Molecular refraction. Refractometric method of analysis in chemistry Atomic refractions and bond refractions

Molar polarization in the optical frequency range is called molar refraction R M . Refraction of a substance provided F=E(rarefied gas) is described by the same expression for both polar and non-polar substances

R M = (n n 2 – 1)V 0 =(N A/e 0)a el, (39)

Where a el – electronic (optical) polarizability of the molecule. Optical polarizability differs from deformational polarizability in that it takes into account only the shift in electron density while the arrangement of molecular nuclei remains unchanged. Expression (39) serves to determine the electronic polarizability of atoms and molecules of any substance (both polar and non-polar).

Sometimes the difference between a def and a e is not taken into account and the expression is used to estimate the value of the dipole moment

P M-R M=(N A/e 0)× ( m 2 / 3kT) (40)

If for some reason it is difficult to measure the molar polarization at different temperatures, then we can limit ourselves to finding the quantities P M And R M at one temperature.

In the case of condensed states of matter, the expression for molar refraction is the Lorentz-Lorentz formula:

R M= (n n 2 – 1)V 0 /(n n 2 + 2)= (N A/3e 0) a el. (41)

It is assumed that the value R M is constant. This means the constancy of the polarizability of molecules ( N A and e 0 are constants), due to which refraction is considered a molecular constant of a given substance (to a first approximation).

Heating a liquid in a closed volume, that is, at a constant density, almost always entails a slight change in refraction; in most cases the value R M decreases. In the process of heating a substance at a constant volume (at a constant concentration n) the energy of interaction between molecules decreases. This means that electronic polarizability depends on state of aggregation substance and changes during phase transitions (solid - liquid and liquid - gas) and when the substance is heated.

The refraction of any substance depends on the wavelength of the transmitted light; in other words, refraction is subject to the phenomenon of dispersion. If we limit ourselves to considering the visible region of the spectrum, then we will almost always observe an increase in the refractive index with decreasing wavelength. This dependence is called normal dispersion. However, there are substances in which an inverse relationship can be observed. In this case the variance is abnormal; it is always associated with the absorption of light. Special methods have been developed to observe anomalous dispersion. Known, for example, is the so-called “hooks” method by D.S. Rozhdestvensky. Anomalous dispersion can be observed in the visible, ultraviolet and infrared regions of the spectrum.

Attempts to represent the refractions of various chemical compounds as the sum of the refractions of individual atoms or bonds have been made for a long time. To gain a deeper understanding of the properties of various chemical compounds, it is necessary to take into account the interaction of bonds and different groups of atoms. The presence of such interactions naturally leads to a deviation from additivity; therefore, the refraction calculated using the additive scheme almost always differs from its experimental value.

Under the assumption of additivity, the molar refraction is written as

R M"S n i R at"S n j R St, (42)

Where R at – refraction of atoms or groups of atoms; R sm – increments of molar refraction of bonds; n i– the number of atoms or groups of atoms in a molecule of a certain type; n j– the number of connections of a certain type.

In table Figure 7 shows the components of the electronic polarizability of atoms belonging to various groups.

Table 7

Components of the electronic polarizability of atoms,
included in various groups ( Viktorov M.M.., 1977)

Table 8

Molar refraction values ​​of certain atoms or bonds
(Viktorov M.M.. 1977)

In table Figure 9 shows the values ​​of molar polarization and refraction of some substances.

Molar refraction

From Maxwell’s electromagnetic theory of light it follows that for wavelengths significantly removed from the region of their absorption by molecules of matter, the equality is true:

where n? - the refractive index of light for certain wavelengths.

Taking this into account, the Clausius-Mosotti equation (15) takes the following form:

[cm 3 /(g mol)] (19)

From the resulting expression it is clear that the RM index, called molar refraction, has the dimension of the volume of molecules contained in 1 mole of a substance.

Equation (15), which is called the Lorentz-Lorentz equation, was derived in 1880 independently by H. Lorentz and L. Lorentz.

In practice, the specific refraction index r is often used, that is, the refraction of one gram of a substance. Specific and molar refractions are related by the relation: R = r M, where M is the molar mass.

Since in equation (19) N is proportional to density, it can be represented in the following form:

[cm 3 /g] (20)

H. Lorentz and L. Lorentz revealed the physical meaning of the concept of refraction - as a measure of electronic polarizability and laid a solid theoretical foundation for the doctrine of refraction.

The value of specific refraction is practically independent of temperature, pressure and the state of aggregation of a substance.

In research practice, in addition to the molar and specific refraction R M and r, other derivatives of the refractive indices n are used (Table 2).

The refractive index of non-polar substances practically does not depend on the frequency of light waves and therefore equation (19) is valid at all frequencies. For example, for benzene n 2 = 2.29 (wavelength 289.3 nm), while e = 2.27. therefore, if for approximate calculations of refraction it is enough to use the refractive index of the visible spectrum, then for accurate calculations it is necessary to extrapolate using the Cauchy formula:

nл = n? + a/l2, (21) where nl is the refractive index at wavelength l;

a is an empirical coefficient.

Table 2 Refractometric constants

For polar substances e > n 2. For water, for example, n 2 = 1.78 (l = 589.3 nm), and e = 78. Moreover, in these cases it is impossible to directly extrapolate n l using the Cauchy formula due to the fact that the refractive index of polar substances often changes anomalously with frequency. However, there is usually no need to make such an extrapolation, since refraction is an additive quantity and is conserved if the refractive indices of all substances are measured at a certain wavelength. The yellow line in the sodium spectrum was chosen for this standard wavelength (l D = 589.3). The reference tables provide data specifically for this wavelength. Thus, to calculate molecular refraction (in cm 3 /mol), use the formula in which n? replaced by n D.

Light waves have a high oscillation frequency; in their electromagnetic field, the permanent dipole of a polar molecule does not have time to orient itself during one oscillation, and the nuclei of atoms do not have time to shift to the side

from the center of concentration of positive charges. Therefore, in the equation, the last two terms are equal to zero and the molecular polarization is determined by inductive (electronic) polarization. In this case, the electronic polarization of the molecule

represents a change in the state of electron clouds that form chemical bonds between atoms. The quantity is an important molecular one, it is called molecular refraction and is designated

From Maxwell's electromagnetic theory of light it is known that for wavelengths very far from the region of their absorption by molecules of matter, the equality is true where n is the refractive index of light for certain wavelengths. From here, equation (III.1) becomes:

From equation (II 1.2) it is clear that it has the dimension of volume, which means that molecular refraction expresses the volume of all molecules contained in a mole of a substance and characterizes the polarizability of all electrons contained in it. Molecular refraction is practically independent of temperature and the state of aggregation of a substance. Unlike the dipole moment, it is scalar quantity.

Molecular refractions of compounds can be presented additively, i.e., as sums of refractions components molecules (additivity rule). The latter can be considered bonds or atoms (ions). Refractions of bonds have a true physical meaning, since the polarizable electron cloud in a chemical compound belongs to the bond, and not to individual atoms. For homeopolar compounds, atomic refractions are more often used in calculations, and ionic refractions are used in calculations of ionic compounds.

Refractive additivity is widely used as a simple, unreliable way to verify the correctness of the proposed structure of a molecule. In this case, they do this: they calculate the theoretical value of refraction for each possible structure using the additivity rule and compare it with the refraction of a given substance found experimentally. To determine the experimental value, one practically has to find only the values ​​of n and d in equation (II 1.2). For example

mor, experience"™5 value of diethyl sulfide is 28.54. The theoretical value is calculated based on the expected structural

Using the bond refraction values ​​(Table 3), we obtain the following value:

Calculation by atomic refractions also leads to a similar result:

The coincidence of the Ya values ​​obtained experimentally and theoretically indicates the correctness of the assumptions of the structural formula of diethyl sulfide.

Table 3

Atomic refractions and bond refractions

When studying compounds with alternating multiple bonds, a difference is observed between the calculated and experimental values ​​of /\m, which goes beyond the limits of experimental errors. This discrepancy is explained by a change in the nature of the bond as a result of the interaction of directly unconnected atoms and is called exaltation of refraction (denoted by ER). The exaltation value is entered as an additional term in the sum of the refractions of atoms. Typically, exaltation increases greatly as the number of conjugated bonds increases, indicating an increase in the mobility of n electrons.

From Maxwell’s electromagnetic theory of light it follows that for wavelengths significantly removed from the region of their absorption by molecules of matter, the equality is true:

where n∞ is the refractive index of light for certain wavelengths.

Taking this into account, the Clausius-Mosotti equation (15) takes the following form:

[ cm3/(g mol)] (19)

From the resulting expression it is clear that the RM index, called molar refraction, has the dimension of the volume of molecules contained in 1 mole of a substance.

Equation (15), which is called the Lorentz-Lorentz equation, was derived in 1880 independently by H. Lorentz and L. Lorentz.

In practice, the specific refraction index r is often used, that is, the refraction of one gram of a substance. Specific and molar refractions are related by the relation: R = r∙M, where M is the molar mass.

Since in equation (19) N is proportional to density, it can be represented in the following form:

[cm3/g] (20)

H. Lorentz and L. Lorentz revealed the physical meaning of the concept of refraction - as a measure of electronic polarizability and laid a solid theoretical foundation for the doctrine of refraction.

The value of specific refraction is practically independent of temperature, pressure and the state of aggregation of a substance.

In research practice, in addition to the molar and specific refraction RM and r, other derivatives of the refractive indices n are used (Table 2).

The refractive index of non-polar substances practically does not depend on the frequency of light waves and therefore equation (19) is valid at all frequencies. For example, for benzene n2 = 2.29 (wavelength 289.3 nm), while ε = 2.27. therefore, if for approximate calculations of refraction it is enough to use the refractive index of the visible spectrum, then for accurate calculations it is necessary to extrapolate using the Cauchy formula:

nλ = n∞ + a/λ2, (21)

where nλ is the refractive index at wavelength λ;

a is an empirical coefficient.

Table 2 Refractometric constants

	Name	Designation	Application area
	Refractive index		Characteristics of the purity of substances. Analysis of binary systems of substances
	Specific refraction		Characteristics of the purity of substances. Determination of substance concentration
	Molecular refraction		Determination of the values ​​of some atomic and molecular constants. Determination of the structure of organic molecules
	Relative dispersion		Analysis of complex mixtures. Determination of the structure of organic molecules

For polar substances ε > n2. For water, for example, n2 = 1.78 (λ = 589.3 nm), and ε = 78. Moreover, in these cases it is impossible to directly extrapolate nλ using the Cauchy formula due to the fact that the refractive index of polar substances often changes anomalously with frequency . However, there is usually no need to make such an extrapolation, since refraction is an additive quantity and is conserved if the refractive indices of all substances are measured at a certain wavelength. The yellow line in the sodium spectrum (λD = 589.3) was chosen for this standard wavelength. The reference tables provide data specifically for this wavelength. Thus, to calculate molecular refraction (in cm3/mol), a formula is used in which n∞ is replaced by nD.

Almost all methods for studying polarizability are based on changes in the characteristics of light during its interaction with matter. The limiting case is a constant electric field.

The internal local field F acting on the molecule is not identical to the external field E imposed on the dielectric. To calculate it, the Lorentz model is usually used. According to this model

F = (e + 2) ∙ E / 3,

where e is the dielectric constant (permeability).

The sum of the dipole moments induced in each of the N 1 molecules contained in a unit volume is the polarization of the substance

P = N 1 ∙a∙F = N 1 ∙a∙E∙(e+2)/3,

where a is polarizability.

Molar polarization (cm 3 /mol) is described by the Clausius-Mossotti equation

P = (e-1) / (e+2)∙M/r = 2.52∙10 24 ∙a,

in the SI system (F∙m 2 ∙mol -1)

P = N∙a / 3∙e 0 = 2.52∙10 37 ∙a

In the case of an alternating electric field, including the field of a light wave, various polarization components appear due to the shift of electrons and atomic nuclei, depending on frequency.

For non-polar dielectrics, according to Maxwell’s theory, e = n 2, therefore, with appropriate replacement, the Lorentz-Lorentz equation of molecular refraction is obtained

R = (n 2 –1) / (n 2 +2)∙M / r = 4/3∙p∙N∙a,

where n is the refractive index; r - density; N is Avogadro's number.

A similar equation can describe the specific refraction

(n 2 –1) / (n+2)∙1/r = 4/3∙p∙N 1 ∙a.

Molecular refraction is the polarization of one mole of a substance in the electric field of a light wave of a certain length. This is the physical meaning of molecular refraction.

When extrapolated to an infinite wavelength, the electronic polarization P e is obtained:

P e = P ¥ = (n 2 ¥ -1)/(n 2 ¥ +2)∙M/r = 4/3∙N∙a e

Calculation from molecular refraction is the only practically used method for finding the average polarizability a, cm 3. Substituting the numerical values ​​of the constants gives

a = 0.3964∙10 24 ∙R ¥ .

The experimental determination of molecular refraction involves measurements of refractive index and density.

The most important property of molecular refraction is its additivity. The possibility of a priori calculation of the value of refraction from the increments of the corresponding atoms and bonds allows, in some cases, to accurately identify a chemical compound, as well as to study the resulting intra- and intermolecular interactions based on the deviations of the experiment from the calculation.

The refraction of the mixture is additive - specific refraction by mass fractions of components w, molecular - by mole fractions x, which makes it possible to calculate the refraction of substances from data for solutions. If we denote the parameters of the solvent by index 1, the dissolved substance by 2, and the solution by 1.2, we get

R 2 =1/f 2 ×[(n 1.2 2 – 1)/(n 1.2 2 + 2) × (M 2 f 2 + M 1 (1 – f 2))/r 1.2 – R 1 × (1 – f 2)] .

When expressing the concentration in moles per 1 liter (C), we have

R 2 =(n 1 2 –1)/(n 1 2 +2)(M 2 /r 1 –1000/C(r 1.2 –r 1)/r 1)+1000/C((n 1, 2 2 –1)/(n 1,2 2 +2)–(n 1 2 –1)(n 1 2 +2)).

The best results are obtained by graphical or analytical extrapolation of the refraction or refractive indices and densities of solutions to infinite dilution. If the concentration dependences of the latter are expressed by the equations

r 1,2 = r 1 ×(1 + b×w 1),

n 1,2 = n 1 × (1 + g×w 2),

then the specific refraction

¥ R 2 = R 1 (1-b) + 3n 1 2 g/r 1 (n 1 2 + 2) 2 .

When carrying out measurements in solutions, it is necessary to fulfill certain experimental conditions, in particular, the use of the maximum possible concentrations of the analyte.

4.1.1. Calculation of polarizability values ​​of atoms and molecules from refractometric data. Boettcher, based on the Onsager model, obtained the equation of molecular refraction in the form

R=4/3pNa9n 2 /((n 2 +2)[(2n 2 +1)–a/r 3 (2n 2 –2)]),

where r is the radius of the molecule.

This equation allows one to simultaneously determine polarizability and molecular sizes.

An approximate calculation of atomic polarizability as a certain fraction of electronic polarizability or molecular refraction has become widespread: P a = kP e, where the coefficient k is 0.1 or 0.05.

4.1.2. Additive nature of molecular refraction and polarizability. The basis on which the use of polarizability to establish chemical structure, the distribution of electrons and the nature of intramolecular interactions, the configuration and conformation of molecules, became the idea of ​​​​the additivity of molecular quantities. According to the principle of additivity, each structural fragment - a chemical bond, an atom, a group of atoms, or even individual electron pairs– a certain value of the parameter under consideration is assigned. The molecular value is represented as a sum over these structural fragments. Any molecule is a system of atoms or bonds interacting with each other. Strict additivity assumes that the parameters of each structural fragment remain unchanged during the transition from one molecule containing it to another. Any interactions lead to changes in the properties of atoms and bonds or to the appearance of additional contributions to molecular quantities. In other words, the additive value of a property assigned to each atom depends not only on its nature, but also on its environment in the molecule. Therefore, no physical property can be strictly additive. In such a situation, the way in which the principle of additivity is used must be adjusted to certain specified conditions.

To date, two main trends have emerged in the development and application of additive polarizability schemes. On the one hand, the dependence of the polarizability parameters of atoms or bonds on their environment forces us to specify the additive scheme, introducing, for example, increments for atoms of any element in different valence states or different types of bonds; then the nature of the substitution at the neighboring atom is taken into account, etc. In the limit, this approach leads to a set of polarizabilities of each fragment or to the calculation of average polarizabilities and anisotropies of large structural units, a kind of “submolecules”, which automatically take into account the interactions within them.

The second tendency is to use some additive scheme and consider all deviations from it as manifestations of interactions.

The first approach is considered more acceptable when studying the spatial structure of molecules, when identifying the effects of mutual influence is unimportant.

The second approach is used mainly in analyzing the electronic structure of rigid molecules.

In 1856, Berthelot pointed out that there is a simple relationship between the molecular refractions of neighboring members of a homologous series:

R n–1 – R n = const = R CH 2

In accordance with this equation, the molecular refraction of the nth member of the homologous series can be considered as the sum of the molecular refractions of the first member and n–1 CH 2 groups:

R n = R 1 + (n–1)∙R CH2 ,

where n is the serial number of the member of the homologous series.

In chemistry, two schemes are used for calculating molecular refraction - by atoms and by bonds that make up the compound under study.

According to the first scheme, molecular refraction for some groups of compounds depends only on the nature and number of atoms in the molecule, and can be calculated by summing the atomic refractions characteristic of a given element:

R(C n H m O p X g)=n×R C +m×R H +p×R O +g×R X ,

where R(C n H m O p X g) is the molecular refraction of the compound with the composition C n H m O p X g ; R C, R H, etc. – atomic refractions of carbon, hydrogen and other elements.

In the second case, molecular refraction is calculated from bonds. The use of this calculation scheme was facilitated by the establishment of the influence of the nature of bonds on molecular refraction, which had great importance, because opened up the possibility of using molecular refraction to determine the structure of organic substances. It was shown that the value of molecular refraction also reflects the nature of the bonds of other elements. In addition to the nature of the atoms forming the bond and the multiplicity of the bond, the influence of strained cycles on molecular refraction was proven and special increments were derived for three-membered and then four-membered carbon rings.

In complex functional groups with multivalent elements (–NO 2 , –NO 3 , –SO 3 , etc.) it is impossible to strictly determine atomic refractions without conditional assumptions, so group refractions of radicals began to be used.

Subsequently, it was found that the values ​​of molecular refraction are determined mainly by the number and properties of higher (valence) electrons involved in the formation of chemical bonds, in addition, the nature of the chemical bonds plays a decisive role. In this regard, Steiger (1920), and then Fajans and Klorr proposed to consider molecular refraction as the sum of bond refractions. For example, for CH 4:

R CH4 = R C + 4R H = 4R C-H

R C-H = R H + 1/4×R C

R CH2 = R C + 2×R H = R C-C + 2×R C-H

R C - C = 1/2×R C

The method of calculating bonds using refractions is more consistent, simpler and more accurate. In chemistry, both bond refractions and atomic refractions are used.