General characteristics of molecular spectra. The structure and spectra of molecules What makes it possible to study molecular absorption spectra

Chemical bonds and structure of molecules.

Molecule - the smallest particle of a substance, consisting of the same or different atoms connected to each other chemical bonds, and being the carrier of its main chemical and physical properties. Chemical bonds are due to the interaction of external, valence electrons of atoms. There are two types of bonds most often found in molecules: ionic and covalent.

Ionic bond (for example, in molecules NaCl, KVR) is carried out by the electrostatic interaction of atoms during the transition of an electron from one atom to another, i.e. in the formation of positive and negative ions.

covalent bond(for example, in H 2 , C 2 , CO molecules) is carried out when valence electrons are shared by two neighboring atoms (the spins of valence electrons must be antiparallel). The covalent bond is explained on the basis of the principle of indistinguishability of identical particles, such as electrons in a hydrogen molecule. The indistinguishability of particles leads to exchange interaction.

The molecule is a quantum system; it is described by the Schrödinger equation, which takes into account the motion of electrons in a molecule, the vibrations of the atoms of the molecule, and the rotation of the molecule. The solution of this equation is a very complex problem, which is usually divided into two: for electrons and nuclei. Energy of an isolated molecule:

where is the energy of motion of electrons relative to nuclei, is the energy of vibrations of nuclei (as a result of which the relative position of nuclei periodically changes), is the energy of rotation of nuclei (as a result of which the orientation of the molecule in space periodically changes). Formula (13.1) does not take into account the translational energy of the center of mass of the molecule and the energy of the nuclei of atoms in the molecule. The first of them is not quantized, so its changes cannot lead to the appearance of a molecular spectrum, and the second can be ignored if the hyperfine structure of the spectral lines is not considered. It is proved that eV, eV, eV, so >>>>.

Each of the energies included in expression (13.1) is quantized (it corresponds to a set of discrete energy levels) and is determined by quantum numbers. During the transition from one energy state to another, energy is absorbed or emitted D E=hv. During such transitions, the energy of electron motion, the energy of vibrations and rotation change simultaneously. It follows from theory and experiment that the distance between rotational energy levels D is much less than the distance between vibrational levels D, which, in turn, is less than the distance between electronic levels D. Figure 13.1 schematically shows the energy levels of a diatomic molecule (for example, only two electronic levels are considered are shown in bold lines).

The structure of molecules and the properties of their energy levels are manifested in molecular spectra– emission (absorption) spectra arising from quantum transitions between the energy levels of molecules. The emission spectrum of a molecule is determined by the structure of its energy levels and the corresponding selection rules.

Thus, different types of transitions between levels give rise to different types of molecular spectra. The frequencies of the spectral lines emitted by molecules can correspond to transitions from one electronic level to another (electronic spectra) or from one vibrational (rotational) level to another ( vibrational (rotational) spectra). In addition, transitions with the same values ​​are also possible and to levels having different values ​​of all three components, resulting in electronic-vibrational and vibrational-rotational spectra.

Typical molecular spectra are banded, which are a combination of more or less narrow bands in the ultraviolet, visible and infrared regions.

Using high-resolution spectral instruments, it can be seen that the fringes are such closely spaced lines that they are difficult to resolve. The structure of molecular spectra is different for different molecules and becomes more complicated with an increase in the number of atoms in a molecule (only continuous broad bands are observed). Only polyatomic molecules have vibrational and rotational spectra, while diatomic ones do not have them. This is explained by the fact that diatomic molecules do not have dipole moments (during vibrational and rotational transitions, there is no change in the dipole moment, which is a necessary condition for the transition probability to differ from zero). Molecular spectra are used to study the structure and properties of molecules, are used in molecular spectral analysis, laser spectroscopy, quantum electronics, etc.

Studies of molecular spectra make it possible to determine the forces acting between atoms in a molecule, the dissociation energy of a molecule, its geometry, internuclear distances, etc. , i.e. provide extensive information about the structure and properties of the molecule.

Under the molecular spectrum, in a broad sense, is understood the distribution of the probability of transitions between two separate energy levels of the molecule (see Fig. 9) depending on the energy of the transition. Since in what follows we will deal with optical spectra, each such transition must be accompanied by the emission or absorption of a photon with energy

E n \u003d hn \u003d E 2 - E 1, 3.1

where E 2 and E 1 are the energies of the levels between which the transition occurs.

If radiation, consisting of photons emitted by gas molecules, is passed through a spectral device, then the emission spectrum of the molecule will be obtained, consisting of individual bright (maybe colored) lines. Moreover, each line will correspond to the corresponding transition. In turn, the brightness and position of the line in the spectrum depend on the transition probability and energy (frequency, wavelength) of the photon, respectively.

If, on the contrary, radiation consisting of photons of all wavelengths (continuous spectrum) is passed through this gas, and then through a spectral device, then an absorption spectrum will be obtained. In this case, this spectrum will be a set of dark lines against the background of a bright continuous spectrum. The contrast and position of the line in the spectrum here also depend on the transition probability and photon energy.

Based on the complex structure of the energy levels of the molecule (see Fig. 9), all transitions between them can be divided into separate types, which give a different character of the spectrum of molecules.

A spectrum consisting of lines corresponding to transitions between rotational levels (see Fig. 8) without changing the vibrational and electronic states of the molecule is called the rotational spectrum of the molecule. Since the energy of rotational motion lies in the range of 10 -3 -10 -5 eV, the frequency of the lines in these spectra should lie in the microwave region of radio frequencies (far infrared region).

A spectrum consisting of lines corresponding to transitions between rotational levels belonging to different vibrational states of a molecule in the same electronic state is called the vibrational-rotational or simply vibrational spectrum of the molecule. These spectra, at energies oscillatory motion 10 -1 -10 -2 eV, lie in the infrared region of frequencies.

Finally, the spectrum, consisting of lines corresponding to transitions between rotational levels belonging to different electronic and vibrational states of the molecule, is called the electronic-vibrational-rotational or simply electronic spectrum of the molecule. These spectra lie in the visible and ultraviolet frequency regions, since the energy of the electronic motion is a few electron volts.

Since the emission (or absorption) of a photon is an electromagnetic process, its necessary condition is the presence or, more precisely, a change in the electric dipole moment associated with the corresponding quantum transition in the molecule. Hence it follows that rotational and vibrational spectra can only be observed for molecules with an electric dipole moment, i.e. composed of dissimilar atoms.

MOLECULAR SPECTRA, spectra of emission and absorption of electromagnet. radiation and combinat. scattering of light belonging to free or weakly bound molecules. They have the form of a set of bands (lines) in the X-ray, UV, visible, IR and radio wave (including microwave) regions of the spectrum. The position of the bands (lines) in the spectra of emission (emission molecular spectra) and absorption (absorption molecular spectra) is characterized by frequencies v (wavelengths l \u003d c / v, where c is the speed of light) and wave numbers \u003d 1 / l; it is determined by the difference between the energies E "and E: those states of the molecule, between which a quantum transition occurs:

(h is Planck's constant). When combined scattering, the value of hv is equal to the difference between the energies of the incident and scattered photons. The intensity of the bands (lines) is related to the number (concentration) of molecules of a given type, the population of the energy levels E "and E: and the probability of the corresponding transition.

The probability of transitions with the emission or absorption of radiation is determined primarily by the square of the matrix element of the electric. dipole moment of the transition, and with a more accurate consideration - and the squares of the matrix elements of the magn. and electric quadrupole moments of the molecule (see Quantum transitions). When combined In light scattering, the transition probability is related to the matrix element of the induced (induced) dipole moment of the transition of the molecule, i.e. with the matrix element of the polarizability of the molecule .

states of the pier. systems, transitions between to-rymi are shown in the form of these or those molecular spectra, have the different nature and strongly differ on energy. The energy levels of certain types are located far from each other, so that during transitions the molecule absorbs or emits high-frequency radiation. The distance between the levels of other nature is small, and in some cases, in the absence of external. field levels merge (degenerate). At small energy differences, transitions are observed in the low-frequency region. For example, the nuclei of atoms of certain elements have their own. magn. torque and electric spin-related quadrupole moment. Electrons also have a magnet. the moment associated with their spin. In the absence of external magnetic orientation fields moments are arbitrary, i.e. they are not quantized and the corresponding energetic. states are degenerate. When applying external permanent magnet. field, degeneracy is lifted and transitions between energy levels are possible, which are observed in the radio-frequency region of the spectrum. This is how NMR and EPR spectra arise (see Nuclear magnetic resonance, Electron paramagnetic resonance).

Kinetic distribution energies of electrons emitted by the pier. systems as a result of irradiation with X-ray or hard UV radiation, gives X-rayspectroscopy and photoelectron spectroscopy. Additional processes in the mall. system, caused by the initial excitation, lead to the appearance of other spectra. Thus, Auger spectra arise as a result of relaxation. electron capture from ext. shells to.-l. atom per vacant ext. shell, and the released energy turned into. in the kinetic energy other electron ext. shell emitted by an atom. In this case, a quantum transition is carried out from a certain state of a neutral molecule to a state they say. ion (see Auger spectroscopy).

Traditionally, only the spectra associated with the optical properties are referred to as molecular spectra proper. transitions between electronic-vibrational-rotate, energy levels of the molecule associated with three main. energy types. levels of the molecule - electronic E el, vibrational E count and rotational E vr, corresponding to three types of ext. movement in a molecule. For E el take the energy of the equilibrium configuration of the molecule in a given electronic state. The set of possible electronic states of a molecule is determined by the properties of its electron shell and symmetry. Swing. the motion of the nuclei in the molecule relative to their equilibrium position in each electronic state is quantized so that at several vibrations. degrees of freedom, a complex system of vibrations is formed. energy levels E col. The rotation of the molecule as a whole as a rigid system of bound nuclei is characterized by rotation. the moment of the number of motion, which is quantized, forming a rotation. states (rotational energy levels) E temp. Usually the energy of electronic transitions is of the order of several. eV, vibrational -10 -2 ... 10 -1 eV, rotational -10 -5 ... 10 -3 eV.

Depending on between which energy levels there are transitions with emission, absorption or combinations. electromagnetic scattering. radiation - electronic, oscillating. or rotational, distinguish between electronic, oscillating. and rotational molecular spectra. The articles Electronic spectra , Vibrational spectra , Rotational spectra provide information about the corresponding states of molecules, selection rules for quantum transitions, methods of pier. spectroscopy, as well as what characteristics of molecules can be. obtained from molecular spectra: St. islands and symmetry of electronic states, vibrate. constants, dissociation energy, molecular symmetry, rotation. constants, moments of inertia, geom. parameters, electrical dipole moments, data on the structure and ext. force fields, etc. Electronic absorption and luminescence spectra in the visible and UV regions provide information on the distribution

In addition to the spectra corresponding to the radiation of individual atoms, there are also spectra emitted by whole molecules (§ 61). Molecular spectra are much more diverse and more complex in structure than atomic spectra. There are thickening sequences of lines, similar to the spectral series of atoms, but with a different frequency law and with lines so closely spaced that they merge into continuous bands (Fig. 279). In view of the peculiar nature of these spectra, they are called striped.

Rice. 279. Striped Spectrum

Along with this, sequences of equidistant spectral lines and, finally, multi-line spectra are observed, in which, at first glance, it is difficult to establish any regularities (Fig. 280). It should be noted that in the study of the spectrum of hydrogen, we always have a superposition of the molecular spectrum of Ha on the atomic spectrum, and special measures must be taken to increase the intensity of the lines emitted by individual hydrogen atoms.

Rice. 280. Molecular spectrum of hydrogen

From the quantum point of view, as in the case of atomic spectra, each line of the molecular spectrum is emitted when the molecule moves from one stationary energy level to another. But in the case of a molecule, there are many more factors on which the energy of the stationary state depends.

In the simplest case of a diatomic molecule, the energy consists of three parts: 1) the energy of the electron shell of the molecule; 2) vibrational energies of the nuclei of atoms that make up the molecule along the straight line connecting them; 3) the energy of rotation of nuclei around a common center of mass. All three types of energy are quantized, that is, they can only take on a discrete range of values. The electron shell of a molecule is formed as a result of the fusion of the electron shells of the atoms that make up the molecule. Energy electronic states of molecules can be considered as a limiting case

a very strong Stark effect caused by the interatomic interaction of atoms that form a molecule. Although the forces that bind atoms into molecules are purely electrostatic in nature, a correct understanding chemical bond turned out to be possible only within the framework of modern wave-mechanical quantum theory.

There are two types of molecules: homeopolar and heteropolar. Homeopolar molecules with increasing distance between the nuclei break up into neutral parts. Hemopolar molecules include molecules. Heteropolar molecules decompose into positive and negative ions as the distance between the nuclei increases. A characteristic example of heteropolar molecules are molecules of salts, for example, etc. (vol. I, § 121, 130, 1959; in the previous ed. § 115 and 124 and vol. II, § 19, 22, 1959 ; in the previous ed. § 21 and 24).

The energy states of the electron cloud of a homeopolar molecule are largely determined by the wave properties of the electrons.

Consider a very rough model of the simplest molecule (an ionized hydrogen molecule representing two potential "wells" located at a close distance from each other and separated by a "barrier" (Fig. 281).

Rice. 281. Two potential wells.

Rice. 282. Wave functions of an electron in the case of distant "holes".

Each of the "pits" depicts one of the atoms that make up the molecule. With a large distance between atoms, the electron in each of them has quantized energy values ​​corresponding to standing electron waves in each of the "wells" separately (§ 63). On fig. 282, a and b depict two identical wave functions describing the state of electrons in isolated atoms. These wave functions correspond to the same energy level.

As the atoms approach the molecule, the "barrier" between the "pits" becomes "transparent" (§ 63), because its width becomes commensurate with the length of the electron wave. As a result of this, there

the exchange of electrons between atoms through the "barrier", and it makes no sense to talk about the belonging of an electron to one or another atom.

The wave function can now have two forms: c and d (Fig. 283). Case c can be approximately considered as the result of the addition of curves a and b (Fig. 282), the case as the difference between a and b, but the energies corresponding to states c and d are no longer exactly equal to each other. The energy of the state is somewhat less than the energy of the state. Thus, two molecular electronic levels arise from each atomic level.

Rice. 283. Wave functions of an electron in the case of close "holes".

Until now, we have been talking about the ion of the hydrogen molecule, which has one electron. There are two electrons in a neutral hydrogen molecule, which leads to the need to take into account mutual arrangement their spins. In accordance with the Pauli principle, electrons with parallel spins seem to “avoid” each other, so the probability density of finding each electron is distributed according to Fig. 284, a, i.e., electrons are most often located outside the gap between the nuclei. Therefore, with parallel spins, a stable molecule cannot form. On the contrary, antiparallel spins correspond to the highest probability of finding both electrons inside the gap between the nuclei (Fig. 294, b). In this case, the negative electron charge attracts both positive nuclei to itself, and the whole system as a whole forms a stable molecule.

For heteropolar molecules, the pattern of electron charge density distribution has a much more classical character. An excess of electrons is grouped around one of the nuclei, and around the other, on the contrary, there is a shortage of electrons. Thus, two ions are formed in the composition of the molecule, positive and negative, which are attracted to each other: in, for example, and

The symbolism of the electronic states of molecules has many similarities with the atomic symbolism. Naturally, the main role in the molecule is played by the direction of the axis connecting the nuclei. Here the quantum number A is introduced, analogous to I in the atom. The quantum number characterizes the absolute value of the projection onto the axis of the molecule of the resulting orbital momentum of the electron cloud of the molecule.

There is a correspondence between the meanings and symbols of molecular electronic states, similar to that in atoms (§ 67):

The absolute value of the projection of the resulting spin of the electron cloud on the axis of the molecule is characterized by the quantum number 2, and the projection of the total rotational momentum of the electron shell is characterized by the quantum number. Obviously,

The quantum number is analogous to the internal quantum number of the atom (§ 59 and 67).

Rice. 284. Probability density of finding an electron at various points of a molecule.

Like atoms, molecules exhibit multiplicity caused by different orientations of the resulting spin with respect to the resulting orbital momentum.

Given these circumstances, the electronic states of molecules are written as follows:

where 5 is the value of the resulting spin, and means one of the symbols or A corresponding to different meanings quantum number A. For example, the normal state of the hydrogen molecule is 2, the normal state of the hydroxyl molecule is the normal state of the oxygen molecule is . During transitions between different electronic states, selection rules take place: .

The vibrational energy of a molecule associated with vibrations of nuclei is quantized based on the wave properties of nuclei. Assuming that the nuclei in a molecule are bound by a quasi-elastic force (the potential energy of a particle is proportional to the square of the displacement, § 63), we obtain from the Schrödinger equation the following permitted values ​​of the vibrational energy of this system (harmonic

oscillator):

where is the frequency of natural oscillations of the nuclei, determined as usual (Vol. I, § 57, 1959; in the previous ed. § 67):

where is the reduced mass of nuclei; the masses of both nuclei; quasi-elastic constant of the molecule; quantum number equal to Due to the large size of the mass, the frequency lies in the infrared region of the spectrum.

Rice. 285. Vibrational energy levels of a molecule.

The quasi-elastic constant depends on the configuration of the electron shell and is therefore different for different electronic states of the molecule. This constant is the greater, the stronger the molecule, i.e., the stronger the chemical bond.

Formula (3) corresponds to a system of equally spaced energy levels, the distance between which is equal to In fact, at large amplitudes of oscillations of the nuclei, deviations of the restoring force from Hooke's law already begin to affect. As a result energy levels approach (Fig. 285). At sufficiently large amplitudes, the dissociation of the molecule into parts occurs.

For a harmonic oscillator, transitions are allowed only at , which corresponds to the emission or absorption of frequency light. Due to deviations from harmonicity, transitions appear corresponding to

According to the quantum condition for frequencies (§ 58), overtones should appear in this case, which is observed in the spectra of molecules.

Vibrational energy is a relatively small addition to the energy of the electron cloud of the molecule. Vibrations of the nuclei lead to the fact that each electronic level is converted into a system of close levels corresponding to different values ​​of vibrational energy (Fig. 286). This does not exhaust the complexity of the system of energy levels of the molecule.

Rice. 286. Addition of vibrational and electronic energy of a molecule.

It is also necessary to take into account the smallest component of molecular energy - rotational energy. Permissible values ​​of rotational energy are determined, according to wave mechanics, based on the principle of torque quantization.

According to wave mechanics, the torque (§ 59) of any quantized system is equal to

In this case, replaces and is equal to 0, 1, 2, 3, etc.

Kinetic energy of a rotating body in prev. ed. § 42) will

where is the moment of inertia, co - angular velocity rotation.

But, on the other hand, the torque is equal. From here we get:

or, substituting expression (5) instead, we finally find:

On fig. 287 shows the rotational levels of the molecule; in contrast to vibrational and atomic levels, the distance between rotational levels increases with increasing transitions between rotational levels are allowed, while lines with frequencies are emitted

where Evrash corresponds corresponds

Formula (9) gives for frequencies

Rice. 287. Levels of rotational energy of a molecule.

We get equidistant spectral lines lying in the far infrared part of the spectrum. Measurement of the frequencies of these lines makes it possible to determine the moment of inertia of the molecule. It turned out that the moments of inertia of the molecules are of the order of magnitude.

centrifugal force increases with increasing speed of rotation of the molecule. The presence of rotations leads to the splitting of each vibrational energy level into a number of close sublevels corresponding to different values ​​of the rotational energy.

During the transitions of a molecule from one energy state to another, all three types of energy of the molecule can change simultaneously (Fig. 288). As a result, each spectral line that would be emitted during an electronic-vibrational transition acquires a fine rotational structure and turns into a typical molecular band.

Rice. 288. Simultaneous change of all three types of energy of a molecule

Such bands of equidistant lines are observed in vapors and water and lie in the far infrared part of the spectrum. They are observed not in the emission spectrum of these vapors, but in their absorption spectrum, because the frequencies corresponding to the natural frequencies of the molecules are absorbed more strongly than the others. On fig. 289 shows a band in the absorption spectrum of vapors in the near infrared region. This band corresponds to transitions between energy states that differ not only in the energy of rotation, but also in the energy of vibrations (at a constant energy of the electron shells). In this case, and and Ekol change simultaneously, which leads to large changes in energy, i.e., the spectral lines have a higher frequency than in the first case considered.

In accordance with this, lines appear in the spectrum that lie in the near infrared part, similar to those shown in Fig. 289.

Rice. 289. Absorption band.

The center of the band (corresponds to the transition at a constant Evrach; according to the selection rule, such frequencies are not emitted by the molecule. Lines with higher frequencies - shorter wavelengths - correspond to transitions in which the change in Europax is added to the change. Lines with lower frequencies (right side) correspond to the inverse relation: change rotational energy has the opposite sign.

Along with such bands, bands are observed corresponding to transitions with a change in the moment of inertia but with. In this case, according to formula (9), the line frequencies should depend on and the distances between the lines become unequal. Each stripe consists of a series of lines, thickening towards one edge,

which is called the strip head. As early as 1885, Delandre gave the following empirical formula for the frequency of an individual spectral line that is part of the band:

where is an integer.

The Delandre formula follows directly from the above considerations. The Delandre formula can be depicted graphically if one plots along one axis and along the other (Fig. 290).

Rice. 290. Graphic representation of the Delandre formula.

The corresponding lines are shown below, forming, as we see, a typical strip. Since the structure of the molecular spectrum strongly depends on the moment of inertia of the molecule, the study of molecular spectra is one of the reliable methods for determining this quantity. The slightest changes in the structure of a molecule can be detected by studying its spectrum. The most interesting circumstance is that molecules containing different isotopes (§ 86) of the same element must have different lines in their spectrum corresponding to different masses of these isotopes. This follows from the fact that the masses of atoms determine both the frequency of their oscillations in the molecule and its moment of inertia. Indeed, the lines of copper chloride bands consist of four components, respectively, to four combinations of copper isotopes 63 and 65 with chlorine isotopes 35 and 37:

Lines corresponding to molecules containing a heavy isotope of hydrogen were also found, despite the fact that the concentration of the isotope in ordinary hydrogen is

In addition to the mass of nuclei, other properties of nuclei also affect the structures of molecular spectra. In particular, the rotational moments (spins) of the nuclei play a very important role. If in a molecule consisting of identical atoms, the rotational moments of the nuclei are equal to zero, every second line of the rotational band falls out. Such an effect, for example, is observed in the molecule

If the angular moments of the nuclei are nonzero, they can cause an alternation of intensities in the rotational band, weak lines will alternate with strong ones.)

Finally, using the methods of radio spectroscopy, it was possible to detect and accurately measure the hyperfine structure of molecular spectra, related to the quadrupole electric moment of the nuclei.

The quadrupole electric moment arises as a result of the deviation of the shape of the nucleus from the spherical one. The nucleus may be in the form of an elongated or flattened ellipsoid of revolution. Such a charged ellipsoid can no longer be replaced by a simple point charge placed in the center of the nucleus.

Rice. 291. Absorbing device of "atomic" clocks: 1 - a rectangular waveguide with a cross-section of length closed on both sides by gas-tight bulkheads 7 and filled with ammonia at low pressure;

2 - crystal diode, which creates harmonics of the high-frequency voltage supplied to it; 3 - output crystal diode; 4 - generator of frequency-modulated high-frequency voltage; 5 - pipeline to the vacuum pump and ammonia gas tank; 6 - output to a pulse amplifier; 7 - bulkheads; And - indicator of the current of the crystal diode; B - vacuum gauge.

In addition to the Coulomb force, an additional force appears in the field of the nucleus, which is inversely proportional to the fourth power of the distance and depends on the angle with the direction of the symmetry axis of the nucleus. The appearance of an additional force is associated with the presence of a quadrupole moment at the nucleus.

For the first time, the presence of a quadrupole moment in the nucleus was established by conventional spectroscopy using certain details of the hyperfine structure of atomic lines. But these methods did not make it possible to accurately determine the magnitude of the moment.

In the radiospectroscopic method, the waveguide is filled with the investigated molecular gas and the absorption of radio waves in the gas is measured. The use of klystrons to generate radio waves makes it possible to obtain oscillations with a high degree of monochromaticity, which are then modulated. The absorption spectrum of ammonia in the region of centimeter waves was studied in particular detail. In this spectrum, a hyperfine structure was found, which is explained by the presence of a connection between the quadrupole moment of the nucleus and the electric field of the molecule itself.

The fundamental advantage of radio spectroscopy is the low energy of photons corresponding to radio frequencies. Due to this, by the absorption of radio frequencies, it is possible to detect transitions between extremely close energy levels of atoms and molecules. In addition to nuclear effects, the method of radiospectroscopy is very convenient for determining the electric dipole moments of the entire molecule from the Stark effect of molecular lines in weak electric fields.

fields. Per last years a huge number of works appeared devoted to the radiospectroscopic method of studying the structure of various molecules. The absorption of radio waves in ammonia was used to build ultra-precise "atomic" clocks (Fig. 291).

The duration of the astronomical day slowly increases and, in addition, fluctuates within the limits. It is desirable to build clocks with a more uniform course. "Atomic" clock is a quartz generator of radio waves with a frequency controlled by the absorption of the generated waves in ammonia. At a wavelength of 1.25 cm resonance occurs with the natural frequency of the ammonia molecule, which corresponds to a very sharp absorption line. The slightest deviation of the generator wavelength from this value breaks the resonance and leads to a strong increase in the transparency of the gas for radio emission, which is recorded by the appropriate equipment and activates the automation that restores the frequency of the generator. "Atomic" clocks have already given a course more uniform than the rotation of the Earth. It is assumed that it will be possible to achieve an accuracy of the order of fractions of a day.

spectrum called the sequence of energy quanta electromagnetic radiation, absorbed, released, scattered or reflected by matter during the transitions of atoms and molecules from one energy state to another.

Depending on the nature of the interaction of light with matter, the spectra can be divided into absorption (absorption) spectra; emissions (emission); scattering and reflection.

For the objects under study, optical spectroscopy, i.e. spectroscopy in the wavelength range 10 -3 ÷10 -8 m subdivided into atomic and molecular.

atomic spectrum is a sequence of lines, the position of which is determined by the energy of the transition of electrons from one level to another.

The energy of an atom can be represented as the sum of the kinetic energy of translational motion and electronic energy:

where - frequency, - wavelength, - wave number, - speed of light, - Planck's constant.

Since the energy of an electron in an atom is inversely proportional to the square of the principal quantum number , then for the line in the atomic spectrum we can write the equation:

.

(4.12)

Here - electron energies at higher and lower levels; - Rydberg constant; - spectral terms, expressed in units of wave numbers (m -1 , cm -1).

All lines of the atomic spectrum converge in the short-wavelength region to a limit determined by the ionization energy of the atom, after which there is a continuous spectrum.

Molecule energy in the first approximation can be considered as the sum of translational, rotational, vibrational and electronic energies:

(4.15)

For most molecules, this condition is satisfied. For example, for H 2 at 291K, the individual components full energy differ by an order of magnitude or more:

309,5 kJ/mol,

=25,9 kJ/mol,

2,5 kJ/mol,

=3,8 kJ/mol.

The values ​​of photon energies in different regions of the spectrum are compared in Table 4.2.

Table 4.2 - Energy of absorbed quanta of different regions of the optical spectrum of molecules

The concepts of "oscillations of nuclei" and "rotation of molecules" are conditional. In fact, such types of motion only very approximately convey ideas about the distribution of nuclei in space, which is of the same probabilistic nature as the distribution of electrons.

A schematic system of energy levels in the case of a diatomic molecule is shown in Figure 4.1.

Transitions between rotational energy levels give rise to rotational spectra in the far IR and microwave regions. Transitions between vibrational levels within the same electronic level give vibrational-rotational spectra in the near-IR region, since a change in the vibrational quantum number inevitably entails a change in the rotational quantum number . Finally, transitions between electronic levels cause the appearance of electronic-vibrational-rotational spectra in the visible and UV regions.

In the general case, the number of transitions can be very large, but in fact, far from all appear in the spectra. The number of transitions is limited selection rules .

Molecular spectra provide a wealth of information. They can be used:

For the identification of substances in a qualitative analysis, as each substance has its own unique spectrum;

For quantitative analysis;

For structural group analysis, since certain groups, such as, for example, >C=O, _ NH 2 , _ OH, etc., give characteristic bands in the spectra;

To determine the energy states of molecules and molecular characteristics (internuclear distance, moment of inertia, natural vibration frequencies, dissociation energies); a comprehensive study of molecular spectra makes it possible to draw conclusions about the spatial structure of molecules;

In kinetic studies, including for the study of very fast reactions.

- energy electronic levels;

Energy of vibrational levels;

Energy of rotational levels

Figure 4.1 - Schematic arrangement of energy levels of a diatomic molecule

Bouguer-Lambert-Beer law

Quantitative molecular analysis using molecular spectroscopy is based on Bouguer-Lambert-Beer law , relating the intensity of the incident and transmitted light with the concentration and thickness of the absorbing layer (Figure 4.2):

or with a proportionality factor:

Integration result:

(4.19)

.

(4.20)

When the intensity of the incident light decreases by an order of magnitude

.

(4.21)

If \u003d 1 mol / l, then, i.e. the absorption coefficient is equal to the reciprocal thickness of the layer in which, at a concentration equal to 1, the intensity of the incident light decreases by an order of magnitude.

The absorption coefficients and depend on the wavelength. The type of this dependence is a kind of “fingerprint” of molecules, which is used in qualitative analysis to identify a substance. This dependence is characteristic and individual for a particular substance and reflects the characteristic groups and bonds included in the molecule.

Optical density D

expressed in %

4.2.3 Rotation energy of a diatomic molecule in the rigid rotator approximation. Rotational spectra of molecules and their application to determine molecular characteristics

The appearance of rotational spectra is due to the fact that the rotational energy of the molecule is quantized, i.e.

0

a

Energy of rotation of a molecule around the axis of rotation

Since the point O is the center of gravity of the molecule, then:

Introduction of the reduced mass notation:

(4.34)

leads to the equation

.

(4.35)

Thus, a diatomic molecule (Figure 4.7 a) rotating around the axis or , passing through the center of gravity, can be simplified as a particle with mass , describing a circle with a radius around the point O(Figure 4.7 b).

The rotation of the molecule around the axis gives the moment of inertia, which is practically equal to zero, since the atomic radii are much smaller than the internuclear distance. Rotation about the axes or , mutually perpendicular to the bond line of the molecule, leads to equal moments of inertia:

where is a rotational quantum number that takes only integer values

0, 1, 2…. In accordance with selection rule for the rotational spectrum of a diatomic molecule, a change in the rotational quantum number upon absorption of an energy quantum is possible only by one, i.e.

transforms equation (4.37) into the form:

20 12 6 2

wavenumber of the line in the rotational spectrum corresponding to the absorption of a quantum upon transition from j energy level per level j+1, can be calculated by the equation:

Thus, the rotational spectrum in the rigid rotator model approximation is a system of lines at the same distance from each other (Figure 4.5b). Examples of rotational spectra of diatomic molecules estimated in the rigid rotator model are shown in Figure 4.6.

a b

Figure 4.6 - Rotational spectra HF (a) and CO(b)

For hydrogen halide molecules, this spectrum is shifted to the far IR region of the spectrum; for heavier molecules, to the microwave.

Based on the obtained patterns of the occurrence of the rotational spectrum of a diatomic molecule, in practice, first determine the distance between adjacent lines in the spectrum, from which then find, and according to the equations:

,

(4.45)

where - centrifugal distortion constant , is related to the rotational constant by the approximate relationship . The correction should be taken into account only for very large j.

For polyatomic molecules, in the general case, the existence of three different moments of inertia is possible . In the presence of symmetry elements in the molecule, the moments of inertia can coincide or even be equal to zero. For example, for linear polyatomic molecules(CO 2 , OCS, HCN, etc.)

where - position of the line corresponding to the rotational transition in an isotopically substituted molecule.

To calculate the isotopic shift of the line, it is necessary to sequentially calculate the reduced mass of the isotopically substituted molecule, taking into account the change in the atomic mass of the isotope, the moment of inertia , rotational constant and the position of the line in the spectrum of the molecule according to equations (4.34), (4.35), (4.39) and (4.43), respectively , or estimate the ratio of the wave numbers of lines corresponding to the same transition in isotopically substituted and non-isotopically substituted molecules, and then determine the direction and magnitude of the isotopic shift using equation (4.50). If the internuclear distance is approximately constant , then the ratio of the wave numbers corresponds to the inverse ratio of the reduced masses:

where is the total number of particles, is the number of particles per i- that level of energy at temperature T, k- Boltzmann's constant, - statistical ve forces degree of degeneracy i-th energy level, characterizes the probability of finding particles at a given level.

For a rotational state, the population of a level is usually characterized by the ratio of the number of particles j- that energy level to the number of particles at the zero level:

,

(4.53)

where - statistical weight j-th rotational energy level, corresponds to the number of projections of the momentum of a rotating molecule on its axis - the communication line of the molecule, , energy of the zero rotational level . The function goes through a maximum when increasing j, as Figure 4.7 illustrates with the CO molecule as an example.

The extremum of the function corresponds to the level with the maximum relative population, the value of the quantum number of which can be calculated by the equation obtained after determining the derivative of the function in the extremum:

.

(4.54)

Figure 4.7 - Relative population of rotational energy levels

molecules CO at temperatures of 298 and 1000 K

Example. In the rotational spectrum of HI, the distance between adjacent lines is determined cm -1. Calculate the rotational constant, the moment of inertia, and the equilibrium internuclear distance in the molecule.

Solution

In the approximation of the rigid rotator model, in accordance with equation (4.45), we determine the rotational constant:

cm -1.

The moment of inertia of the molecule is calculated from the value of the rotational constant according to equation (4.46):

kg . m 2.

To determine the equilibrium internuclear distance, we use equation (4.47), taking into account that the masses of hydrogen nuclei and iodine expressed in kg:

Example. In the far IR region of the spectrum of 1 H 35 Cl, lines were found whose wavenumbers are:

Determine the average values ​​of the moment of inertia and the internuclear distance of the molecule. Attribute the observed lines in the spectrum to rotational transitions.

Solution

According to the rigid rotator model, the difference between the wave numbers of adjacent lines of the rotational spectrum is constant and equal to 2 . Let us determine the rotational constant from the average value of the distances between adjacent lines in the spectrum:

cm -1 ,

cm -1

We find the moment of inertia of the molecule (equation (4.46)):

We calculate the equilibrium internuclear distance (equation (4.47)), taking into account that the masses of hydrogen nuclei and chlorine (expressed in kg):

Using equation (4.43), we estimate the position of the lines in the rotational spectrum of 1 H 35 Cl:

We correlate the calculated values ​​of the wave numbers of the lines with the experimental ones. It turns out that the lines observed in the rotational spectrum of 1 H 35 Cl correspond to the transitions:

N lines
, cm -1	85.384	106.730	128.076	149.422	170.768	192.114	213.466
	3 4	4 5	5 6	6 7	7 8	8 9	9 10

Example. Determine the magnitude and direction of the isotopic shift of the absorption line corresponding to the transition from energy level, in the rotational spectrum of the 1 H 35 Cl molecule when the chlorine atom is replaced by the 37 Cl isotope. The internuclear distance in 1 H 35 Cl and 1 H 37 Cl molecules is considered to be the same.

Solution

To determine the isotopic shift of the line corresponding to the transition , we calculate the reduced mass of the 1 H 37 Cl molecule, taking into account the change in the atomic mass of 37 Cl:

then we calculate the moment of inertia, the rotational constant and the position of the line in the spectrum of the 1 H 37 Cl molecule and the value of the isotopic shift according to equations (4.35), (4.39), (4.43) and (4.50), respectively.

Otherwise, the isotopic shift can be estimated from the ratio of the wave numbers of lines corresponding to the same transition in molecules (we assume that the internuclear distance is constant) and then the position of the line in the spectrum using equation (4.51).

For 1 H 35 Cl and 1 H 37 Cl molecules, the ratio of the wave numbers of a given transition is:

To determine the wave number of the line of an isotopically substituted molecule, we substitute the value of the transition wave number found in the previous example j → j+1 (3→4):

We conclude: the isotopic shift to the low-frequency or long-wave region is

85.384-83.049=2.335 cm -1 .

Example. Calculate the wave number and wavelength of the most intense spectral line of the rotational spectrum of the 1 H 35 Cl molecule. Match the line to the corresponding rotational transition.

Solution

The most intense line in the rotational spectrum of the molecule is associated with the maximum relative population of the rotational energy level.

Substituting the value of the rotational constant found in the previous example for 1 H 35 Cl ( cm -1) into equation (4.54) allows you to calculate the number of this energy level:

.

The wave number of the rotational transition from this level is calculated by equation (4.43):

We find the transition wavelength from the equation (4.11) transformed with respect to:

4.2.4 Multivariant task No. 11 "Rotational spectra of diatomic molecules"

1. Write a quantum mechanical equation to calculate the rotational energy of a diatomic molecule as a rigid rotator.

2. Derive an equation for calculating the change in the rotation energy of a diatomic molecule as a rigid rotator when it passes to the next, higher quantum level .

3. Derive an equation for the dependence of the wave number of rotational lines in the absorption spectrum of a diatomic molecule on the rotational quantum number.

4. Derive an equation for calculating the difference between the wave numbers of adjacent lines in the rotational absorption spectrum of a diatomic molecule.

5. Calculate the rotational constant (in cm -1 and m -1) of a diatomic molecule A by the wave numbers of two adjacent lines in the long-wavelength infrared region of the rotational absorption spectrum of the molecule (see Table 4.3) .

6. Determine the rotational energy of the molecule A at the first five quantum rotational levels (J).

7. Draw schematically the energy levels of the rotational motion of a diatomic molecule as a rigid rotator.

8. Plot on this diagram the rotational quantum levels of a molecule that is not a rigid rotator.

9. Derive an equation for calculating the equilibrium internuclear distance based on the difference in the wavenumbers of adjacent lines in the rotational absorption spectrum.

10. Determine the moment of inertia (kg. m 2) of a diatomic molecule A.

11. Calculate the reduced mass (kg) of the molecule A.

12. Calculate the equilibrium internuclear distance () of a molecule A. Compare the resulting value with the reference data.

13. Assign the observed lines in the rotational spectrum of the molecule A to rotational transitions.

14. Calculate the wavenumber of the spectral line corresponding to the rotational transition from the level j for a molecule A(see table 4.3).

15. Calculate the reduced mass (kg) of an isotopically substituted molecule B.

16. Calculate the wave number of the spectral line associated with the rotational transition from the level j for a molecule B(see table 4.3). Internuclear distances in molecules A and B consider equal.

17. Determine the magnitude and direction of the isotopic shift in the rotational spectra of molecules A and B for the spectral line corresponding to the rotational level transition j.

18. Explain the reason for the nonmonotonic change in the intensity of absorption lines as the rotational energy of the molecule increases

19. Determine the quantum number of the rotational level corresponding to the highest relative population. Calculate the wavelengths of the most intense spectral lines of the rotational spectra of molecules A and B.