POLARITY AND STRUCTURE OF MOLECULES

Ya. Syrkin, Ivanovo-Voznesensk

Polarization of Dielectrics

The long-known phenomenon of polarization of dielectrics is explained in modern physics from the point of view of the electrical structure of atoms. Ultimately, an atom consists of a nucleus and electrons, combined according to the laws of quantum mechanics. Between the individual constituent parts of a molecule or atom there exist bonds that vary from electron to electron in one and the same molecule.

At the present time we distinguish:

1) Intranuclear bonds. These are those which determine the stability of the nucleus, which is a complex formation of electrons and protons. The problem of the stability of the nucleus and the clarification of the dependence between the rate of its disintegration and the energy of the products of decomposition (helium nuclei) is posed by modern wave mechanics.

2) Intra-atomic bonds. Here the question concerns the interaction between the nucleus and the inner electrons of the atom. A quantitative equivalent of these bonds may be the energy required to detach an electron from the nucleus. In the case, for example, of the silver atom, containing 47 electrons, the ionization potential of the last, peripheral electron is equal to 7.5 volts, whereas the ionization potential of the first, nearest electron is equal to 30,000 volts. In the case of ordinary chemical reactions with a heat effect of about 100–300 small calories (4–12 volts), the inner electrons apparently remain untouched.

3) Bonds determined by peripheral electrons.

a) heteropolar—when there is a displacement of one or several electrons toward the more strongly electronegative atom. This creates an electrical asymmetry of the molecule, expressed in the appearance of a dipole moment.

b) homeopolar, which according to wave mechanics are a consequence of \(\psi\)-oscillations (a rough analogy may be seen in the coupled oscillations of two pendulums suspended from one board, where the energy of one pendulum is resonantly transferred to the other, setting it into oscillatory motion).1

Separate parts of a molecule—ions or organic radicals (\(\mathrm{CH}_2\), etc.)—constitute, to a certain extent, a single whole; they in a sense “self-determine” and exist in the molecule with the rights of an autonomous state within a state. A quantitative sign of this autonomy is the well-known additivity of the properties contributed to the molecule by groups. The heats of combustion or formation in homologous series increase approximately by the same amount with the introduction of each new \(\mathrm{CH}_2\) group. In a number of cases definite values of heats of formation may be assigned to the bonds \(\mathrm{C—C}\), \(\mathrm{C=O}_2\), etc. However, this autonomy is relative. Additivity is preserved in homologous series of similar compounds. A \(\mathrm{CH}_2\) group adjacent to a hydroxyl in alcohols, to a carboxyl in acids, to \(\mathrm{COH}\) in aldehydes, or \(\mathrm{NH}_2\) in amines, acquires different properties depending on the neighboring group. Sometimes an electrically polar group imposes a definite stamp upon the entire remaining part of the molecule. We shall see below that all alcohols, irrespective of the length of the hydrocarbon chain, possess a constant moment. Apparently, in this case the polarity, i.e. the electrical asymmetry of the molecule, is determined entirely by the hydroxyl group.

An external electric field, or the field of neighboring molecules, may exert a twofold action upon a given molecule.

POLARITY AND STRUCTURE OF MOLECULES

It can cause a displacement of charges, in other words deform the molecule. The electrons will then be displaced toward the positive side, and the positive charges toward the negative side. An induced additional moment will result, the magnitude of which may be taken as proportional to the field strength. It can be shown that this effect (deformation polarization), in the first approximation, does not depend on temperature. Under the action of the field the charges assume a certain new position of equilibrium, being displaced by a distance \(x\). If the force tending to return the displaced charges is proportional to the displacement (a quasi-elastic bond), then the potential energy is obtained from the equation

\[ \int f x\, dx = \frac{f x^{2}}{2}, \]

where \(f\) is the coefficient of proportionality. The potential energy in a field of strength \(F\) is equal to \(- e x F\). Thus the total potential energy is expressed as

\[ U = \frac{f x^{2}}{2} - e x F. \]

The number of charges displaced by an amount lying between \(x\) and \(x + dx\) is equal to \(A e^{-\frac{U}{kT}} dx\) (\(k\) is Boltzmann’s constant — \(1.37 \cdot 10^{[[unclear: exponent]]}\), \(A\) is a constant obtained from the condition that integration over all values of \(x\) must give the total number of displaced particles. But displacement of a charge by an amount \(x\) means the appearance of an additional moment \(ex\),

therefore the expression \(A e^{-\frac{U}{kT}} dx\) gives the number of molecules with moments \(ex\). To find the mean moment, the sum of all moments must be divided by the sum of all molecules. Thus

\[ m = \frac{\int A e x e^{-\frac{U}{kT}}\, dx} {\int A e^{-\frac{U}{kT}}\, dx} = \frac{e^{2}F}{f} \tag{1} \]

This result, correct in the first approximation, has essential significance. It shows that, since polarization is conditioned by the displacement of charges, the effect (aver-

... arising moment) independent of temperature. It is erroneous to ascribe deformation only to the displacement of electrons relative to positive charges. A molecule may contain the above-mentioned autonomous groups or ions, which also may be displaced under the action of the field. Therefore one must distinguish the electron-deformation effect (the induced moment \(m_e\)) and the ion-atomic-group effect—\(m_a\).

The action of the field is not always limited to the appearance of an additional moment. Such an asymmetric arrangement of charges in the molecule is possible, owing to which a dipole moment already exists in finished form even before the action of the field. Let us renumber the charges of the molecule and note the position of each in some coordinate system rigidly connected with the given molecule. In this there is no need to regard the charges as immobile. In the case of motion of the charges, apparently, one may speak of some average position in time, considering the charge in each region proportional to the time during which it stays there. Therefore the acceptance of electronic waves and “smeared” charges does not contradict the calculations below. Let us show how the electrostatic treatment leads to dipole moments. Let us imagine a molecule with charges \(e_1, e_2, e_3,\ldots e_i\). Each charge has coordinates \(\xi_i, \eta_i, \zeta_i\). We ask about the magnitude of the electric potential of the molecule at a point \(P\), situated at a distance \(r\) from the origin of coordinates. If the distances of the charges from the point \(P\) are \(r_1, r_2,\ldots r_i\), then the potential

\[ E=\sum_i \frac{e_i}{r_i}. \]

If \(P\) is far from the origin of coordinates, then \(r\) differs little from the values \(r_i\). We write

\[ r_i=\sqrt{(x-\xi_i)^2+(y-\eta_i)^2+(z-\zeta_i)^2}, \]

where \(x, y\), and \(z\) denote the coordinates of the point \(P\).

Taking into account that \(x^2+y^2+z^2=r^2\), we may write

\[ \frac{1}{r_i} = \frac{1}{r} \left[ 1-\frac{2}{r^2}\left(\xi_i x+\eta_i y+\zeta_i z\right) +\frac{1}{r^2}\left(\xi_i^2+\eta_i^2+\zeta_i^2\right) \right]^{-\frac12}. \]

Expanding in a series, we find

\[ \frac{1}{r_i} = \frac{1}{r} \left[ 1+\frac{\xi_i x+\eta_i y+\zeta_i z}{r^2} -\frac{1}{2}\frac{\xi_i^2+\eta_i^2+\zeta_i^2}{r^2} + \frac{3}{2}\frac{\left(\xi_i x+\eta_i y+\zeta_i z\right)^2}{r^4} -\ldots \right]. \]

If we neglect terms decreasing faster than \(\dfrac{1}{r^3}\), then for the potential we obtain

\[ \begin{aligned} E &= \sum \frac{e_i}{r_i} = \frac{1}{r}\sum e_i + \frac{1}{r^2} \left[ \frac{x}{r}\sum e_i\xi_i + \frac{y}{r}\sum e_i\eta_i + \frac{z}{r}\sum e_i\zeta_i \right] \\ &\quad + \frac{1}{r^3} \left[ \left(\frac{3}{2}\frac{x^2}{r^2}-\frac{1}{2}\right)\sum e_i\xi_i^2 + \left(\frac{3}{2}\frac{y^2}{r^2}-\frac{1}{2}\right)\sum e_i\eta_i^2 + \left(\frac{3}{2}\frac{z^2}{r^2}-\frac{1}{2}\right)\sum e_i\zeta_i^2 \right] \\ &\quad + \frac{3}{r^3} \left[ \frac{yz}{r^2}\sum e_i\eta_i\zeta_i + \frac{zx}{r^2}\sum e_i\zeta_i\xi_i + \frac{xy}{r^2}\sum e_i\xi_i\eta_i \right] +\ldots . \end{aligned} \tag{2} \]

The first term of the series is not equal to zero if \(\sum e_i\) is different from zero, i.e. when charges of one sign predominate. In this case there is an ion. If \(\sum e_i\) is equal to zero, the molecule is electrically neutral.

The quantities \(\sum e_i\xi_i,\ \sum e_i\eta_i,\ \sum e_i\zeta_i\) (the sums of the products of the charges by the corresponding coordinates) are the components of the dipole moment of the molecule. If \(\sum e_i\xi_i\) is not equal to zero, this shows that the centers of gravity of the plus and minus charges do not coincide and that the molecule is polar. In simplified form the molecule may be represented by an electric rod (by analogy with a magnet), at whose two ends opposite charges are concentrated. The product of the length of the dipole by the magnitude of the charge gives the moment. The coordinates of the electric centers of gravity of the plus and minus charges are, apparently, equal to

\[ \overset{+}{x} = \frac{\overset{(+)}{\sum} e_i\xi_i}{\overset{(+)}{\sum} e_i}; \qquad \overset{+}{y} = \frac{\overset{(+)}{\sum} e_i\eta_i}{\overset{(+)}{\sum} e_i}; \qquad \overset{+}{z} = \frac{\overset{(+)}{\sum} e_i\zeta_i}{\overset{(+)}{\sum} e_i} \]

\[ \overset{-}{x} = \frac{\overset{(-)}{\sum} e_i\xi_i}{\overset{(-)}{\sum} e_i}; \qquad \overset{-}{y} = \frac{\overset{(-)}{\sum} e_i\eta_i}{\overset{(-)}{\sum} e_i}; \qquad \overset{-}{z} = \frac{\overset{(-)}{\sum} e_i\zeta_i}{\overset{(-)}{\sum} e_i}. \]

\[ m_x = (\overset{+}{x} - \overset{-}{x}) \sum e_i;\qquad m_y = (\overset{+}{y} - \overset{-}{y}) \sum e_i; \]

\[ m_z = (\overset{+}{z} - \overset{-}{z}) \sum e_i. \tag{3} \]

In the case where the components of the dipole moment are equal to zero, the molecule is characterized by \(\sum e_i \xi_i^2\); these are the components of the so-called quadrupole moment (electric moment of inertia). Next come octupoles. At present nothing is known about moments of higher orders. In this article we shall consider the properties of molecules of polar character. Atomic dimensions are of the order of \(10^{-8}\) cm, and elementary charges are \(10^{-10}\). Therefore an appreciable dipole moment has the value \(10^{-18}\). Experiment gives, in the best cases, an accuracy of \(0.03\)—\(0.1 \cdot 10^{-18}\). In this connection, the modern methodology cannot serve as a criterion of polarity if the moments are below the value just indicated.

Under the action of a field, a dipole molecule will rotate, tending to assume the position of least potential energy. Such an alignment is impeded by disordered thermal motion. It is natural, therefore, that the orientational polarization of rigid dipoles should depend on temperature (in contrast to deformation polarization). At absolute zero, or close to it, even an insignificant field will be capable of orienting the molecules completely. At very high temperatures it will be difficult for the field to cope with the thermal chaos, and the orientation will be insignificant. Apparently, the effect must depend on the magnitude of the field, the temperature, and the magnitude of the moment—so to speak, on the readiness of the molecule itself for orientation. To find the quantitative dependence, Debye transferred entirely to electric dipoles the analogous reasoning of Langevin as applied to paramagnetic gases. In the latter case, a magnetic moment is assigned to the molecule and its distribution in a magnetic field is investigated with allowance for temperature disorientation.

Let the dipole moment be equal to \(m\), the field strength \(F\), and the angle between the axis of the dipole and the direction of the field \(\alpha\); then the compo-

POLARITY AND STRUCTURE OF MOLECULES

The moment’s component in the direction of the field is equal to \(m\cos\alpha\), and the relative potential energy is \(-mF\cos\alpha\). The number of dipoles forming with the field a solid angle between \(\omega\) and \(\omega+d\omega\) (where \(\omega=\sin\alpha\,d\alpha\,d\varphi\)) will be

\[ Ae^{\frac{mF\cos\alpha}{kT}}\sin\alpha\,d\alpha\,d\varphi . \]

Again, \(A\) is determined from the condition that, upon integrating over all possible directions (\(\alpha\) from \(0\) to \(\pi\) and \(\varphi\) from \(0\) to \(2\pi\)), we obtain the total number of dipoles. The mean moment is equal to

\[ \bar m= \frac{\iint Ae^{\frac{mF\cos\alpha}{kT}}\,m\cos\alpha\sin\alpha\,d\alpha\,d\varphi} {\iint Ae^{\frac{mF\cos\alpha}{kT}}\sin\alpha\,d\alpha\,d\varphi}. \tag{4} \]

or

\[ \frac{\bar m}{m}= \frac{e^{\frac{mF}{kT}}+e^{-\frac{mF}{kT}}} {e^{\frac{mF}{kT}}-e^{-\frac{mF}{kT}}} -\frac{kT}{mF}, \]

\[ \frac{\bar m}{m} = \frac{1}{3}\cdot\frac{mF}{kT} -\frac{1}{45}\left(\frac{mF}{kT}\right)^3 . \tag{5} \]

If

\[ \frac{mF}{kT}\ll 1, \]

i.e. the field strength is small or the temperature is high, then one may restrict oneself to the first term.

\[ \frac{\bar m}{F}=\frac{1}{3}\frac{m^2}{kT}. \tag{6} \]

In the Langevin equation, instead of \(F\) there enters the strength of the magnetic field and, instead of \(m\), the magnetic moment. Let us note that the coefficient \(1/3\) appears in formula (6) as the mean value of \(\cos^2\alpha\).

The mean moment caused by the displacement of electrons, atoms, and groups, as well as by orientation, is equal to

\[ m=\left(m_{\mathrm e}+m_a+\frac{1}{3}\frac{m^2}{kT}\right)F \tag{7} \]

or

\[ m=a'F. \tag{8} \]

The classical Clausius–Mossotti equation gives

\[ \frac{D-1}{D+2}=\frac{4}{3}\pi n a'. \tag{9} \]

Here \(D\) is the dielectric constant (DEK), \(n\) is the number of molecules per unit volume, and \(\alpha'\) is the polarizability. If the molecular weight is \(M\), Avogadro’s number is \(N\), and the specific gravity is \(d\), then

\[ n=\frac{dN}{M}. \]

Hence

\[ \frac{D-1}{D+2}\,\frac{M}{D} = \frac{4}{3}\pi N\left(m_v+m_a+\frac{1}{3}\frac{m^2}{kT}\right). \tag{10} \]

Usually the molecular polarization is denoted as

\[ P=\frac{4}{3}\pi N\alpha'. \tag{11} \]

First of all, the question naturally arises how legitimate it is to replace the classical \(\alpha'\) by the value

\[ m_v+m_a+\frac{1}{3}\frac{m^2}{kT}. \]

Expression (9) is obtained from the following considerations. One imagines a molecule situated between the plates of a large capacitor. Around the molecule one conceives a certain spherical volume, large in comparison with the dimensions of the molecule. The force caused by the charges on the plates of the capacitor is taken into account, and then also the additional force acting on the molecule from that part of the dielectric which lies outside the described sphere, i.e. between the sphere and the plates. This force is caused by charges induced outside the sphere and on its surface. Such a treatment is macroscopic, for the action of molecules separated from the one under consideration by distances comparable with molecular distances is not taken into account. Therefore expression (10) is valid above all for gases of low density. In the case of association, when two neighboring molecules possess sufficiently large moments and can form temporary complexes of interlacing dipoles, expression (10) is already insufficient. Likewise for solutions, where the molecules are close to one another and where interaction with the solvent is possible (especially if the latter itself has a dipole structure), formula (10) is not always applicable. However, this danger is not especially great, for, as we shall see below,

dipole moments of one substance in different nonpolar solvents \((\mathrm{C_6H_6},\ \mathrm{CS_2},\ \mathrm{CCl_4},\ \mathrm{C_6H_4})\) have close values, with an accuracy up to \(0.04—0.1 \cdot 10^{-18}\). The application of formula (10) for a single substance in the gaseous or liquid state may give fluctuating values of the moment.

Gans made an attempt to refine expression (10) by taking into account the action of fields caused by nearby molecules. The neighbors interact with the molecule and hinder its alignment in the field. As a consequence, this disorienting action will be greater than that which is taken into account in the Langevin formula. This deviation, according to Debye, is too insignificant. Let us imagine two neighboring molecules in a liquid. The first creates a field acting at the location of the second. This field changes all the time in magnitude and direction. Therefore one may speak of a time-average value of the field intensity and of deviations from this average value. The additional field (fluctuating) changes rapidly, for it depends on the rotation of the molecule. The latter is measured by a time of the order of \(10^{-13}\) sec. Such a short interval is insufficient for the additional Gans effect to manifest itself. In other words, the alignment in the field required by the Langevin formula is only slightly distorted by the influence of rapidly rotating neighboring molecules. Gans’s theory leads to a complicated formula into which a new quantity enters, namely the distance between the centers of two molecules at their maximum approach to one another. Without entering into the question of how exhaustively Gans’s theory takes into account the correction for the interaction of molecular fields, we shall point out that in practice this correction is inessential. Where Debye’s formula is insufficient (the case of association, strong solutions), Gans’s formula gives no better results. In other cases the deviation from Debye’s formula is by no means sufficient to prefer Gans’s formula on this basis. The point is that the temperature coefficient of polarization is sometimes positive and sometimes negative. Some authors, such as Kröö, Boguslavsky, Pukker, and Lundblad, point out that equation (10), for a number of

of reasons may be inaccurate. First of all, the proportionality of the moment to the field is only an approximation. The terms of formula (10) that reflect the influence of the induced moments only in the first approximation do not contain the temperature. Therefore slight deviations may be attributed not only to Gans’s correction. Equation (10) may be regarded, for example, as the equation of state of ideal gases. At low pressure, for gases and dilute solutions, it is undoubtedly the most reliable method for determining the magnitude of dipole moments.

The validity of formula (6) has also been discussed from another standpoint. In deriving the mean value of the moment in a field we used the classical statistics of Boltzmann. Under the action of the field a molecule rotates about its center of gravity, but the rotational energy can assume only certain values permitted by quantum theory. It turned out that the old quantum mechanics (with integral quantum numbers) gives, for the rotator, formula (6) with the coefficient 1.54 instead of \(1/3\). Another method of calculation with half-integral quanta led to the number 4.57. It is interesting that the new quantum mechanics of Schrödinger–de Broglie, as well as of Heisenberg–Born, gives the formula with the coefficient \(1/3\), obtained from classical statistics. With respect to dipole moments, only in Schrödinger’s mechanics is the “correspondence” condition fulfilled, i.e. the moment at high temperatures passes into the value required by the classical theory. In the following small table the moments of HCl for different coefficients are given.

Table 1

	Coeff.	\(m \cdot 10^{18}\) for HCl
Classical theory, Debye, 1912	\(1/3\)	1.03
Integral quantum numbers, Pauli, 1921	1.54	0.492
Half-quantum numbers, Pauling, 1925	4.57	0.32
New quantum mechanics, 1926, Mensing and Pauli, de Kronig, Manneback	\(1/3\)	1.03

The separate terms of formula (7), which subdivide the polarization into electronic, atomic, and rigid-dipole parts, are not all known with the same degree of accuracy.

When passing from a constant field to an alternating one, the total polarizability changes. It decreases with increasing frequency, showing an irregular course in the absorption regions. The polarization caused by electronic displacement is generally identical with molecular refraction in the visible part of the spectrum. There are almost no measurements of refraction in the infrared region; therefore knowledge of atomic polarization is especially difficult. An approximate estimate can be made as follows. For gases and liquids without a permanent dipole moment, \(P_{\text{total}} = P_{\text{electr.}} + P_{\text{atomic}}\). But \(P_{\text{el.}}\) is determined from refraction. This makes it possible to calculate \(P_{\text{atomic}} = P_{\text{total}} - P_{\text{electr.}}\). In general this is also applicable to solids with permanent dipoles, since in the solid state the dipoles may be regarded as oriented and the additional alignment under the action of the external field may be neglected. With some reservation, supercooled liquids may also be included here.

If gases and liquids have a permanent moment, then in order to judge the atomic polarization it is necessary to make measurements for the substance in the solid state. Then \(P_{\text{solid}} = P_{\text{electr., solid}} + P_a{}_{\text{ solid}}\). Next, for the same substance, \(P_{\text{el., liquid}}\) is found from refraction and, taking \(P_{\text{el., liquid}} \cong P_{\text{el., solid}}\), one finds \(P_a\).

Such approximate calculations show that atomic polarization constitutes only an insignificant part of the whole effect.

Substance	\(P - P\) el.
Methylbenzene	\(2.9\ \text{cm}^3\)
Ethylbenzene	\(3.4\) "
normal propylbenzene	\(3.0\) "
iso-propylbenzene	\(3.5\) "
iso-butylbenzene	\(3.0\) "

Substance	\(P\)	\(P\) el.
\(\mathrm{CH_4}\) gas	7.15	\(6.45\ \text{cm}^3\)
\(\mathrm{CCl_4}\) liquid		\(25.8\) "
\(\mathrm{CCl_4}\) solid	27
\(\mathrm{CS_2}\) liquid		\(20\) "
\(\mathrm{CS_2}\) solid	19.8
\(\mathrm{C_6H_6}\) liquid		\(25.65\) "
\(\mathrm{C_6H_6}\) solid	26.95
\(\mathrm{H_2O}\) vapor	\(\sim 60\)	\(3.7\) "
\(\mathrm{H_2O}\) liquid	17.5	\(3.7\) "
\(\mathrm{H_2O}\) ice \((<-2.1^\circ)\)	9.5	\(3.7\) "
\(\mathrm{CH_3OH}\) liquid		\(8.2\) "
\(\mathrm{CH_3OH}\) solid	13.1
\(\mathrm{C_2H_5OH}\) liquid		\(12.7\) "
\(\mathrm{C_2H_5OH}\) solid	16.2

A similar calculation can be carried out for crystal hydrates. If \(P\) of the anhydrous salt and of the hydrate is known, \(P\) for water can be found. Dividing this quantity by the number of water molecules, one obtains \(P_{h_2o}\). Subtracting \(P_{\text{electr.}} = 3.7\), one finds \(P_a\) in the hydrate.

\(\mathrm{Na_2CO_3\,10H_2O}\)	\(P_{h_2o}=8.45\)	\(P_a=4.75\ \mathrm{cm^3}\)
\(\mathrm{BaCl_2\,2H_2O}\)	\(7.95\)	\(4.25\)
\(\mathrm{CuSO_4\,5H_2O}\)	\(7.76\)	\(4.07\)

The obtained \(P_a\) are smaller than the corresponding value for ice (\(5.8\ \mathrm{cm^3}\)) and, moreover, differ from one another. This depends on the strength of the bonding of water in the lattice. Different water molecules in one molecule of salt are not bound in the same way. For the first molecule \((\mathrm{CuSO_4H_2O})\) an even smaller value of \(P_a\) is found, which is explained by strong deformation. Comparing a large amount of material, Smyth found that 1) \(P_a\) is not additive, 2) \(P_a\) is the greater, the more atomic nuclei and groups there are in the molecule, 3) \(P_a\) is the greater, the more asymmetrically the dipoles are arranged in the molecule.

Determination of dipole moments

Measurement of the dielectric constant of gases at different temperatures makes it possible to determine dipole moments by formula (10):

\[ \frac{D-1}{D+2}\frac{M}{d}=A+\frac{B}{T} \tag{12} \]

where

\[ B=\frac{4}{9}\pi\frac{N}{K}m^2 \tag{13} \]

Hence substitution of \(k=1.37\cdot10^{-16}\), \(N=6.06\cdot10^{23}\) gives

\[ m=0.0127\cdot10^{-18}\sqrt{B}. \]

If the expression \(\dfrac{D-1}{D+2}\dfrac{M}{d}\) does not change with temperature \((B=0)\), then there is no moment in the molecule. Experiment shows that in atoms of the zero group and in the molecules

\[ \mathrm{H_2,\ N_2,\ O_2,\ CO_2,\ CH_4,\ CS_2,\ CCl_4,\ C_6H_6,\ CH_{14}} \]

permanent dipoles are absent. For better confidence in the accuracy of formula (12), measurements of the D.E.C. must be made at not too low temperatures and in not too strong fields, when dielectric saturation is possible, i.e. an approximation to the most complete orientation. In the latter case the D.E.C. decreases (Ratnowsky, Terwerg).

As methods one uses: 1) the improved Nernst method, 2) the resonance method, 3) the method of undamped oscillations, 4) the ellipsoidal method according to Fürth, and others. Recently the molecular-beam method has been improved; it consists in transferring the Stern—Gerlach experiment to the case of an electric field. This method, refined by Estermann, is related to Stern’s method in approximately the same way as Debye’s theory is to Langevin’s.

If the measurement of the D.E.C. is given only for one temperature, then, in order to determine the moment, it is necessary to eliminate the term \(A\) in formula (12). For this the Lorentz–Lorenz equation is used

\[ \frac{n^2 - 1}{n^2 + 2}\,\frac{M}{d} = \frac{4}{3}\pi N\alpha, \tag{14} \]

where \(n\) is the refractive index.

If the frequency of the incident wave is chosen in the appropriate way, it is possible to arrange that, of the electronic, atomic-nuclear, or permanent dipole polarization, one or two will be eliminated. In essence, one may speak of the D.E.C. for a given wavelength. Orientation and the corresponding alignment in the field require a certain time to attain the stationary state. This is what happens in a static field. If, however, the field is variable, then after a change in direction the dipoles reorient themselves, which requires a certain time. If the alternating field has a high frequency, then the molecules do not in general have time to follow the field, and then the permanent dipoles will not affect the polarization. Debye applied to this phenomenon the concept of relaxation time, which Maxwell introduced into the kinetic theory of gases. There the question is, after how much time

molecules that have deviated for some reason from the normal distribution of velocities will return to it, if the cause that produced the deviation is removed. By relaxation time is meant, in this case, the time in which the deviation falls to \(e^{-1}=0.36\) of its former value.

If particles of radius \(a\) rotate in a liquid with viscosity \(\mu\), then for the relaxation time Debye finds

\[ \tau=\frac{4\pi \eta a^3}{kT} \]

For water this gives, at \(a=2\cdot 10^{-8}\), \(\tau=0.25\cdot 10^{-10}\) sec. For air one obtains approximately 8–10 times more. The corresponding frequencies lie in the region of short radio waves. Thus it becomes possible to explain the phenomenon of anomalous dispersion and absorption in the electrical part of the spectrum.

In the case of colloidal particles in the form of long dipole rods, even at a frequency of \(10^6\) orientation will no longer occur. At frequencies \(10^{15}\) only the effect of displacement of peripheral electrons will appear; at \(\nu=10^{17}\) (the region of X-rays) polarization will be due to the internal electrons of the atom. Thus the refractive index, by the Lorentz–Lorenz formula, makes it possible to determine the deformational polarization. The obtained \(A\) may be substituted into formula (12) and the moment calculated. This method no longer has the same accuracy as the first, since polar molecules give absorption bands both in the ultraviolet and in the infrared part of the spectrum. The latter are explained partly by the action of rigid dipoles, partly by intra-atomic vibrations in the molecule. If the deformational polarization is small in comparison with the total polarization, then a small error in the value of the first has little effect on the result of the calculation. In this case the indicated method (a combination of the DEK formula and refraction) gives satisfactory agreement with the moment found from the temperature coefficient of DEK.

Equation (12) is applicable also to solutions, if one assumes,

that the polarization of the solution obeys the additivity of the component parts. The results may be distorted by the interaction of molecular fields. In dilute solutions the dissolved molecules are relatively far from one another, but close to the molecules of the solvent. If the latter is nonpolar, it may be regarded as a quadrupole, octupole, etc. In this case the force of interaction decreases inversely proportional to a higher power of the distance than in the interaction of dipoles. Neglecting the action of the electro-symmetric multipoles of the solvent on the dipole, one may apply equation (12), extrapolating the polarization to the case of infinite dilution. Let the dielectric constant of the solution be \(D'\); in a unit volume there are \(n_1\) molecules of the first kind (molecular weight \(M_1\)) and \(n_2\) molecules of the second kind (molecular weight \(M_2\)). Then

\[ \frac{D' - 1}{D' + 2} = \frac{4}{3}\pi (n_1\alpha_1 + n_2\alpha_2) \tag{15} \]

Let us introduce the notation

\[ \frac{n_1}{n_1+n_2}=f;\quad \frac{n_2}{n_1+n_2}=f_2 \tag{16} \]

We obtain

\[ \frac{D' - 1}{D' + 2}\ \frac{1}{(n_1+n_2)} = \frac{4}{3}\pi (f_1\alpha_1 + f_2\alpha_1) \tag{17} \]

The density of the solution \(d_1\)

\[ d_1=\frac{n_1M_1}{N}+\frac{n_2M_2}{N} \tag{18} \]

From (18) and (16) we have

\[ \frac{1}{n_1+n_2}=\frac{f_1M_1+f_2M_2}{Nd_1} \]

Substituting this into equation (17), we find

\[ \frac{D' - 1}{D' + 2}\ \frac{M_1f_1+M_2f_2}{d_1} = \frac{4}{3}\pi N(\alpha_1f_1+\alpha_2f_2) \tag{19} \]

But \(\frac{4}{3}\pi N\alpha_1\) is the polarization of the first component \((P_1)\), and correspondingly \(\frac{4}{3}\pi N\alpha_2\) that of the second \((P_2)\). This gives

\[ \frac{D' - 1}{D' + 2}\ \frac{M_1f_1+M_2f_2}{d_1} = P_1+P_2=P_{12} \tag{20} \]

\(P_{12}\)—the polarization of the mixture—is determined experimentally.

If the polarization of the pure solvent is known, it is possible to calculate \(P_2\)—that of the dissolved substance. This latter is plotted for different concentrations of the solution and the limiting value is taken.

\[ P_2 \text{ at } \lim f_2 = 0 \]

Subtracting from the obtained \(P_2\) the temperature-independent part according to formula (14), one finds the value of \(B\) in formula (12), and hence the moment.

The DEK of solutions can be determined at different temperatures, and thus the term \(B\) (formula 12) is obtained without the aid of refraction. Unfortunately, in this case the temperature interval is more limited than for gases.

Finally, the moments can be calculated from the equation of state of real gases. The mutual attraction (van der Waals forces) apparently has an electrostatic character. In this case mutual deformation and orientation are possible.

Representing the equation of state in expanded form with virial coefficients and comparing the corresponding terms of the series with the results of statistical calculation, one can, as Keesom showed, calculate the moment. However, this method is unreliable for detecting polarity, since in the cases studied the results can sometimes be interpreted both from the standpoint of dipole interaction and of quadrupole interaction.

Different methods of determining moments—such as measurement of the DEK of gases at different temperatures, at constant or variable density, at one temperature with allowance for refraction, determination of the polarization in solution—sometimes lead to several differing values of the moments. In a liquid and in a gas the moments are by no means obliged to coincide. The phenomenon of association of dipoles also leads to a change in the mean observed moment. However, the methodology has recently been developed to such an extent that, for many substances, the magnitudes of the moments are well known.

Polar Groups and Dipoles

The dipole moment is a physical constant characterizing a molecule with respect to electrosymmetry. However, only in the case of comparatively simple molecules \((\mathrm{H_2O}, \mathrm{NH_3})\) is it possible, by combining data from molecular spectra with the moments found, to come close to the question of structure. Often the interpretation is hypothetical in character. For the greater part of organic substances the moments have been obtained from measurements of the dielectric constant of the solution. The solvent must necessarily be nonpolar \((\mathrm{CS_2}, \mathrm{C_6H_6}\), etc.). Only in this case is the influence of the medium negligible, as is seen from the table.

Table 2

Solvent	Dissolved substance	\(m \cdot 10^{18}\)
\(\mathrm{C_6H_6}\)	\(\mathrm{C_6H_5NO_2}\)	3.90
\(\mathrm{CS_2}\)	"	3.89
\(\mathrm{C_6H_{14}}\)	"	3.89
\(\mathrm{C_6H_6}\)	\(\mathrm{C_6H_5Cl}\)	1.55
\(\mathrm{CS_2}\)	"	1.52
\(\mathrm{C_6H_{14}}\)	"	1.55
\(\mathrm{C_6H_6}\)	\(\mathrm{C_6H_5OH}\)	1.70
\(\mathrm{CS_2}\)	"	1.64
\(\mathrm{C_6H_{14}}\)	\(\mathrm{C_{10}H_8}\)	0.72
\(\mathrm{CS_2}\)	\(\mathrm{C_{10}H_8}\)	0.69
\(\mathrm{C_6H_6}\)	\(\mathrm{C_6H_4(OC_2H_5)_2}\)	1.76
\(\mathrm{CCl_4}\)	\(\mathrm{C_6H_4(OC_2H_5)_2}\)	1.72

Between the moment determined in the gas and in solution there may be a somewhat greater difference. For ethyl ether \((\mathrm{C_2H_5})_2\mathrm{O}\), in vapor, \(1.1 \cdot 10^{-18}\) is found, and in solution \(1.22 \cdot 10^{-18}\). Among triatomic molecules, \(\mathrm{CO_2}\) and \(\mathrm{CS_2}\) do not have a permanent moment. One may assume that in them the O and S atoms are arranged linearly and symmetrically. Fatty saturated hydrocarbons do not have dipoles. In them the moments of the \(\mathrm{C-H}\) bonds apparently mutually compensate one another. Of the unsaturated compounds, ethylene \(\mathrm{C_2H_4}\) is not dipolar. However, a moment appears when radicals are present on both sides of the double bond.

are not identical. Already α-butylene, \(\mathrm{CH_2=CH-CH_2-CH_3}\), has a small moment, \(\sim 0.37 \cdot 10^{-18}\). In general, however, the experimental material concerning unsaturated hydrocarbons is so scanty that it does not permit any unambiguous conclusions. Replacement of hydrogen by some group—\(\mathrm{COH}\), \(\mathrm{COOH}\), \(\mathrm{OH}\), \(\mathrm{NO_2}\), \(\mathrm{NH_2}\), \(\mathrm{Cl}\), \(\mathrm{Br}\), \(\mathrm{J}\)—entails the appearance of a moment. In this case it sometimes happens that the magnitude of the moment is determined mainly by the polar group, and does not depend on the rest of the molecule. Practically all alcohols have \(m=\sim 1.70 \cdot 10^{-18}\), independently of the length of the hydrocarbon chain, the character of the radical (aliphatic or aromatic), and the position of the \(\mathrm{OH}\) group (\(-\mathrm{CH_2OH}\) bond, \(-\mathrm{CHOH}-\), \(\equiv\mathrm{C}-\mathrm{OH}\)).

Table 3

Substance	\(m \cdot 10^{18}\)	Substance	\(m \cdot 10^{18}\)
\(\mathrm{CH_3OH}\)	1,68	Dimethyl \(\mathrm{C_5H_{11}O_5}\)
\(\mathrm{C_2H_5OH}\)	1,69	Ethyl carbinol	1,8
\(\mathrm{C_3H_4OH}\)	1,66	Normal \(\mathrm{C_8H_{17}OH}\)	1,7
\(\mathrm{C_6H_5OH}\)	1,66	\(\mathrm{C_7H_{15}OH}\) normal heptanol	1,7
\(\mathrm{C_4H_9OH}\)	1,66	\(\mathrm{CH_3 \cdot CHOH \cdot C_5H_{11}}\)	1,7
iso-\(\mathrm{C_4H_9OH}\)	1,72	\(\mathrm{C_2H_5CHOH \cdot C_4H_9}\)	1,7
” \(\mathrm{C_5H_{11}OH}\)	1,76	\((\mathrm{C_4H_7})_2\mathrm{CHOH}\)	1,7

An analogous phenomenon also occurs in ketones. The presence of the \(=\mathrm{CO}\) group determines a constant moment.

Table 4

Substance	\(m \cdot 10^{18}\)
\(\mathrm{CH_3COCH_3}\)	2,71
\(\mathrm{CH_3COC_2H_5}\)	22,79
\(\mathrm{CH_3COC_3H_7}\)	2,72
\(\mathrm{C_2H_5COC_2H_5}\)	2,72
\(\mathrm{CH_3COC(CH_3)_3}\)	2,79
\((\mathrm{CH_3})_3\mathrm{C \cdot CO \cdot C(CH_3)_3}\)	2,76

This regularity was, to be sure, noticed rather in qualitative form by J. Thomson even before the moments had been measured with sufficient accuracy. Thomson found that the ratio \(DEK\) of certain liquids to the number of polar groups per unit volume has an almost constant value. This dependence in alcohols is explained by the fact that the large

part of the polarization is due to the etheric defect.

The above-mentioned cases of alcohols and ketones are the only ones in which only the polar group leaves its imprint on the molecular dipole. More often the moment changes in homologous series. But this change does not always proceed in one direction with the lengthening of the hydrocarbon chain.

Table 5

Nitriles		Ethers
HCN	2.65	(CH₃)₂O	1.85
CH₃HN	3.11	(C₂H₅)₂O	1.10
C₂H₅CN	3.34	(C₃H₇)₂O	0.85
C₃H₇CN	3.46	HOH	1.85
C₄H₉CN	3.84

Alkyl halides
HC₃Cl	1.86
C₂H₅Cl	1.99
C₃H₇Cl	1.87

The assumption of the ionicity of the molecule led to a simplified conception of dipole moments. The theories of Kossel and then of Langmuir proceeded from the view that the electronegative atom “draws” the electron toward itself. Therefore the molecule was treated as a combination of ions. These views must now be abandoned. If gaseous HCl consisted of H⁺ and Cl⁻, then the moment would be equal to the charge multiplied by the distance between the centers of the plus and minus charges. From spectral data it is known that the distance between the nuclei in HCl is equal to \(1.27 \cdot 10^{-8}\) cm. Hence a moment is obtained equal to \(6 \cdot 10^{-18}\), whereas the experimental value is \(1.03 \cdot 10^{-18}\). Evidently, the question must be one of a displacement of electrons or of intramolecular polarization. The action of a polar group can extend far into the molecule and affect not only the nearest neighbor. It must be said that the course of moments in homologous series cannot always be interpreted simply. In ethers, as in a number of other cases in which oxygen is a bridge between two groups, one may assume that it possesses angular valency. The two strokes with which chemistry endows oxygen,

lie at an angle. As the number of $\mathrm{CH_3\cdot CH_2\ldots}$ groups increases, the angle apparently increases; this entails a “straightening” of the molecule, an increase in symmetry, and a decrease in the dipole moment. This is indicated by the fall of the moment from 1.85 for water to 0.85 for propyl ether.

\[ \begin{array}{ccccc} \mathrm{H} &&&& \mathrm{H}\\ & \diagdown && \diagup &\\ && \mathrm{O} && \end{array} \qquad \begin{array}{ccccc} \mathrm{H_7C_3} &&&& \mathrm{C_3H_7}\\ & \diagdown && \diagup &\\ && \mathrm{O} && \end{array} \]

The explanation of the course of the values of the moments in homologous series is connected by many authors with the concept of “free rotation,” borrowed from organic chemistry. At the basis of this lies the absence of isomeric derivatives of ethane. The combination $\mathrm{C}$ and $4\mathrm{H}$ can be represented in the simplest way as a tetrahedron with carbon at the center. The molecule $\mathrm{C_2H_6}$ is likened to two tetrahedra with a common vertex. If this is represented schematically, then for dichloro-substituted $\mathrm{C_2H_4Cl_2}$ the following configurations are conceivable:

\[ \begin{array}{ccc} \begin{array}{c} \mathrm{H}\\ |\\ \mathrm{H{-}C{-}H}\\ |\\ \mathrm{H{-}C{-}H}\\ |\\ \mathrm{H} \end{array} & \begin{array}{c} \mathrm{H}\\ |\\ \mathrm{H{-}C{-}Cl}\\ |\\ \mathrm{H{-}C{-}Cl}\\ |\\ \mathrm{H} \end{array} & \begin{array}{c} \mathrm{H}\\ |\\ \mathrm{H{-}C{-}Cl}\\ |\\ \mathrm{Cl{-}C{-}H}\\ |\\ \mathrm{H} \end{array} \end{array} \]

Experiment does not yield two isomers. Among the possible explanations of this difficulty, rotation of the tetrahedra is admitted. The dipole moment, being a vector property, depends on the angle of inclination and rotation of the component tetrahedra. This creates a basis for various hypothetical spatial configurations, as a result of which the experimentally observed moments appear. Free rotation may have as its result an arrangement in the position of minimum potential energy. Single bonds rotate about the axis of valence, striving to bring opposite charges closer together. The rotating radical oscillates with an amplitude that is the larger, the less it is acted upon by neighboring charged groups.

The assumption that the carriers of the moments are polar groups is insufficient. It is more correct to speak of interatomic

POLARITY AND STRUCTURE OF MOLECULES

bonds that change within one molecule from group to group. Symmetric methane, $\mathrm{CH_4}$, has no dipole moment. Replacement of hydrogen by chlorine gives $\mathrm{CH_3Cl}$ with $m = 1.9 \cdot 10^{-18}$. Displacement of the first electron occurs more readily than that of the second and third. The moments of $\mathrm{CH_2Cl_2}$ and $\mathrm{CHCl_3}$ are smaller than the vector sums, for the different $\mathrm{C—Cl}$ bonds are not equivalent. An attempt to ascribe a constant moment to each bond fails to achieve its purpose already because the accepted magnitude cannot be used for calculating the moment that appears when several bonds are present in the molecule. Further, different values are obtained for one and the same bond in the aromatic and aliphatic series. Thus $m$ for $\mathrm{CH_3Cl}$ is equal to $1.86 \cdot 10^{-18}$, while for $\mathrm{C_6H_5Cl}$ it is $1.56$. The change of moment depending on the polar group or atom is seen from the table.

Table 6

X	$\mathrm{CH_3X}$	HX
Cl	1.86	1.03
Br	1.82	0.79
J	1.6	0.38
CN	3.1
$\mathrm{NO_2}$	3.1
$\mathrm{NH_2}$		1.55

Table 7

Compound	Moment
$\mathrm{NH_3}$	1.55
$\mathrm{PH_3}$	0.5
$\mathrm{AsH_3}$	0.15
$\mathrm{H_2O}$	1.85
$\mathrm{H_2S}$	1.1*

The moments of analogous compounds of one group in the periodic system have the trend indicated in Table 7.

For benzene derivatives, the appearance of a moment is characteristic when hydrogen is replaced by any group.

Table 8

X	$\mathrm{C_6H_5X}$	X	$\mathrm{C_6H_5X}$
$\mathrm{CH_3}$	$+0.4$	OH	$-1.7$
$\mathrm{NH_2}$	$+1.6$	$\mathrm{OCH_3}$	$-1.2$
$\mathrm{NO_2}$	$-3.9$	COOH	$-0.9$
COH	$-2.3$	Br	$-1.5$
Cl	$-1.56$	CN	$-3.85$
J	$-1.25$
$\mathrm{COOCH_3}$	$1.8$

Of particular interest are the moments of di- and tri-substituted compounds as functions of the relative positions of the groups (ortho, meta, or para). For dinitrobenzenes the following values have been obtained.

If we assume: 1) the equivalence of the bonds of all groups with the nucleus, and 2) the planar character of \(C_6H_6\), then the moment can be calculated by the rule of vector addition. Relative to the benzene nucleus, ortho groups are at an angle of \(60^\circ\), meta—\(120^\circ\), and para—\(180^\circ\). Therefore

\[ m_{\text{ortho}}=\sqrt{m^2+m^2+2m^2\cos 60^\circ}=m\sqrt{3} \]

\[ m_{\text{meta}}=\sqrt{m^2+m^2+2m^2\cos 120^\circ}=m \]

\[ m_{\text{para}}=0. \]

In dinitro-substituted compounds, as is evident, this regularity is obeyed. If both substituent groups are different, the resulting moment is

\[ m_{\text{ortho}}=\sqrt{m_1^2+m_2^2+2m_1m_2\cos 60^\circ} \]

Let us take nitrotoluenes as an example.

Para-dinitrobenzene has no moment. Both nitro groups, identical in sign, mutually compensate each other, being located at opposite ends of the molecule. Conversely, para-nitrotoluene has a moment approximately equal to

sum of the moments of NO₂ and CH₃. Here mutual reinforcement occurs, caused by the fact that the groups are opposite in sign. This makes it possible to assign the corresponding signs to the substituent groups. Taking chlorine to have a negative charge, from the sign of mutual reinforcement or weakening we find the signs of the other groups.

The rule of vector additivity of moments was expressed by J. Thomson and tested on a small amount of material by Kerr (for OH- and CN-substituted compounds). As we shall see, the applicability of this rule is very limited.

The following table gives experimental and calculated values.

Table 9.

	Found	Calculated		Found	Calculated
Symm. C₆H₃(NO₂)₃	0,7	0	Ortho C₆H₄Br₂	1,6	2,6
α nitronaphthalene	3,62	3,75	Meta »	1,1	1,5
1–5 dinitronaphthalene	0,6	0	Para »	0	0
1–8 »	7,1	7,5	Ortho C₆H₄J₂	1,8	2,3
Ortho ClC₆H₄NO₂	4,3	4,78	Meta »	1	1,3
Meta »	3,4	3,26	Ortho NH₂C₆H₄CO₂CH₃	1	1,7
Para »	2,6	2,1	Meta »	2,4	2,9
Symm. C₆H₃Br₃	0,2	0	Para »	3,3	3,4
Para BrC₆H₄NO₂	2,53	2,19	» NO₂C₆H₄COH	2,4	1,1
Ortho NO₂C₆H₄NH₂	4,45	3,27	» NO₂C₆H₄CO₂H	3,5	3
Meta NO₂C₆H₄NH₂	4,72	4,69	» ClC₆H₄OH	2,4	0,2
Para »	7,1	5,26	» NO₂C₆H₄OH	5,1	2,2
Ortho C₆H₄Cl₂	2,3	2,6	Ortho C₆H₄(CH₃)₂	0,63	0,6
Meta »	1,6	1,5	Meta »	0,4	0,46
Para »	0	0	Para »	0,2	0

Sometimes better values are obtained when, in disubstituted compounds, each group is assigned not its moment in the monosubstituted compound, but some other one. This is seen from Table 10.

Table 10.

	Found	Calculated from OH = −1,73; CH₃ = +0,1	Calculated from OH = −1,63; CH₃ = +0,2
Ortho cresol	1,54	1,57	1,54
Meta »	1,76	1,97	1,74
Para »	1,86	2,13	1,83

From the data presented it is evident that it is not possible to construct an elementary theory of the vector additivity of moments. It is interesting that the deviations are often considerable for ortho-

compounds. In this case the groups are closest to one another and mutually influence one another. Such an influence, however, is also observed in certain para-compounds, where the groups are farthest apart. Para-\(\mathrm{NO_2C_6H_4NH_2}\), according to the rule of additivity, should have a moment of \(5.3\), whereas experiment gives \(7.1\). The question of how far the influence of a substituent group on the benzene nucleus extends can be clarified from absorption spectra in the ultraviolet region. It is possible that the groups are not situated in the direction from the center of the benzene ring, but form angles. For the detection of angular valence, X-ray diagrams of the compounds under study should be of help. Characteristic are the values of the moments when both groups are attached to the nucleus through an oxygen atom. Such a bond, independently of position, gives rise to a strong moment. In this respect hydroquinones and their derivatives are of interest.

\[ \mathrm{H_3CO{-}C_6H_4{-}OCH_3} \qquad 1.81\cdot 10^{-18} \qquad \mathrm{H_5C_2O{-}C_6H_4{-}OC_2H_5} \qquad 1.76 \]

\[ \begin{aligned} &\mathrm{C_6H_4(OCH_3)_2}\quad 1.58, \qquad \mathrm{C_6H_4(OCH_3)_2}\quad 1.31,\\ &\mathrm{C_6H_4(OC_2H_5)_2}\quad 1.7, \qquad \mathrm{C_6H_4(OC_2H_5)_2}\quad 1.37 \end{aligned} \]

\[ \mathrm{H_9C_4O{-}C_6H_4{-}OC_4H_9} \qquad 1.79 \]

Symmetrical diethylbenzene \(\mathrm{C_6H_4(C_2H_5)_2}\) is almost nonpolar
\((m=\sim 0.2\cdot 10^{-18})\). Thus, a bond through oxygen, even in para-compounds, accounts for the presence of a moment. Diphenyls behave analogously. Dichloro- and dinitrodiphenyls are nonpolar.

\[ \mathrm{Cl{-}C_6H_4{-}C_6H_4{-}Cl} \]

Meanwhile

\[ \mathrm{H_3CO}-\begin{matrix} \text{benzene ring} \end{matrix}-\ \cdots\ -\mathrm{OCH_3}\quad \ldots\quad 1.52 \]

\[ \mathrm{C_2H_5O}-\begin{matrix} \text{benzene ring} \end{matrix}-\ \cdots\ -\mathrm{OC_2H_5}\quad \ldots\quad 1.9 \]

The mutual induction of groups, which chemically corresponds to secondary valence, increases the asymmetry. This property is especially characteristic of oxygen and nitrogen with their angular valences.

\[ \begin{array}{cccc} \begin{matrix} & \mathrm{O^-} & - & \mathrm{O} \\ / & & & \backslash \\ \mathrm{R} & & & \mathrm{R} \end{matrix} & \begin{matrix} & \mathrm{O} \\ / & \backslash \\ \text{ring} & \mathrm{CH_3} \end{matrix} & \begin{matrix} & \mathrm{O} \\ / & \backslash \\ \text{benzene ring} & \mathrm{H} \\ & & \mathrm{CH_3} \end{matrix} \end{array} \]

Substituted groups are sometimes regarded (Estermann) as strongly deformed ions that cause a displacement in the hydrocarbon chain. Under such an interpretation, the moment is ascribed to the molecule as a whole.

If diphenyl derivatives are nonpolar, this may be an argument in favor of a planar distribution of the benzene nuclei. On the other hand, the presence of a moment is associated with the spatial configuration, as is seen in the figure, where the benzene nuclei lie one upon the other.

\[ \begin{matrix} \text{benzene ring}\\[-2pt] \text{benzene ring} \end{matrix} \]

Among the compounds that are of interest from the standpoint of the relation between the moment and the arrangement of polar groups are the cis- and trans-isomers of ethylene derivatives.

For compounds of the type $\mathrm{C_2H_2Cl_2}$ the following structures are also possible:

\[ \begin{array}{ccc} \begin{array}{c} \mathrm{Cl}\quad\quad \mathrm{Cl}\\[-2pt] \diagdown\quad\diagup\\[-2pt] \mathrm{C}\\[-2pt] \Vert\\[-2pt] \mathrm{C}\\[-2pt] \diagup\quad\diagdown\\[-2pt] \mathrm{H}\quad\quad \mathrm{H} \end{array} & \begin{array}{c} \mathrm{H}-\mathrm{C}-\mathrm{Cl}\\[-2pt] \Vert\\[-2pt] \mathrm{H}-\mathrm{C}-\mathrm{Cl}\\[-2pt] \text{cis} \end{array} & \begin{array}{c} \mathrm{H}-\mathrm{C}-\mathrm{Cl}\\[-2pt] \Vert\\[-2pt] \mathrm{Cl}-\mathrm{C}-\mathrm{H}\\[-2pt] \text{trans} \end{array} \end{array} \]

The structure of a molecule of the type $\mathrm{CCl_2=CH_2}$ is unambiguously determined by chemical methods. In the case of cis- and trans-compounds, chemistry knows two isomers, but it is not always possible to indicate which of them is cis and which is trans. Usually a higher boiling point and a larger refractive index are ascribed to the cis-compound. However, between these properties and the composition no connection can be indicated a priori. Sometimes one group of authors regards as the cis-compound that which by others is considered the trans-molecule. The moment in the given case makes it possible to determine the configuration. In trans-compounds the polar groups must partially or wholly compensate one another, which is not the case in cis-isomers; therefore the former must have an insignificant moment (or zero), and the latter a noticeable one. This is confirmed by experiment.

The solution of the question of the cis- and trans-isomer, on the basis of measurement of the moment, is impossible in the case of the presence of oxygen or nitrogen in the substituent groups, since the latter distort the effect.

Table 11.

Compound	Isomer	Moment
$\mathrm{C_2H_2Cl_2}$	cis	1.9
$\mathrm{C_2H_2Cl_2}$	trans	0
$\mathrm{C_2H_2Br_2}$	cis	1.4
$\mathrm{C_2H_2Br_2}$	trans	0
$\mathrm{C_2H_2J_2}$	cis	0.8
$\mathrm{C_2H_2J_2}$	trans	0
$\mathrm{C_2H_2ClBr}$	cis	1.6
$\mathrm{C_2H_2ClBr}$	trans	0

For chloroethylene, Errera gives the value cis — 0.6 and trans — 13. The indicated author explains this by the fact that in $\mathrm{C_2H_2ClJ}$ iodine is positive with respect to chlorine, which

Indirect proof is provided by the ability of \(\mathrm{C_2H_2ClJ}\) to give addition products.

\[ \begin{array}{ccccc} & \mathrm{H} & & \mathrm{H} & \\ & \backslash & & / & \\ \mathrm{J} & - & \mathrm{C}=\mathrm{C} & - & \mathrm{J} \\ & & & \backslash & \backslash \\ & & & \mathrm{Cl} & \mathrm{Cl} \end{array} \]

On the other hand, in para-dibromobenzene \(\mathrm{J}\) and \(\mathrm{Br}\) have the same signs, for they mutually weaken one another, as a result of which the moment decreases. It is possible that Errera regards as the cis-isomer what is in reality the trans compound.

The determination of moments may serve as a criterion for the identification of molecules in a number of cases where chemical differentiation is impracticable. This includes the benzil oximes.

\[ \begin{array}{c} \text{benzene ring} - \mathrm{C}-\mathrm{C}- \text{benzene ring}\\[-2mm] \phantom{\text{benzene ring} -}\Vert\ \ \Vert\\[-1mm] \phantom{\text{benzene ring} -}\mathrm{HNOH}\ \ \mathrm{O} \end{array} \qquad \begin{array}{c} \text{benzene ring} - \mathrm{C}-\mathrm{C}- \text{benzene ring}\\[-2mm] \phantom{\text{benzene ring} -}\Vert\ \ \Vert\\[-1mm] \phantom{\text{benzene ring} -}\mathrm{OHNH}\ \ \mathrm{O} \end{array} \]

Terephthalic derivatives, quinols, decalins, etc.

\[ \begin{array}{ccc} \begin{array}{c} \mathrm{H}\backslash\mathrm{C}/\mathrm{Cl}\\ \mathrm{H_2C}\quad\quad\mathrm{CH_2}\\ \mathrm{H_2C}\quad\quad\mathrm{CH_2}\\ \mathrm{H}/\mathrm{O}\backslash\mathrm{Cl} \end{array} & \begin{array}{c} \mathrm{Cl}\backslash\mathrm{C}/\mathrm{H}\\ \mathrm{H_2C}\quad\quad\mathrm{CH_2}\\ \mathrm{H_2C}\quad\quad\mathrm{CH_2}\\ \mathrm{H}/\mathrm{O}\backslash\mathrm{Cl} \end{array} & \begin{array}{c} \mathrm{H}\backslash\mathrm{C}/\mathrm{OH}\\ \mathrm{HC}\quad\quad\mathrm{CH}\\ \mathrm{HC}\quad\quad\mathrm{CH}\\ \mathrm{H}/\mathrm{C}\backslash\mathrm{OH} \end{array} \end{array} \]

The study of dipole moments can facilitate the task of elucidating structural features. Substituted benzils give, for example, two isomers—yellow and colorless. To explain this, Schenberg considers the yellow form to be the normal 1–2 diketone

\[ \mathrm{R\cdot C_6H_4 - C - C - C_6H_4R} \]
\[ \phantom{\mathrm{R\cdot C_6H_4 -}}\Vert\quad\Vert \]
\[ \phantom{\mathrm{R\cdot C_6H_4 -}}\mathrm{O}\quad\mathrm{O} \]

whereas the colorless one is the peroxide isomer

\[ \begin{gathered} \mathrm{R\cdot C_6H_4 — C — C — C_6H_4\cdot R}\\ \phantom{\mathrm{R\cdot C_6H_4 —}}\vert\quad \vert\\ \phantom{\mathrm{R\cdot C_6H_4 —}}\mathrm{O — O} \end{gathered} \]

It is possible, however, that what is involved here is cis- and trans-isomerism about the single bond \(\mathrm{C—C}\). This is supported by the fact that the colorless form does not exhibit peroxide properties.

The study of moments, closely connected with the degree of symmetry in the structure of molecules, has not passed by the fundamental question of organic chemistry concerning tetrahedral carbon. It turns out that tetra-substituted methanes \(\mathrm{CCl_4}\), \(\mathrm{C(CH_2Cl)_4}\), \(\mathrm{C(NO_2)_4}\), \(\mathrm{C(CH_2Br)_4}\), \(\mathrm{C(CH_2J)_4}\) are not dipolar.

Such a result is in agreement with the classical theory.

But other tetra-derivatives such as

\[ \mathrm{C(CH_2ONO_2)_4,\quad C(COOC_2H_5)_4} \]

show the presence of a moment.

Let us give examples:

\[ \begin{aligned} \mathrm{C(OCH_3)_4} &\;—\; 0.8\cdot 10^{-18}\\ \mathrm{C(OC_2H_5)_4} &\; 1.1\\ \mathrm{C(CH_2O_2C\cdot CH_3)_4} &\; 2.6 \end{aligned} \]

Although the numerical values given are not especially accurate, for in calculating these moments in solutions no extrapolation to infinite dilution was made, still there can be no doubt as to the dipolar character of the indicated molecules. The presence of moments was interpreted by Weissenberg as confirmation of his view that carbon compounds are built not on the tetrahedral type, but as a tetragonal pyramid with the carbon at the apex and the substituent groups in the plane. The molecular spectrum of \(\mathrm{CH_4}\) and the moments of inertia associated with it also do not accord with a tetrahedral structure. Investigation of the X-ray photographs of pentaerythritols led Mark and Weissenberg to the conclusion that the substituent groups lie in one plane.

Knagg doubts that the X-ray diagrams have been interpreted correctly. In any case, as Debye points out, the presence of a moment is still insufficient for postulating a tetragonal structure. But, on the other hand, the absence of a moment (for example, in $\mathrm{C}(\mathrm{CH}_2\mathrm{Cl})_4$) is not yet proof of tetrahedrality. The absence of a moment in compounds of the type $\mathrm{CA}_4$ speaks only in favor of a centric symmetry of the molecule. Thus, in this question dipole moments are insufficient for a final decision. It is characteristic that in pentaerythritols the appearance of oxygen atoms bonded to carbon is accompanied by the appearance of a moment.

A detailed calculation of the structure can be carried out only for molecules that are not too complex. If one takes into account the polarization of oxygen in the water molecule, then the condition of a linear asymmetric model requires that one hydrogen be farther from the oxygen than the other by a factor of 1.85. But such a model is unstable even under small vibrations of the hydrogen. As for the symmetric linear model, it is ruled out, since the absence of a moment is associated with it. A calculation made under the assumption of a triangular structure leads to a stable configuration. The molecular spectrum of water, according to Hund, admits two solutions. According to the first, the angle between the two hydrogens and the oxygen is $64^\circ$; according to the second, $110^\circ$. In both cases the distance between O and H is approximately the same—$1\cdot 10^{-8}\ \text{cm}$. If, using these data and taking into account the polarization of oxygen, one calculates the possible moments, then they should be equal to $-1.34\cdot 10^{-18}$ or $4.32\cdot 10^{-18}$. The experimental value $1.84\cdot 10^{-18}$ (Zenger) lies between the two values, closer to the first. Therefore the triangular structure of water is the best substantiated. Similar considerations lead to the structure of ammonia in the form of a triangular pyramid with nitrogen at the apex and three hydrogens at the base. According to quantum mechanics, molecules containing more than three nuclei may have two energy minima. The probability of mutual transition depends on the height of the barrier separating the two minima. Hence the possibility is not excluded of obtaining in the future mutually interconverting

isomers of such molecules, which chemical structural theory does not provide for. Experience with ortho- and parahydrogen teaches us that physical theories fructify chemistry and introduce new content into it.

ASSOCIATION

The terms association and associated liquid have long since acquired citizenship rights in physical chemistry. Usually the matter stood as follows: various deviations (increased molecular weight, anomalous elasticity of the vapor of mixtures, a peculiar course of the viscosity curve, etc., etc.) were ascribed to association. The question is above all one of the physical content of this concept. If for an ideal gas kinetic theory gives a simple dependence between \(p\), \(T\), and \(V\), if the equations of Born, Grüneisen, and others are applicable to a crystal lattice, then we still have no clear approach to the “ideal” liquid. Intermolecular van der Waals forces in a liquid reduce to di-, quadru-, and in general multipole interaction. Expressions for the potential energy of a system of dipoles are obtained comparatively simply from a statistical calculation only when the distances between the dipoles considerably exceed the dimensions of the dipoles. In the case of a liquid this condition is not fulfilled. In connection with this, our knowledge in this field is of a qualitative character. In the simplest case one may speak of a combination of several molecules in different forms, as is seen in Fig. 1.

Fig. 1.

In the first case two dipoles, approaching with unlike parts, give a quadrupole; in the second, a chain bond is obtained. This latter is characteristic of some colloids (vanadium pentoxide sol), and an increase of the moment causes an increase of \(DEK\). Combinations of molecules may lead to a decrease of the moment (even to its complete annulment) or, in other cases, to an increase of dipoles. Both cases must be regarded as asso-

association. The molecular combinations obtained as a result are by no means always already-formed complexes. Here a statistical treatment is applicable, in the sense that microscopic observation of the liquid over a long period should give average values for molecular complexes of different degrees of complexity, different forms of combination, and different stability. The question of the character of the association (increase or decrease of the moment) must be decided on the basis of an investigation of the polarization of solutions. To obtain a simple picture in which one can find one’s way, the solvent must be nonpolar. From equation 20 one can find the polarization of the dissolved substance as a function of concentration. First of all, let us note that, when calculating \(P_2\), extrapolating to the case of dilute solutions, and calculating the moment, we do not always find the same value as for the gas. This is explained by the impossibility of eliminating the interaction of the dissolved substance with the solvent. Association of dissolved dipoles with quadrupoles of the solvent gives a distorted value of the polarization. Nonpolar mixtures \(CS_2—C_6H_6\), \(CCl_4—C_6H_6\), \(CS_2—CCl_4\) give the normal additive course. The curves of polarization of the dissolved substances (abscissa axis \(f_2\), ordinate axis \(P_2\)) are of three kinds.

1) \(P_2\) is constant, there is no association; \(P_2\) is additive.

2) \(P_2\) changes monotonically, e.g., falls continuously, reaching the value of the polarization of the pure dissolved substance (\(f_2 = 1\)). This means that, with increasing concentration, the molecules associate more and more with a decrease of the dipole moment. This is observed for nitrobenzene, acetic acid, ether, chlorobenzene, and quinoline dissolved in benzene.

3) \(P_2\) at first increases, reaches a maximum, and then falls. This is the most complicated case, connected with the presence of twofold association. At first there predominates the union of molecules with a resultant moment of larger magnitude; then, with increasing concentration, complexes are formed with mutual compensation of moments, which causes a fall of the polarization. This is observed for alcohol in benzene. That part of the polariza-

of the polarization effect caused by electronic displacement does not change under molecular interaction. Thus refraction is insensitive to association. The latter affects only the orientational term. In the case of two nonpolar liquids, association is also possible (quadrupole interaction), which cannot be determined from the dielectric constant.

If both liquids are polar, then the course of the total polarization is so complex that there are not sufficient grounds for judging the behavior of individual dipoles.

A special case occurs in solutions when the dipoles of the solvent are polarized by ions. The field strength of an individual ion is equal to \(\frac{e}{Dr^2}\), which, for \(e = 4.77 \cdot 10^{-10}\), \(DEK = 80\), and \(r = 10^{-7}\), gives \(180\,000\) volts/cm. In such fields the dipoles are oriented almost completely, and Langevin’s approximate formula is no longer sufficient. The dielectric constant ceases to be constant. Ions orient neighboring dipoles to such an extent that the latter no longer obey the external field. Dielectric saturation sets in; part of the molecules escapes the control of the field, and the dielectric constant falls. The surrounding of the ion by a shell of polarized and oriented dipoles accounts for hydration or solvation. Unfortunately, the data on the dielectric constants of ionic solutions are such that it is not possible to obtain a criterion for associative capacity from hydration data.

Association depends not only on the magnitude of the dipole, but also on its geometrical position in the molecule. From the moment alone one cannot draw a conclusion about the character of association. The small (not long) molecules of the first alcohols of the homologous series are strongly associated. With lengthening of the hydrocarbon chain, the capacity for association decreases. In connection with this is also the decrease in solubility in water. In the elementary theory of association it is assumed that in solution there is a mixture of complexes and ordinary molecules, the equilibrium between which obeys the law of mass action. In the simplest case the existence of dimers and single molecules is admitted. The formation of an associated molecule is accompanied by a decrease in

of osmotic pressure. These same solutions should give an abnormal course of the polarization curve. Such an interpretation is no more than a first approximation, for association even in dilute solutions gives molecules of different complexity. To explain the dual behavior of alcohol, which first shows an increase in polarization, association of the type

\[ \begin{array}{ccc} \mathrm{R} & \mathrm{R} & \mathrm{R}\\ | & | & |\\ \mathrm{H}-\mathrm{O}\;-\;-\;-\;-\;\mathrm{H}-\mathrm{O}\;-\;-\;-\;\mathrm{H}-\mathrm{O} \end{array} \]

is assumed. In strong solutions the decrease in polarization may possibly be connected with another configuration

\[ \begin{array}{cc} \mathrm{O}-\mathrm{H}\;-\;-\;-\;\mathrm{R}\\ | & |\\ \mathrm{R}\;-\;-\;-\;\mathrm{H}-\mathrm{O} \end{array} \]

The numerical expression of the degree of association is very complicated. In the simplest case, if association is taken to be the union of two dipolar molecules into one non-dipolar dimer, the degree of association can be expressed as the ratio of the number of molecules in the complex state to the total number of molecules. Under such an interpretation, the orientational term of formula 12 includes not the total number of molecules, but only the free, non-associated ones. This gives

\[ \frac{D-1}{D+2}\frac{M}{d} = \frac{4}{3}\pi N a' + \frac{4}{9}\pi N_0 \frac{m^2}{kT} \tag{21} \]

From equation 21 one finds \(N_0\) and then the degree of association

\[ \eta=\frac{N-N_0}{N} \]

The results of such a calculation are given in Table 12 for ether in benzene.

Table 12.

Concentration of ethyl ether, percent	\(100\eta\)
9.02	0.5
29.77	2.1
60.27	3.6
90.49	3.9
100	3.9

According to formula 21 one may compare \(\eta\) with the magnitude of the moment for pure dipole liquids. In this case, as the moment increases, association also increases.

Table 13

Substance	\(\eta\)	\(m\,10^{18}\)
Nitrobenzene	80.7	3.84
Quinoline	59.2	2.25
Pyridine	57.9	2.11
Chlorobenzene	41.0	1.55
Ethyl ether	3.9	1.22

The method of calculation by formula 21 is valid only under the assumption that, as a result of association of two dipoles, a quadrupole is obtained. In the general case, as a numerical measure one may take the ratio of the mean mass moment to the single, exact moment.

An increase in temperature promotes disruption of the configuration of associated molecules. Therefore concavities and convexities in the polarization curves should straighten out, approaching the normal additive course. Hence, in associated liquids the polarization may increase with temperature, owing to the formation of an ever greater number of free dipoles, instead of decreasing according to Debye. This is one of the uncertainties in determining moments in liquids.

The statistical theory of association is still in an embryonic state. The difficulty is that, for the calculation, it is necessary: 1) to assume the existence of a certain elementary volume into which the molecules must fall in order that they may be considered associated, 2) to introduce an energy factor, assuming, for example, that molecules with energy above a certain minimum do not associate, but again fly apart, 3) to make an assumption concerning the character and probability of association. All this complicates the quantitative treatment of the theory.

Despite the preliminary character of our knowledge on this question, it must be said that the study of the deviation of polarization from the additive value may serve as a supporting basis for the construction, in the future, of a theory of association.

Literature

General:

1. Debye. Handb. d. Radiologie, 6, 597 (1925).
2. Debye. Polare Molekeln (1929). (A Russian translation is in preparation.)
3. Dipolmoment und chemische Struktur, Leipziger Vorträge (1929).
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5. K. Höjendahl. Studies of Dipole-Moment, Copenhagen, 1928 (dissertation).
6. Smyth. Electric Moment; Chemical Reviews, 6, 550, 1929.
7. J. Williams. The Structure of Molecules; ibid. 6, 589 (1929).
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10. Sack. Dipolmoment und Molekularstruktur, ibid., 307.

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1. Cf. the article by F. London and Ya. I. Frenkel: Advances in the Physical Sciences, 9, 1929. ↩