresponding tables. According to the tabular data, the proton mass is approximately 1.0073 a. e.m., neutron mass - 1.0087 a. e.m., the mass of the deuterium nucleus is 2.0141 a.m. a.m. So, Δm = (1.0073 a.u.m. + 1.0087 a.u.m.) - 2.0141 a.u. e.m. = 0.0019 a. eat.

To obtain the binding energy in joules, the mass defect must be expressed in kilograms.

Considering that 1 a. e.m. = 1.6605 10 -27 kg, we get:

Δm = 1.6605 10 -27 kg 0.0019 = 0.0032 10 -27 kg.

Substituting this value of the mass defect into the binding energy formula, we obtain:

The energy released or absorbed during any nuclear reactions can be calculated if the masses of interacting nuclei and particles formed as a result of this interaction are known.

Questions

1. What is the binding energy of a nucleus?
2. Write down the formula for determining the mass defect of any nucleus.
3. Write down the formula for calculating the binding energy of a nucleus.

1 The Greek letter Δ (“delta”) usually denotes a change in the physical quantity whose symbol is preceded by this letter.

Atomic nuclei are strongly bound systems of a large number of nucleons.
To completely split the nucleus into its component parts and remove them at large distances from each other, it is necessary to expend a certain amount of work A.

Binding energy is the energy equal to the work that must be done to split a nucleus into free nucleons.

E connection = - A

According to the law of conservation, the binding energy is simultaneously equal to the energy that is released during the formation of a nucleus from individual free nucleons.

Specific binding energy

This is the binding energy per nucleon.

Apart from the lightest nuclei, the specific binding energy is approximately constant and equal to 8 MeV/nucleon. The maximum specific binding energy (8.6 MeV/nucleon) is found in elements with mass numbers from 50 to 60. The nuclei of these elements are the most stable.

As the nuclei are overloaded with neutrons, the specific binding energy decreases.
For elements at the end of the periodic table it is equal to 7.6 MeV/nucleon (for example, for uranium).

Release of energy as a result of nuclear fission or fusion

In order to split a nucleus, a certain amount of energy must be expended to overcome nuclear forces.
In order to synthesize a nucleus from individual particles, it is necessary to overcome the Coulomb repulsive forces (for this, energy must be expended to accelerate these particles to high speeds).
That is, in order to carry out nuclear fission or nuclear synthesis, some energy must be expended.

When a nucleus is fused at short distances, nuclear forces begin to act on the nucleons, which cause them to move with acceleration.
Accelerated nucleons emit gamma rays, which have an energy equal to the binding energy.

At the exit of a nuclear fission or fusion reaction, energy is released.

It makes sense to carry out nuclear fission or nuclear synthesis if the resulting, i.e. the energy released as a result of fission or fusion will be greater than the energy expended
According to the graph, a gain in energy can be obtained either by the fission (splitting) of heavy nuclei, or by the fusion of light nuclei, which is what is done in practice.

Mass defect

Measurements of nuclear masses show that the nuclear mass (Nm) is always less than the sum of the rest masses of the free neutrons and protons composing it.

During nuclear fission: the mass of the nucleus is always less than the sum of the rest masses of the free particles formed.

During nuclear synthesis: the mass of the resulting nucleus is always less than the sum of the rest masses of the free particles that formed it.

The mass defect is a measure of the binding energy of an atomic nucleus.

The mass defect is equal to the difference between the total mass of all nucleons of the nucleus in the free state and the mass of the nucleus:

where Mya is the mass of the nucleus (from the reference book)
Z – number of protons in the nucleus
mp – rest mass of a free proton (from the reference book)
N – number of neutrons in the nucleus
mn – rest mass of a free neutron (from the reference book)

A decrease in mass during the formation of a nucleus means that the energy of the nucleon system decreases.

Calculation of nuclear binding energy

The binding energy of a nucleus is numerically equal to the work that must be expended to split a nucleus into individual nucleons, or the energy released during the synthesis of nuclei from nucleons.
A measure of the binding energy of a nucleus is the mass defect.

The formula for calculating the binding energy of a nucleus is Einstein's formula:
if there is some system of particles that has mass, then a change in the energy of this system leads to a change in its mass.

Here the binding energy of the nucleus is expressed by the product of the mass defect and the square of the speed of light.

In nuclear physics, the mass of particles is expressed in atomic mass units (amu)

in nuclear physics it is customary to express energy in electronvolts (eV):

Let's calculate the correspondence of 1 amu. electronvolts:

Now the calculation formula for binding energy (in electronvolts) will look like this:

EXAMPLE OF CALCULATION OF THE BINDING ENERGY OF THE NUCLEUS OF A HELIUM Atom (He)

By studying the composition of matter, scientists came to the conclusion that all matter consists of molecules and atoms. For a long time, the atom (translated from Greek as “indivisible”) was considered the smallest structural unit of matter. However, further research showed that the atom has a complex structure and, in turn, includes smaller particles.

What does an atom consist of?

In 1911, the scientist Rutherford suggested that the atom has a central part with a positive charge. This is how the concept of the atomic nucleus first appeared.

According to Rutherford's scheme, called the planetary model, the atom consists of a nucleus and elementary particles with a negative charge - electrons, moving around the nucleus, just as the planets orbit the Sun.

In 1932, another scientist, Chadwick, discovered the neutron, a particle that has no electrical charge.

According to modern ideas, the nucleus corresponds to the planetary model proposed by Rutherford. The nucleus carries most of the atomic mass. It also has a positive charge. The atomic nucleus contains protons - positively charged particles and neutrons - particles that do not carry a charge. Protons and neutrons are called nucleons. Negatively charged particles - electrons - move in orbit around the nucleus.

The number of protons in the nucleus is equal to those moving in orbit. Therefore, the atom itself is a particle that does not carry a charge. If an atom gains electrons from others or loses its own, it becomes positive or negative and is called an ion.

Electrons, protons and neutrons are collectively called subatomic particles.

Charge of the atomic nucleus

The nucleus has a charge number Z. It is determined by the number of protons that make up the atomic nucleus. Finding out this quantity is easy: just turn to Mendeleev’s periodic table. The atomic number of the element to which the atom belongs is equal to the number of protons in the nucleus. Thus, if the chemical element oxygen has an atomic number of 8, then the number of protons will also be eight. Since the number of protons and electrons in an atom is the same, there will also be eight electrons.

The number of neutrons is called the isotopic number and is designated by the letter N. Their number can vary in an atom of the same chemical element.

The sum of protons and electrons in the nucleus is called the mass number of the atom and is denoted by the letter A. Thus, the formula for calculating the mass number looks like this: A = Z + N.

Isotopes

When elements have equal numbers of protons and electrons, but different numbers of neutrons, they are called isotopes of a chemical element. There can be one or more isotopes. They are placed in the same cell of the periodic table.

Isotopes are of great importance in chemistry and physics. For example, an isotope of hydrogen - deuterium - in combination with oxygen gives a completely new substance called heavy water. It has a different boiling and freezing point than normal. And the combination of deuterium with another isotope of hydrogen, tritium, leads to a thermonuclear fusion reaction and can be used to generate huge amounts of energy.

Mass of the nucleus and subatomic particles

The size and mass of atoms are negligible in human perception. The size of the nuclei is approximately 10 -12 cm. The mass of an atomic nucleus is measured in physics in the so-called atomic mass units - amu.

For one amu take one twelfth of the mass of a carbon atom. Using the usual units of measurement (kilograms and grams), mass can be expressed by the following equation: 1 amu. = 1.660540·10 -24 g. Expressed in this way, it is called the absolute atomic mass.

Despite the fact that the atomic nucleus is the most massive component of an atom, its size relative to the electron cloud surrounding it is extremely small.

Nuclear forces

Atomic nuclei are extremely stable. This means that protons and neutrons are held in the nucleus by some force. These cannot be electromagnetic forces, since protons are similarly charged particles, and it is known that particles with the same charge repel each other. Gravitational forces are too weak to hold nucleons together. Consequently, particles are held in the nucleus by another interaction - nuclear forces.

Nuclear force is considered the strongest of all existing in nature. Therefore, this type of interaction between the elements of the atomic nucleus is called strong. It is present in many elementary particles, just like electromagnetic forces.

Features of nuclear forces

1. Short action. Nuclear forces, unlike electromagnetic ones, appear only at very small distances, comparable to the size of the nucleus.
2. Charge independence. This feature is manifested in the fact that nuclear forces act equally on protons and neutrons.
3. Saturation. The nucleons of the nucleus interact only with a certain number of other nucleons.

Nuclear binding energy

Another thing closely related to the concept of strong interaction is the binding energy of nuclei. Nuclear bond energy refers to the amount of energy required to split an atomic nucleus into its constituent nucleons. It equals the energy required to form a nucleus from individual particles.

To calculate the binding energy of a nucleus, it is necessary to know the mass of subatomic particles. Calculations show that the mass of a nucleus is always less than the sum of its constituent nucleons. A mass defect is the difference between the mass of a nucleus and the sum of its protons and electrons. Using the relationship between mass and energy (E=mc 2), one can calculate the energy generated during the formation of a nucleus.

The strength of the binding energy of a nucleus can be judged by the following example: the formation of several grams of helium produces the same amount of energy as the combustion of several tons of coal.

Nuclear reactions

The nuclei of atoms can interact with the nuclei of other atoms. Such interactions are called nuclear reactions. There are two types of reactions.

1. Fission reactions. They occur when heavier nuclei, as a result of interaction, decay into lighter ones.
2. Synthesis reactions. The reverse process of fission: nuclei collide, thereby forming heavier elements.

All nuclear reactions are accompanied by the release of energy, which is subsequently used in industry, the military, the energy sector, and so on.

Having familiarized ourselves with the composition of the atomic nucleus, we can draw the following conclusions.

1. An atom consists of a nucleus containing protons and neutrons, and electrons around it.
2. The mass number of an atom is equal to the sum of the nucleons in its nucleus.
3. Nucleons are held together by strong interactions.
4. The enormous forces that give stability to the atomic nucleus are called nuclear binding energies.