I. I. KOTYUKOV and M. N. YAKIMOV.

ON THE JOINT INFLUENCE

OF RADICALS ON OPTICAL

ACTIVITY.

Tomsk
1929.

I. I. Kotyukov and M. N. Yakimov.

On the Joint Influence of Radicals on Optical Activity

There exists a whole series of optically active substances in which substitution is possible at several positions in the molecule. Thus, for example, in tartaric acid radicals may enter both the carboxyl and the hydroxyl groups, either one of them or both simultaneously. In the case of the simultaneous presence of several substituents, their influence on the magnitude of rotation must in one way or another be summed. Taking into account the influence of each radical separately on the parent substance, we decided to trace in what relation the magnitude of rotation of the di- and, in general, polysubstituted compound would stand to the rotation of both monosubstituted compounds. At the same time we dwelt in some detail on the optical activity of tartaric acid diamide, owing to its unusually high rotation and the absence in the literature of data on its dispersion. It also seemed of interest to compare, as far as possible, all the literature data on the influence of amidation of acids on rotation. The corresponding selection from the “Index of Optically Active Substances” gave a very variegated picture of the influence of amidation. The angle of rotation of an acid, when hydroxyl is replaced by \(\mathrm{NH_2}\), usually changes both its magnitude, in one direction or the other, and not infrequently its sign. All known cases are compared in the appended table. Here the first column gives the serial number of the comparison, the second the series number in the “Index of Optically Active Substances,” and the third the empirical formula. Then follow the molecular weight, name and structure. Next come the boiling or melting point, solvent, specific and molecular rotation, coefficient of influence, and finally the citation. The coefficient of influence was calculated by the usual formula

\[ \frac{[M]'-[M]}{[M]} \]

where \((M)'\) is the molecular rotation of the amide and \((M)\) the rotation of the acid. Cases with a change in the sign of rotation we have all assigned to a decrease in the angle of rotation of the acid, assuming that the rotation here falls to zero and then changes sign.

Then \((M)'\) and \((M)\), in their absolute magnitude, are added together and the sum is divided by \((M)\). (See Table No. 1.)

The table contains in all 43 cases of known amides together with their acids, and in 30 comparisons we observe a decrease in the magnitude of the angle of rotation. No special regularity, either in the strength of the influence or in its direction, is naturally detected.

We also selected from the literature cases in which it is possible to compare the joint influence of radicals on optical activity.

For structural reasons such a comparison is possible chiefly for derivatives of tartaric, malic, and lactic acids, and in isolated cases for other active substances. Four preparations were compared simultaneously: the parent substance, its monosubstituted ...

by radical R, its monosubstituted derivative by radical R′, and finally the substance with both radicals R and R′ present. In some cases the radicals R and R′ could be identical, and then only three substances were compared. For all cases we calculated the active magnitude of the influence according to the scheme \(M_4 = M_1 + (M_2 - M_1) + (M_3 - M_1) = M_2 + M_3 - M_1\), where \(M_1\) is the molecular rotation of the parent substance, \(M_2\)—of its substance monosubstituted by radical R, \(M_3\)—by radical R′, and \(M_4\)—simultaneously by radicals R and R′.

In the enormous majority of cases the influence does not obey simple additivity, and therefore here, as in the preceding table, the coefficient of influence was calculated according to the same scheme. (See Table No. 2).

Thus, we assembled a total of 48 comparisons, and additivity is observed in only 13 of them: Nos. I 1, 2, II 1, 2, 3, 4, 5, for tartaric; I 1, 2, II 1, 2 for malic; II 1 for lactic and for diphenyltartaric acid. If one takes into account that in the remaining comparisons the different preparations were not polarized in identical solvents, then this number will evidently increase considerably (tartaric I 6, 8, 10—12; II 7, 17—19. Malic II 3, 4; lactic I 3, etc.).

It must be supposed that, in cases where additivity is absent, the meso-formula of the total preparation differs substantially from the meso-formula of the starting products. Conversely, for the additive cases we are entitled to regard the meso-structures in all the preparations as analogous.

Tartaric acid diamide was investigated for rotation by Frankland and Slator\(^{1}\), who obtained it from methyl tartrate by the action of alcoholic \(NH_3\). The preparation melted at \(195^\circ\) and had

\[ [\alpha]\frac{20}{D} = +106.5 \]

and

\[ [M] = +158 \]

in water \((p = 1,\ 305)\), and \([\alpha] = 144^\circ\) and \([M] = 213\) in \(CH_3OH\) \((p = 0.797)\).

We prepared the compound by the same method; from 250 g of tartaric ester, 180 g of crude product with \((\alpha) = 95^\circ\) were obtained. The diamide was purified by recrystallization from glacial acetic acid, and after only two recrystallizations optical purity was attained.

I crystallization —

\[ [\alpha]\frac{20}{D} = 99.99, \]

II crystallization —

\[ [\alpha]\frac{20}{D} = 110.74, \]

III crystallization

\[ [\alpha]\frac{20}{D} = 110.64. \]

It melts at \(204—205^\circ\) with decomposition. Thus, recrystallization from acetic acid achieves a very complete purification of the preparation, raising its melting temperature by almost \(10^\circ\), in comparison with Frankland and Slator.

Pure tartaric acid diamide is readily soluble in water, glacial acetic acid, formic acid, and formamide. It dissolves poorly in alcohols and pyridine. It is completely insoluble in \(CHCl_3\), \(CS_2\), \(C_6H_6\), petroleum ether, acetic anhydride, paraldehyde, \(CH_3COH\), \(C_6H_5OH\), acetic ester, etc. It dissolves very well in a mixture of methyl alcohol and glacial acetic acid.

Dibenzoyldiamide of tartaric acid was first obtained by Einhorn\(^{2}\) by the action of benzoyl chloride on tartramide in the presence of NaOH. We prepared it by the same method, specifying the procedure somewhat. 1 g of tartramide is dissolved in 30 cm\(^3\) \(H_2O\), and 3 cm\(^3\) benzoyl chloride and 3 cm\(^3\) of 10% NaOH are added.

\(^{1}\) Soc. 83, 1349 (1903) Frankland und Slator.
\(^{2}\) A. 361, 144 (1908). Einhorn.

The mixture is shaken vigorously for 2–3 minutes, whereupon a white crystalline precipitate separates, containing both the preparation and benzoic acid. The precipitate is filtered off and treated twice with a solution of soda; the dibenzoyl derivative does not dissolve. It is filtered, washed with water, alcohol, and ether, and dried in a desiccator. It melts at about \(240^\circ\). It was purified by recrystallization from hot alcohol, 10 g per 1 liter; upon cooling, 6 g separated. After two recrystallizations it is optically pure. \([\alpha]^{20}_{D}=110.9,\ 114^\circ 9,\ 114.77^\circ\), respectively. It is readily soluble in acetic acid and formic acid. In formamide and alcohol only on heating. In the other ordinary solvents it does not dissolve.

We worked with an excellent Landolt–Lippich polarimeter, with a three-part field, manufactured by F. Schmidt und Haensch, allowing readings to be made with an accuracy up to \(0.01^\circ\). The sources of light used were the sodium flame of a Bunsen burner, a mercury quartz lamp, and a large lamp with a tungsten incandescent filament. As light filters we used a saturated solution of \(K_2Cr_2O_7\) for the sodium flame, several solutions proposed by Winther¹), and glass light filters obtained by us from the firm Goldberg in Berlin. Winther’s solutions served partly for preparing gelatin light filters, and were partly used directly in cuvettes. The gelatin light filters were prepared in the following way. Take \(72\ \mathrm{cm.^3}\) of a 15% solution of gelatin and dilute with \(48\ \mathrm{cm.^3}\) of \(H_2O\). To this are added 0.12 g of crystal violet 5BO. Measure out \(55.7\ \mathrm{cm.}\) of the solution and pour it onto a horizontal glass plate of area \(55.7\ \mathrm{cm.^2}\), previously carefully cleaned with alcohol and rubbed with talc. We obtain crystal-violet light filter No. 1.

From the remaining solution, measure out 60 cm. and add to it \(60\ \mathrm{cm.^3}\) of a 9% solution of gelatin. From this again take \(55.7\ \mathrm{cm.^3}\) and make a casting. This is crystal-violet light filter No. 2. Again from the remainder measure out 60 cm., add \(20\ \mathrm{cm.^3}\) of 9% gelatin, and make the last casting. This is crystal-violet light filter No. 3.

Take \(36\ \mathrm{cm.^3}\) of 15% gelatin \(+\ 24\ \mathrm{cm.^3}\ H_2O\). Dissolve 2.75 g of metanil yellow (Victoria gelb) and cast, on the same plate, \(55.7\ \mathrm{cm.^3}\). A metanil light filter is obtained.

For obvious reasons the salt solutions had to be kept in glass cuvettes, which we partly made ourselves and partly obtained from the firm Goldberg.

In this way we obtained the following regions of the spectrum:

¹) Winther Ph. ch. (1902) 41, 166.

To obtain regions of the spectrum, the transmission resultants (so-called centers of gravity) through quartz were determined.

We had at our disposal a plate of right-handed quartz with an area of \(1\ \mathrm{cm}^2\) and a thickness of about \(1\ \mathrm{mm}\).

At a temperature of \(24.6^\circ\mathrm{C}\) it rotated by \(23^\circ 73\) with light filter No. 2.

We used Lowry’s dispersion formula for quartz

\[ [\alpha]^{20}_{D\,1\,\mathrm{mm}.} = \frac{11\cdot 6064}{\lambda^2-0\cdot 010627} + \frac{13\cdot 42}{\lambda^2-78\cdot 21} + \frac{4\cdot 3685}{\lambda^2} \]

If for light filter No. 2 the center of gravity is \(=539\cdot 48\), i.e. \(0\cdot 58948\), then

\[ \alpha = 34\cdot 4578 - 0\cdot 17236 - 12\cdot 5728 = 21\cdot 71267 \]

The dependence of the magnitude of the angle of rotation of quartz on temperature is expressed by Gumlich’s formula

\[ \alpha_t = \alpha_o \left(1 + 0\cdot 000147\,t\right) \]

Then a plate \(1\ \mathrm{mm}\) thick at a temperature of \(24\cdot 6^\circ\) should rotate by

\[ 21\cdot 71264 \left(1 + 0\cdot 000147 . 24\cdot 6\right) = 21\cdot 7911 \]

Hence the thickness of our plate is

\[ \frac{23\cdot 73}{21\cdot 7911} = 1\cdot 0889\ \mathrm{mm}. \quad (\text{error } \pm 0\cdot 0015) \]

Now determining the angle of rotation of the plate with various light filters, we calculate, by the same Lowry formula, \(\lambda\), i.e. the center of gravity for the given light filter. The results of these determinations are brought together in the following table:

Thus, in the light filters chosen by us, Nos. 1, 2, 4, and 9 very closely approached the corresponding spectral lines and served for us as reference points. In polarization they showed impeccable monochromaticity.

For the Landolt apparatus we had a whole series of tubes, of which we used three: 1, 2, and 3 dm. Their length was measured by us on a dividing machine against the normal meter of the local physical laboratory of the STI. It proved to be 99.50, 199.87, and 300.04 mm, respectively.

Solutions for polarization were prepared by weighing both the solvent and the substance, with subsequent determination of density in Boot pycnometers.

The density was calculated by the Landolt formula

\[ d_t=\frac{FQ_0}{W_0}+\frac{FQ_0}{W_0}(t_0-t)\,3\beta-\frac{\lambda(F-W_0)}{W_0} \]

\(F\)—weight of the liquid at temperature \(t\).
\(Q_0\)—density of water at the same temperature.
\(W_0\)—weight of water at temperature \(t_0\).
\(3\beta=0.000024\) (coefficient of expansion of glass).
\(\lambda=0.0012\)—density of air at temperature \(t\).

Thus, the second term on the right-hand side gives the correction for the change in volume of the pycnometer with temperature, and the third the correction for buoyancy.

All tubes of the polarimeter were fitted with jackets through which vaseline oil from a thermostat circulated. It was delivered into the jackets by a pump set in motion by a motor, which also turned the stirrer of the thermostat. The thermostat was heated by a current passing through a thin iron wire placed on a glass frame directly in the oil. Constancy of temperature was maintained by a nitrobenzene thermoregulator. The temperature could be regulated with an accuracy up to \(0.1^\circ\).

With this apparatus we carried out a series of determinations of rotational spectra for both preparations in various solvents and at various temperatures.

Diamide of tartaric acid.

\[ \mathrm{H_2O};\quad p=1.6291;\quad l=2\ \mathrm{dm}.\quad d=1.0047;\quad t=22\cdot 6^\circ\mathrm{C}. \]

\[ \text{Dispersion coefficient }\frac{8}{1}=\frac{7.28}{2.84}=2.56 \]

Graph No. 1.

Diamide of tartaric acid.

\[ \mathrm{HCOOH};\quad p=0.911;\quad l=1\ \mathrm{dm};\quad d=1.008;\quad t=25^\circ\mathrm{C}. \]

\[ \text{Dispersion coefficient }\frac{9}{1}=\frac{2.32}{0.75}=3.0 \]

Graph No. 2.

Tartaric acid diamide.

\[ \mathrm{HCONH_2};\quad p = 1.607;\quad l = 1\ \mathrm{dm.}\quad d = 1.1539;\quad t = 22.3^\circ \mathrm{C}. \]

\[ \text{Dispersion coefficient } \frac{9}{1}=\frac{5.27}{1.76}=3.0. \]

Graph No. 3.

Tartaric acid diamide.

\[ \mathrm{C_2H_4O_2};\quad p = 0.16;\quad l = \mathrm{dm};\quad d = 1.0497;\quad t = 25^\circ \mathrm{C}. \]

\[ p = 0.085;\quad d = 1.0497;\quad l = \mathrm{dm.}\quad t = 25^\circ \mathrm{C}. \]

\[ p = 0.085;\quad d = 1.006;\quad l = 1\ \mathrm{dm.};\quad t = 60^\circ \mathrm{C}. \]

Graph No. 4.

Tartaric acid diamide.

\[ \mathrm{H_2O}\cdot p = 1.6291. \]

Polarization of tartramide with additions of $\mathrm{HCl}$, $\mathrm{HgCl_2}$, $\mathrm{C_6H_5NH_2}$ gives the same angles, and therefore these additions exert little influence.

Dibenzoyldiamide of tartaric acid.

\[ \mathrm{C_2H_5OH};\ p = 0.03;\ l = 4\ \mathrm{dm}.\ d = 0.7987;\ t = 22^\circ,\ m = 356 \]

\[ \text{Dispersion coefficient}\ \frac{9}{2} = \frac{0.38}{0.16} = 2.3. \]

Graph No. 5.

Dibenzoyldiamide of tartaric acid.

\[ \mathrm{C_2H_4O_2};\ p = 0.302;\ l = 1\ \mathrm{dm}.\ d = 1.0500;\ t = 20^\circ\mathrm{C}. \]

\[ \text{Dispersion coefficient}\ \frac{9}{2} = \frac{1.34}{0.59} = 2.3. \]

Graph No. 6.

Dibenzoyldiamide of tartaric acid.

\[ \mathrm{HCOOH};\ p = 1.126;\ l = 1\ \mathrm{dm}.\ d = 1.232;\ t = 21^\circ5.\ \mathrm{C}. \]

\[ \text{Dispersion coefficient}\ \frac{9}{2} = \frac{6.08}{2.50} = 2.4 \]

Graph No. 7.

From the material presented it is evident that tartramide does indeed possess high rotation and normal dispersion. At present a number of authors consider the cause of anomalous rotational disper-

… the presence in solution of a mixture of two differently rotating mesoisomers; moreover, for tartaric acid the American authors give the following meso-formulas:

\[ \begin{array}{cc} \text{[meso-formula of tartaric acid, with } \mathrm{H\cdots OH}\text{ bridges]} & \text{[alternative cyclic meso-formula of tartaric acid, with } \mathrm{H\cdots OH}\text{ bridges]} \end{array} \]

From our point of view the following meso-formulas are more rational:

\[ \begin{array}{cc} \text{[meso-formula with carboxyl groups and } \mathrm{H\cdots O}\text{ links]} & \text{[cyclic meso-formula with two } \mathrm{C\cdot OH}\text{ groups and } \mathrm{H\cdots O}\text{ links]} \end{array} \]

That is, they are completely analogous to those proposed by one of us for fumaric and maleic acids¹.

The replacement of \(\mathrm{OH}\) by \(\mathrm{NH_2}\) according to formulas I should not have destroyed mesoisomerism, for \(\mathrm{N}\) could still be connected by a mesovalence with \(\mathrm{H}\). The normal rotatory dispersion of tartramide and the high angle argue for the fact that both nitrogens in it are connected with one another by a mesovalence.

\[ \begin{array}{c} \mathrm{CO}\\[-2pt] / \quad \backslash\\[-2pt] \mathrm{CHOH}\quad \mathrm{NH_2}\\[-2pt] \vert \qquad \vdots\\[-2pt] \mathrm{CHOH}\quad \mathrm{NH_2}\\[-2pt] \backslash \quad /\\[-2pt] \mathrm{CO} \end{array} \]

In this structure there are no possibilities for the manifestation of mesoisomerism, and, in addition, it is cyclic, which also accounts for the strong rotation.

Apparently we must assume the same for other optically active diamides as well, the angle of rotation of the diamide being small—

differ from the corresponding imide, if such is known. This is in fact observed.

In a previous communication, one of us indicated that the lowering of the rotation of camphoric acid anhydride, as compared with the acid itself, is explained by the greater mesocyclicity of the acid than of the anhydride. If it is assumed that the nitrogens of both amide groups are joined to one another by mesovalence, then the rotation of camphoric-acid diamide should be less than that of the acid and should differ little from the rotation of a manifestly cyclic derivative, i.e. the imide. The figures in the first table fully correspond to these assumptions.

That is, for camphoric-acid diamide we must adopt the following meso-formula

\[ \begin{array}{c} \mathrm{CO}\\[-2mm] /\ \ \ \ \backslash\\[-1mm] \mathrm{CH}\ \ \ \ \mathrm{NH_2}\\ /\ \backslash\ \ \ \ \vdots\\ \mathrm{H_2C}\ \ \mathrm{C(CH_3)_2}\\ \vdots\ \ \ \ \backslash\\ \mathrm{H_2C}\ \ \ \mathrm{C}\\ \ \ \ \ \ \ \ \ |\\ \ \ \ \ \ \ \ \mathrm{CH_3}\\ \ \ \ \ \ \ \ \backslash\ \ \ /\\ \ \ \ \ \ \ \ \mathrm{CO}\ \ \mathrm{NH_2} \end{array} \]

If we now compare the rotations of all known diamides with the corresponding acids, we obtain the following figures:

\[ \begin{array}{ccccc} (\mathrm{CH_3})_2\mathrm{C} & - & \mathrm{CH.COOH}\\ | & & |\\ \mathrm{CH_2} & - & \mathrm{C} & - & \mathrm{COOH}\\ & & |\\ & & \mathrm{CH_3} \end{array} \]

\[ \begin{array}{ccccc} \mathrm{C_6H_5.CH} & - & \mathrm{CH.C_6H_5}\\ | & & |\\ \mathrm{HOOC} & . & \mathrm{CH} & - & \mathrm{CH.COOH} \end{array} \]

The character of the change in rotation upon amidation here can be wholly motivated by the theory of mesostructure. Thus, the high rotation of malic diamide, of deltruxinic-acid diamide, and of trans-isophen-

that of oxycamphoric acid has the same cause as in tartramide—the joining of both nitrogens by mesovalency and the increase in the cyclicity of the molecule. In cis-isophenoxycamphoric diamide the rotation is likewise the same as that of the acid. It is possible that the proximity of the two carboxyls in the latter produces mesovalency between the two carbonyl oxygens, and therefore the cyclicity remains unchanged upon amidation.

The identical rotation of the acid and the diamide in methoxytartaric acid is quite inexplicable. It should be noted that the introduction of a methoxy or ethoxy group into the molecule always raises the rotation very sharply and unexpectedly, as can be traced in a large number of preparations. From the standpoint both of the classical theory and of the theory of mesostructure, this circumstance appears very puzzling, and a special systematic investigation is needed to clarify it.

As for the rotation of dibenzoyldiamide, its angle proves to obey the rule of strict additivity of influence. Here we compare the rotation in alcohol of the following preparations:

That is, the introduction of two amide groups gives an increase of \(213 - 5 = 208^\circ\). Adding this influence to the rotation of dibenzoyltartaric acid, we have \(208 + 397 = 605^\circ\). The discrepancy with experiment is only \(2\%\). If one takes into account that molecular rotations are being compared, obtained as the result of multiplying the reading of the limb by a large number (300), then the agreement should be regarded as complete. Naturally, the question of the cause of one or another direction of rotation remains open for the time being.

If additivity of rotation is observed, then we are entitled to consider the mesoformulas of all the preparations identical. Then dibenzoyldiamide has the following mesoformula:

It also excellently explains the increase of rotation in the pair dibenzoyltartaric acid—dibenzoyldiamide of tartaric acid—from \(-397\) to \(-594.52^\circ\).

The organization of scientific work at the present time is an extremely difficult task. Both the apparatus and the auxiliary materials were assembled by us only thanks to the enormous kindness of our colleagues. The Landolt apparatus with its inventory and the auxiliary preparations were received by us from Prof. B. V. Tronov. Prof. N. V. Tantsov lent us the quartz lamp. The dyes for the light filters were obtained from Prof. I. V. Gebler. The quartz plate was received from Prof. I. A. Sokolov, who also allowed us to use their dividing machine for measuring polarimetric tubes. In addition, Ilya Arkad′evich personally checked our setup and gave a number of valuable instructions. Prof. Goryachev greatly assisted us in calculating the verniers.

To all the persons named above we express our profound gratitude.

Tomsk, 23 XII, 28.

TABLE A No. 1

INFLUENCE OF AMIDATION ON THE ANGLE OF ROTATION

TABLE No. 2

JOINT INFLUENCE OF RADICALS

II. Angle greater than the additive one.

Malic Acid.

1. The angle is less than additive.

II. Angle greater than the additive.

DIPHENYL—TARTARIC ACID.

I. Angle greater than the additive.

To the article by Prof. I. I. Kanonnikov.

Graph No. 1

Dimide of tartaric acid.
\(H_2O;\ p = 1.6291;\ l = 2\ \mathrm{cm};\ d = 1.004;\ t = 22.5^\circ C\)

\[ \frac{8}{1} = \frac{7.28}{2.84} = 2.56 \]

dispersion coefficient.

Graph No. 2

Dimide of tartaric acid.
\(HCOOH;\ p = 0.911;\ l = 1\ \mathrm{cm};\ d = 1.008;\ t = 25^\circ C\)

\[ \frac{9}{1} = \frac{2.32}{0.75} = 3 \]

dispersion coefficient.

To the article by Prof. I. I. Kotenko.

Graph No. 3.

\([M]\)

Cinnamic acid diamide.

\[ \mathrm{HCONH_2};\ p=1.607;\ l=1\ \mathrm{cm.};\ d=1.1539;\ t=22.3^\circ\mathrm{C} \]

Dispersion coefficient

\[ \frac{g}{l}=\frac{5.27}{1.76}=3.0 \]

\(\lambda\)

Graph No. 4.

\([M]\)

Cinnamic acid diamide.

\[ \mathrm{C_2H_4O_2};\ p=0.16;\ l=1\ \mathrm{cm.};\ d=1.0497;\ t=25^\circ\mathrm{C} \]

Dispersion coefficient:

\[ \frac{g}{l}=\frac{0.54}{0.17}=3.0 \]

\(\lambda\)

For the article of Prof. V. V. Ponomarev.

Graph No. 5

\[ [\alpha] \]

Dibenzoyl tartrate

\[ \mathrm{C_2H_5OH};\ p = 0.03;\ l = 4\ \mathrm{dm};\ d = 0.7987;\ t = 22^\circ\mathrm{C} \]

\[ m = 356 \]

Dispersion coefficient:

\[ \frac{\varphi}{2} = \frac{0.38}{0.16} = 2.3 \]

\[ [\alpha]\quad 0.7\quad 0.6\quad 0.5\quad 0.4\quad \lambda \]

Graph No. 6

\[ [\alpha] \]

Dibenzoyl tartrate

\[ \mathrm{C_2H_4O_2};\ p = 0.3102;\ l = 1\ \mathrm{dm};\ d = 1.0500;\ t = 20^\circ\mathrm{C} \]

Dispersion coefficient:

\[ \frac{\varphi}{2} = \frac{1.34}{0.59} = 2.3 \]

\[ 0.7\quad 0.6\quad 0.5\quad 0.4\quad \lambda \]

Article by Prof. U. U. Kotonov.

[α]

Graph No. 7.

Dibenzoyldiamide of tartaric acid.

\(\mathrm{HCOOH};\ p = 1.195;\ l = 1\ \mathrm{gm};\ d = 1.232;\ t = 21.5^{\circ}\mathrm{C}\)

Dispersion coefficient

\[ \frac{9}{2}=\frac{6.08}{2.50}=2.4 \]

\(\lambda\)

1. Izvestiya STI, vol. 48 (1928). ↩