Quantum numbers n l m. Quantum numbers

Electron quantum numbers

The principal quantum number n determines the total energy of an electron in a given orbital. It can take any integer value, starting from one (n = 1,2,3, ...). Under principal quantum number, equal to ∞, it is implied that the electron has been given energy sufficient for its complete separation from the nucleus (ionization of the atom).

In addition, it turns out that within certain energy levels, electrons can differ in their energy sublevels. The existence of differences in the energy state of electrons belonging to different sublevels of a given energy level is reflected by the side (sometimes called orbital) quantum number l. This quantum number can take integer values ​​from 0 to n - 1 (l = 0.1, ..., n - 1). Typically, numerical values ​​of l are usually denoted by the following alphabetic symbols:

Value l 0 1 2 3 4
Letter designation s p d f g

In this case, we talk about s -, p-, d -, f -, g - states of electrons, or about s -, p -, d -, f -, g - orbitals.

Orbital is a set of positions of an electron in an atom, i.e. region of space in which an electron is most likely to be found.

The side (orbital) quantum number l characterizes the different energy states of electrons at a given level, determines the shape of the electron cloud, and also orbital moment p- the angular momentum of an electron as it rotates around the nucleus (hence the second name for this quantum number - orbital)

Thus, an electron, having the properties of a particle and a wave, most likely moves around the nucleus, forming an electron cloud, the shape of which is different in the s-, p-, d-, f-, g-states.

We emphasize that the shape of the electron cloud depends on the value of the side quantum number l. So, if l = 0 (s-orbital), then the electron cloud has a spherical shape (spherical symmetry) and has no directionality in space

At l = 1 (p-orbital), the electron cloud has the shape of a dumbbell, i.e. the shape of a body of rotation obtained from the figure eight . The shapes of electron clouds of d -, f - and g - electrons are much more complex.

The movement of an electron in a closed orbit causes the appearance of a magnetic field. The state of the electron, determined by the orbital magnetic moment of the electron (as a result of its orbital motion), is characterized by third quantum number - magnetic m 1. This quantum number characterizes the orientation of the orbital in space, expressing the projection of the orbital angular momentum onto the direction of the magnetic field.

According to the orientation of the orbital relative to the direction of the external magnetic field strength vector, the magnetic quantum number m 1 can take on the values ​​of any integers, both positive and negative, from –l to +l, including 0, i.e. total (2l + 1) values.

Thus, m 1 characterizes the magnitude of the projection of the vector of the orbital angular momentum onto the selected direction. For example, a p-orbital (“dumbbell”) in a magnetic field can be oriented in space in three different positions, since in the case of l = 1 the magnetic quantum number can have three values: -1, 0, +1. Therefore, the electron clouds are elongated along the x, y and z axes, with the axis of each of them perpendicular to the other two.

To explain all the properties of the atom, in 1925 a hypothesis was put forward that the electron has a so-called spin (at first, for clarity, it was believed that this phenomenon was similar to the rotation of the Earth around its axis as it moves in orbit around the Sun). In fact, spin is a purely quantum property of an electron that has no classical analogues. Strictly speaking, spin is the electron's own angular momentum, not associated with motion in space. For all electrons, the absolute value of the spin is always s = 1/2. The projection of the spin onto the axis = (magnetic spin number m s) can have only two values: m s = 1/2 or m s = -1/2.

Since the electron spin s is a constant quantity, it is usually not included in the set of quantum numbers that characterize the motion of an electron in an atom, and they speak of four quantum numbers.

Instructions

The main quantum number takes on integer values: n = 1, 2, 3, …. If n=∞, this implies that the electron has been given ionization energy - energy sufficient to separate it from the nucleus.

Within the same level, sublevels may differ. Such in the energy state of one level are reflected by the side quantum number l (orbital). It can take values ​​from 0 to (n-1). The values ​​of l are usually represented symbolically by letters. The shape of the electron depends on the value of the side quantum number.

The movement of an electron along a closed path provokes the appearance of a magnetic field. The state of the electron, due to the magnetic moment, is characterized by the magnetic quantum number m(l). This is the third quantum number of the electron. It characterizes its orientation in magnetic field space and takes a range of values ​​from (-l) to (+l).

In 1925, scientists proposed the presence of an electron. Spin is understood as the electron's own angular momentum, which is not related to its motion in space. The spin number m(s) can take only two values: +1/2 and -1/2.

According to the Pauli principle, there cannot be two electrons with the same set of four quantum numbers. At least one of them must be different. So, if it is in the first orbit, its main quantum number is n=1. Then clearly l=0, m(l)=0, and for m(s) two options are possible: m(s)=+1/2, m(s)=-1/2. That is why at the first energy level there can be no more than two electrons, and they have different spin numbers.

In the second orbital the main quantum number is n=2. The side quantum number takes two values: l=0, l=1. Magnetic quantum number m(l)=0 for l=0 and takes values ​​(+1), 0 and (-1) for l=1. For each option there are two more spin numbers. So, the maximum possible number of electrons in the second energy level is 8.

For example, the noble gas neon has two energy levels completely filled with electrons. The total number of neon electrons is 10 (2 from the first level and 8 from the second). This gas is inert and does not interact with other substances. Other substances, when entering into reactions, tend to acquire the structure of noble ones.

Helpful advice

To fully explain the structure of the electron shells of atoms for all cases except the Pauli principle, you also need to know the principle of least energy and Hund’s rule.

Sources:

• “The Beginnings of Chemistry”, N.E. Kuzmenko, V.V. Eremin, V.A. Popkov, 2008.

Main quantum number- this is a whole number, which is a definition of the state of an electron at the energy level. An energy level is a set of stationary states of an electron in an atom with similar energy values. Main quantum number determines the distance of an electron from the nucleus, and characterizes the energy of the electrons that occupy this level.

The set of numbers that characterize the state are called quantum numbers. The wave function of an electron in an atom, its unique state is determined by four quantum numbers - principal, magnetic, orbital and splin - the moment of motion of an elementary, expressed in a quantitative value. Main quantum number has n .If the principal quantum number increases, then the orbit and energy of the electron increase accordingly. How less value n, the greater the value of the energy interaction of the electron. If the total energy of electrons is minimal, then the state of the atom is called unexcited or ground. The state of an atom with a high energy value is called excited. At the highest level number electrons can be determined by the formula N = 2n2. When an electron transitions from one energy level to another, the main quantum number.In quantum theory, the statement is that the energy of an electron is quantized, that is, it can take only discrete, specific values. To know the state of an electron in an atom, it is necessary to take into account the energy of the electron, the shape of the electron and other parameters. From the area natural numbers, where n can be equal to 1 and 2 and 3 and so on, the main quantum number can take any value. In quantum theory, energy levels are denoted by letters, and the value n is denoted by numbers. Number of the period where the element is located, equal to the number energy levels in an atom in its ground state. All energy levels consist of sublevels. The sublevel consists of atomic orbitals, which are determined and characterized by the main quantum number m n, orbital number m l and quantum number m ml. The number of sublevels of each level does not exceed the value n. The Schrödinger wave equation is the most convenient for the electronic structure of the atom.

The quantum numerical value of any quantized variable of a microscopic object, characterizing the state of the particle, is called a quantum number. Atom chemical element consists of a core and an electron shell. The state of an electron is characterized by its quantum numbers.

You will need

• Mendeleev table

Instructions

The quantum orbital number 2 can take values ​​from 0 to n-2, characterizing the shape of the orbitals. It also characterizes the subshell on which the electron and . The quantum number 2 also has a letter number. Quantum 2 = 0, 1, 2, 3, 4 correspond to the designations 2 = s, p, d, f, g... Letter designations in the entry denoting electronic configuration chemical element are also present. The quantum number is determined from them. Thus, a subshell can have up to 2*(2l+1) electrons.

Magnetic is the quantum number ml, with l written below as an index. Its data shows the atomic orbital, taking values ​​from 1 to -1. Total (21+1) value.

The electron will be a fermion, having a half-integer spin, which is equal to ½. Its quantum number will take two values, namely: ½ and –½. And also amount to two electrons per axis and be considered the quantum number ms.

Video on the topic

An atom consists of a nucleus and surrounding electrons, which revolve around it in atomic orbitals and form electronic layers (energy levels). The number of negatively charged particles at external and internal levels determines the properties of elements. Number electrons contained in atom, can be found by knowing some key points.

You will need

• - paper;
• - pen;
• - Mendeleev's periodic system.

Instructions

To determine the quantity electrons, use the periodic system D.I. Mendeleev. In this table, the elements are arranged in a specific sequence, which is closely related to their atomic structure. Knowing that positive is always equal to the ordinal number of the element, you can easily find the number of negative particles. After all, it is known that the atom as a whole is neutral, which means the number electrons will be equal to the number and element number in the table. For example, equal to 13. Therefore, the quantity electrons it will have 13, sodium - 11, (Sc), located in the 4th period, in the 3rd group, a secondary subgroup, there are 2 of them. While the Three postulates

All quantum mechanics consists of the principle of relativity of measurements, Heisenberg's uncertainty principle and N. Bohr's principle of complementarity. Everything further in quantum mechanics is based on these three postulates. The laws of quantum mechanics are the basis for studying the structure of matter. With the help of these laws, scientists figured out the structure of atoms and explained periodic table elements, studied the properties of elementary particles, and understood the structure of atomic nuclei. Using quantum mechanics, scientists explained the temperature dependence and calculated the value solids and heat capacity of gases, determined the structure and understood some properties of solids.

The principle of relativity of measurements

This principle is based on measurement results physical quantity depending on the measurement process. In other words, an observed physical quantity is the eigenvalue of the corresponding physical quantity. It is believed that measurement accuracy does not always increase with the improvement of measuring instruments. This fact was described and explained by W. Heisenberg in his famous uncertainty principle.

Uncertainty principle

According to the uncertainty principle, as the accuracy of the measurement of movement speed increases elementary particle, the uncertainty of its location in space increases, and vice versa. This discovery of W. Heisenberg was put forward by N. Bohr as an unconditional methodological position.

So, measurement is the most important research process. To carry out a measurement, a special theoretical and methodological explanation is required. And its absence causes uncertainty. The measurement is characterized by accuracy and objectivity. Modern scientists believe that it is the measurement made with the required accuracy that serves as the main factor theoretical knowledge and eliminates uncertainty.

The principle of complementarity

The means of observation are relative to quantum objects. The principle of complementarity is that the data obtained under experimental conditions cannot be described in a single picture. These data are complementary in the sense that the totality of phenomena gives a complete picture of the properties of the object. Bohr tried on the principle of complementarity not only to the physical sciences. He believed that the capabilities of living beings are multifaceted and depend on each other, and that when studying them, one has to turn to the complementarity of observational data again and again.

The wave function that is a solution to the Schrödinger equation is called orbital. To solve this equation, three quantum numbers are introduced ( n, l And m l )

Principal quantum numbern. it determines the energy of the electron and the size of the electron clouds. The energy of an electron mainly depends on the distance of the electron from the nucleus: the closer the electron is to the nucleus, the lower its energy. Therefore we can say that the principal quantum number n defining

determines the location of the electron at a particular energy level. The principal quantum number has values ​​of a number of integers from 1 before ∞ . When the principal quantum number is equal to 1 (n = 1 ), the electron is at the first energy level, located at the minimum possible distance from the nucleus. The total energy of such an electron is the lowest.

The electron located at the energy level furthest from the nucleus has the maximum energy. Therefore, when an electron moves from a more distant energy level to a closer one, energy is released. Energy levels are indicated in capital letters according to the diagram:

Meaning n…. 1 2 3 4 5

Designation K L M N Q

Orbital quantum numberl . According to quantum mechanical calculations, electron clouds differ not only in size, but also in shape. The shape of the electron cloud is characterized by the orbital or side quantum number. The different shapes of electron clouds determine the change in electron energy within one energy level, i.e. its splitting into energy sublevels. Each shape of the electron cloud corresponds to a certain value of the mechanical momentum of the electron  , determined by the orbital quantum number:

A certain shape of the electron cloud corresponds to a very specific value of the orbital angular momentum of the electron  . Because  can only take discrete values ​​given by the quantum number l, then the shapes of electron clouds cannot be arbitrary: for each possible value l corresponds to a very specific shape of the electron cloud.

Rice. 5. Graphic interpretation of the moment of electron motion, where μ - orbital angular momentum

electron movement

The orbital quantum number can range from 0 before n - 1 , Total n– values.

Energy sublevels are indicated by letters:

Meaning l 0 1 2 3 4

Designation s p d f g

Magnetic quantum numberm l . From the solution of the Schrödinger equation it follows that electron clouds are oriented in a certain way in space. The spatial orientation of electron clouds is characterized by a magnetic quantum number.

The magnetic quantum number can take any integer value, both positive and negative, ranging from – l to + l, and in total this number can take (2l+1) values ​​for a given l, including zero. For example, if l = 1, then three possible values m (–1,0,+1) orbital moment  , is a vector whose magnitude is quantized and determined by the value l. From the Schrödinger equation it follows that not only the quantity µ , but the direction of this vector, which characterizes the spatial orientation of the electron cloud, is quantized. Each direction of the vector given

length corresponds to a certain value of its projection onto the axis z, characterizing a certain direction of the external magnetic field. The value of this projection characterizes m l .

Electron spin. The study of atomic spectra showed that three quantum numbers n, l And m l are not a complete description of the behavior of electrons in atoms. With the development of spectral research methods and an increase in the resolution of spectral instruments, the fine structure of spectra was discovered. It turned out that the spectral lines were split. To explain this phenomenon, a fourth quantum number was introduced, related to the behavior of the electron itself. This quantum number was named spin with designation m s and taking only two values +½ And –½ depending on one of two possible orientations of the electron spin in a magnetic field. Positive and negative spin values ​​are related to its direction. Because the spin Since the quantity is vector, it is conventionally denoted by an arrow pointing up or down ↓. Electrons having the same spin direction are called parallel, for opposite values ​​of spins – antiparallel.

The presence of spin in an electron was proven experimentally in 1921 by W. Gerlach and O. Stern, who managed to divide a beam of hydrogen atoms into two parts corresponding to the orientation of the electron spin. The design of their experiment is shown in Fig. 6. When hydrogen atoms fly through a region of strong magnetic field, the electron of each atom interacts with magnetic field, and this causes the atom to deviate from the original straight path. The direction in which the atom deviates depends on the orientation of the spin of its electron. The electron spin does not depend on external conditions and cannot be destroyed or altered.

Thus, it was finally established that the entire state of an electron in an atom is characterized by four quantum numbers n, l, m l . And m s ,

Rice. 6. Scheme of the Stern-Gerlach experiment

Quantum mechanical description of an electron in an atom

Bohr's theory made it possible to accurately calculate the frequencies in the spectra of the hydrogen atom and other single-electron systems, i.e., ions such as helium, lithium, and beryllium.

However, when moving to more complex electronic systems- multi-electron - Bohr's theory turned out to be insufficient.

Therefore, there was a need to create more general theory, which was named quantum mechanics. This theory described the behavior of microworld objects (for example, an electron).

In 1923-1927 the basic principles of quantum mechanics were formulated.

Quantum mechanical theory contains two main provisions.

1. The electron has a dual nature. It has the properties of both a particle and a wave at the same time. As a particle, an electron has mass and charge, but the movement of electrons is a wave process. Electrons are characterized by the phenomenon of diffraction (the flow of electrons bends around an obstacle).

2. The position of the electron in the atom is uncertain. This means that it is impossible to simultaneously accurately determine both the speed of the electron and its coordinates in space.

An electron that moves at a very high speed can be located in any part of the space around the nucleus, and its various instantaneous positions form the so-called electron cloud with uneven negative charge density(drawing). The shape and size of the electron cloud can vary depending on the energy of the electron.

For chemical characteristics element, which is determined by the state of electrons in the electron shell of its atom, as well as to explain the bonds that an atom of a given element can form with other atoms, you need to know:

- the energy of an electron in an atom (more precisely, the energy of a system consisting of this electron, other electrons and a nucleus;

- the shape of the electron cloud formed by a given electron.

The state of an electron in an atom is characterized by a set of four quantum numbers.

According to energy, electrons in an atom are distributed according to energy levels and sublevels.

4.2.1. Principal quantum number (P) characterizes the energy level and determines the size of the electron cloud, i.e., the average distance of the electron from the nucleus; accepts integer values ​​1, 2, 3, ..., P, which correspond to the energy level number. The more P, the higher the electron energy, therefore, the minimum energy corresponds to the first level (P= 1).

4.2.2 Orbital or side quantum number(l) characterizes the energy sublevel and determines the shape of the electron cloud; accepts integer values ​​from 0 to (P - 1). Its meanings are usually indicated by letters:

l = 0 1 2 3

Number of possible values l corresponds to the number of possible sublevels at a given level, equal to the level number (P).

At P= 1 l= 0 (1 value)

P= 2 l= 0.1 (2 values)

P = 3 l= 0, 1, 2 (3 values)

n = 4 l= 0, 1, 2, 3 (4 values)

The energy of electrons at different sublevels of the same level varies depending on l as follows: for each value l corresponds to a certain shape of the electron cloud: s - sphere, R - three-dimensional figure eight, d f - three-dimensional four-petal rosette or a more complex shape (Fig.).

Table 1.1 – Shape of electron clouds

Quantum numbers are energy parameters that determine the state of an electron and the type of atomic orbital in which it is located. Quantum numbers are needed to describe the state of each electron in an atom. There are 4 quantum numbers in total. This is: the main quantum number - n , l , magnetic quantum number – m l and spin quantum number – m s.

The main quantum number is n .

The main quantum number – n – determines the energy level of the electron, the distance of the energy level from the nucleus and the size of the electron cloud. The principal quantum number can take any integer value, starting from n =1 ( n =1,2,3,...) and corresponds to the period number.

Orbital quantum number – l .

Orbital quantum number – l – determines the geometric shape of the atomic orbital. The orbital quantum number can take any integer value, starting from l =0 ( l =0,1,2,3,… n -1). Regardless of the energy level number, each value of the orbital quantum number corresponds to an orbital of a special shape. A “set” of such orbitals with the same values The principal quantum number is called the energy level. Each value of the orbital quantum number corresponds to an orbital of a special shape. The value of the orbital quantum number l =0 corresponds s -orbital (1-in type). The value of the orbital quantum number l =1 correspond p -orbitals (3 types). The value of the orbital quantum number l =2 correspond d -orbitals (5 types). The value of the orbital quantum number l =3 correspond f -orbitals (7 types).

f orbitals have an even more complex shape. Each type of orbital is the volume of space in which the probability of finding an electron is maximum.

Magnetic quantum number – m l.

Magnetic quantum number – m l – determines the orientation of the orbital in space relative to the external magnetic or electric field. The magnetic quantum number takes any integer value from –l to +l, including 0. This means that for each orbital shape there are 2l+1 energetically equivalent orientations in space - orbitals.

For s orbital:

l=0, m=0 – one equivalent orientation in space (one orbital).

For p-orbital:

l=1, m=-1,0,+1 – three equivalent orientations in space (three orbitals).

For d-orbital:

l=2, m=-2,-1,0,1,2 – five equivalent orientations in space (five orbitals).

For f orbital:

l=3, m=-3,-2,-1,0,1,2,3 – seven equivalent orientations in space (seven orbitals).

Spin quantum number – m s.

The spin quantum number – m s – determines the magnetic moment that occurs when an electron rotates around its axis. The spin quantum number can take only two possible values: +1/2 and –1/2. They correspond to two possible and opposite directions of the electron’s own magnetic moment - spins. The following symbols are used to denote electrons with different spins: 5 And 6 .