Molar refraction and determination of the structure of molecules. Molecular refraction Find molecular refraction
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 case of variable electric field, including the field of a light wave, various components of polarization 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 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 - is assigned a certain value of the parameter in question. 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 none physical property cannot 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 “submolecule” that automatically takes 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 researching spatial structure molecules, when identifying the effects of mutual influence is unimportant.
The second approach is used mainly in the analysis electronic structure hard 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 – serial number 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 organic matter. 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.
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 the 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 valid, 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 state of aggregation substances. Unlike the dipole moment, it is a scalar quantity.
Molecular refractions of compounds can be presented additively, that is, as the sum of refractions of the constituent parts of the molecule (additivity rule). The latter can be considered bonds or atoms (ions). Bond refractions have a true physical meaning, since a polarizable electron cloud in chemical compound belongs to bonds 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 Y values obtained experimentally and theoretically indicates the correctness of the assumptions structural formula 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 also 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 |
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Refractive index |
Characteristics of the purity of substances. Analysis of binary systems of substances |
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Specific refraction |
Characteristics of the purity of substances. Determination of substance concentration |
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Molecular refraction |
Determination of the values of some atomic and molecular constants. Determination of the structure of organic molecules |
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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.
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 relationship: 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.
3. Refractive index dispersion. Dependence of refractive indices on temperature and pressure. Molar refraction.
Maxwell's electromagnetic theory for transparent media relates the refractive index n and the dielectric constant by the equation: =n 2 (1). The polarization P of the molecule is related to the dielectric constant of the medium: P = P def + P op = (-1)/(+ 2) (M /d) = 4/3 N A , (2) where P def is the deformation polarization ; P or – orientational polarization; M is the molecular weight of the substance; d-density of the substance; N A - Avagadro number; is the polarizability of the molecule. Substituting n 2 into equation (2) instead of and el, instead of , we obtain (n 2 - 1)/ (n 2 + 2) (M /d) = 4/3 N A el = R el = R M ( 3) This formula is called the Lorentz-Lorentz formula, the value of R M in it is the molar refraction. From this formula it follows that the value of RM, determined through the refractive index of a substance, serves as a measure of the electronic polarization of its molecules. In physicochemical studies, specific refraction is also used: g = R M / M = (n 2 1)/ (n 2 + 2) (1/d) (4)
Molar refraction has the dimension of volume per 1 mole of a substance (cm 3 /mol), specific refraction has the dimension of volume per 1 gram (cm 3 /g). Approximately considering the molecule as a sphere of radius g m with a conducting surface, it is shown that el = g M 3. In this case, R M = 4/3 N A g 3 (5), i.e. molar refraction is equal to the intrinsic volume of molecules of 1 mole of a substance. For non-polar substances R M =P, for polar substances R M is less than P by the amount of orientation polarization.
As follows from equation (3), the value of molar refraction is determined only by polarizability and does not depend on the temperature and state of aggregation of the substance, i.e. is a characteristic constant of a substance.
Refraction is a measure of the polarizability of the molecular electron shell. The electron shell of a molecule is made up of the shells of the atoms that form the molecule. Therefore, if we assign certain refraction values to individual atoms or ions, then the refraction of the molecule will be equal to the sum of the refractions of atoms and ions. When calculating the refraction of a molecule through the refractions of its constituent particles, it is necessary to take into account the valence states of atoms, the features of their arrangement, for which special terms are introduced - increments of multiple (double and triple carbon-carbon) and other bonds, as well as corrections for the special position of individual atoms and groups in the molecule : Rm= Ra+Ri, (6), where R A and Ri are atomic refractions and multiple bond increments, respectively, which are given in reference books.
Equation (6) expresses the rule of additivity of molar refraction. A physically more justified method is to calculate the molar refraction as the sum of the refractions not of atoms, but of bonds (C-H, O-H, N-H, etc.), since it is the valence electrons that are polarized by light , forming a chemical bond.
The molar refraction of compounds built from ions is calculated as the sum of ionic refractions.
The additivity rule (6) can be used to establish the structure of molecules: compare Rm, found from experimental data using equation (3), with that calculated using equation (6) for the expected structure of the molecule.
In some cases, the so-called exaltation of refraction, consisting in a significant excess of the experimental value R M no compared to that calculated by equation (6). Exaltation of refraction indicates the presence of conjugated multiple bonds in the molecule. Exaltation refraction in molecules with such bonds is due to the fact that the -electrons in them belong to all atoms forming the conjugation system and can move freely along this system, i.e. have high mobility and, therefore, increased polarizability in an electromagnetic field.
Additivity also occurs for the refraction of liquid mixtures and solutions - the refraction of the mixture is equal to the sum of the refractions of the components divided by their shares in the mixture. For molar refraction binary mixture in accordance with the additivity rule, we can write: R=N 1 R 1 +(1 N 1)R 2 , (7)
for specific refraction r = f 1 r 1 + (lf 1)r 2 (8), where N 1 and f 1 are the mole and weight fractions of the first component.
These formulas can be used to determine the composition of mixtures and the refraction of components. In addition to the chemical structure of a substance, the value of its refractive index is determined by the wavelength of the incident light and the measurement temperature. As a rule, with increasing wavelength the refractive index decreases, but for some crystalline substances an anomalous behavior of this dependence is observed. Most often shown, refractions are determined for wavelengths (yellow Na line - D-589nm line, red hydrogen line - C-656nm line, blue hydrogen line - F-486nm line).
The dependence of refraction or refractive index of light on wavelength is called dispersion. A measure of dispersion can be the difference between the values of refractive indices measured at different wavelengths, the so-called. average variance. The measure of dispersion is relative dispersion: F , C , D =(n f – n C)/(n D -l)]10 3 (9), where n f , n C , n D are refractive indices measured for lines F and C hydrogen and sodium D lines. The relative dispersion F, C, D is very sensitive to the presence and position of double bonds in the molecule.
The value of the refractive index of a substance also depends on the measurement temperature. As the temperature decreases, the substance becomes more optically dense, i.e. the refractive index increases. Therefore, when carrying out refractometric measurements, it is necessary to thermostat the refractometer. For gases, the refractive index also depends on pressure. The general dependence of the refractive index of a gas on temperature and pressure is expressed by the formula: n-1=(n 0 -1)(P/760)[(1+P)/(1+t) (10), where n is the refractive index at pressure P and temperature t° C; n 0 - refractive index under normal conditions; P - pressure k mm Hg. Art.; and - coefficients depending on the nature of the gas .
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