DETERMINATION OF THE DEGREE OF ASSOCIATION OF MOLECULES FROM THE TEMPERATURE DEPENDENCE OF LIQUID VISCOSITY

V. S. Smorodinov

(Presented by the scientific-methodological seminar of the KhTF)

For the study of hydrogen bonding (H-bonding) in individual liquids, IR spectroscopy, combination scattering of light, and absorption of sound and ultrasound are used. The application of other methods [1] is hindered by the enormous difficulties encountered by any theoretical treatment of the liquid state.

In the present work, the possibility is considered of determining the average molecular weight of associated liquids (or the degree of association of molecules) from the temperature dependence of viscosity.

The dependence of the viscosity of liquids \(\eta\) on the temperature \(T\) is expressed by the empirical formula of Ya. I. Frenkel,

\[ \eta = Ae^{E/RT}, \tag{1} \]

where \(A\) and \(E\) are empirical constants, \(R\) is the universal gas constant; and by the theoretical equation of G. M. Panchenkov [2], which explicitly includes the molecular weight of the liquid. The theoretical equation cannot be used to solve the problem posed, chiefly because, in deriving the theoretical dependence of liquid viscosity on temperature, molecular associations that are preserved during the process of viscous flow of liquids are not taken into account. Therefore, determining the molecular weight of associated liquids from the theoretical equation should lead to the molecular weight of the monomer. To clarify the possibility of determining the molecular weight of a liquid from the coefficient \(E\), its physical meaning has been considered from the standpoint of transition-state theory [3] and the theory of liquid viscosity [2]. In transition-state theory, the coefficient \(E\) has the meaning of the activation heat of the viscous flow of a liquid [3]. In the process of viscous flow of low-molecular liquids, chains of molecules move as a whole, and one may expect a proportional dependence between the activation heat of viscous flow of a liquid \(E\) and the length of a uniform molecular chain (or the molecular weight of such a chain).

The mechanism of momentum transfer in liquids is considered in the theory of G. M. Panchenkov [2]. The basic premises of the theory reduce to the following. Momentum transfer from layer to layer in a liquid occurs through the temporary association of molecules at the boundary of layers at extreme deviations from the equilibrium position during oscillations, and is carried out only by those molecules whose kinetic energy—

values smaller than a certain value \(E\), called the energy of molecular bonding (approximately coinciding with the heat of activation of viscous flow of a liquid).

The calculation of the energy of molecular bonding from the internal heat of vaporization [2] is quite justified for “normal” liquids, since both in the process of their viscous flow and during their vaporization the same van der Waals bonds are broken. In the case of associated liquids such a calculation should not be justified. This conclusion can be reached by comparing the period of oscillation of liquid molecules and the “lifetime” of associated molecules.

The period of oscillation of molecules in a liquid is usually taken to be \(10^{-13}\) sec [4]. The mean lifetime of an associate depends on the relaxation time \(\tau\). The longer the relaxation time, the longer the lifetime of the associate. For low-molecular-weight liquids \(\tau \sim 10^{-10}\) sec [3], and the time required for molecular rearrangements in a liquid is \(10^{-11} \div 10^{-10}\) sec and more [1]. During this period the molecules undergo about 10–100 translational oscillations. Comparison of the period of oscillation of molecules in a liquid with the time required for molecular rearrangements shows that, in the theoretical equation for viscosity, associated molecules should be regarded as independent particles in the liquid. The large values of the heat of activation \(E\) of associated liquids, compared with the heats of activation \(E\) of hydrocarbons of the same molecular weight, characterize not the strength of the H-bond, but the actual increase in the molecular weight of the particles as a result of molecular association. The coefficient \(E\), determined from the temperature dependence of the viscosity of associated liquids, should be regarded as the mean energy of ordinary van der Waals bonds between associated molecules.

Table 1 gives the results of calculating the heat of activation \(E\) from the temperature dependence of viscosity (1) for certain hydrocarbons

Table 1

Comparison of the specific heat of activation of viscous flow of hydrocarbons and their halogen derivatives at 15°

and their halogen derivatives. As follows from the data given in Table 1, when the group \(-\mathrm{CH_3}\) is replaced by a fluorine atom, the specific heat of activation of viscous flow does not change, while the viscosities and densities of the corresponding liquids differ. It is not possible to verify this rule when the terminal group \(-\mathrm{CH_3}\) is replaced by other groups close to it in weight (\(-\mathrm{OH}\), \(=\mathrm{O}\), \(-\mathrm{NH_2}\)), since such liquids are associated to a greater or lesser degree.

Figure 1 presents graphically the dependence of the heat of activation of viscous flow of hydrocarbons of normal structure \(E\) on the mo-

lecular weight at different temperatures. As can be seen from the data presented in Fig. 1, the value of \(E\) increases with increasing molecular weight of the hydrocarbons, and at \(25^\circ\) this dependence is directly proportional. The specific heat of activation of hydrocarbons is, on average, \(18.4\ \text{cal/g}\).

An approximate calculation of the molecular weight of associated liquids from the heat of activation of viscous flow is based on the following assumptions: 1) associated molecules are regarded as independent particles in the process of viscous flow of the liquid; 2) the average specific heat of activation of viscous flow of hydrocarbons of normal structure and of their derivatives with the terminal group \(-\mathrm{CH}_3\) replaced by groups close to it in weight is \(18.4\ \text{cal/g}\) at \(25^\circ\).

Fig. 1. Dependence of the heat of activation of viscous flow of hydrocarbons of normal structure on molecular weight at different temperatures: \(-70^\circ\)—1; \(25^\circ\)—2; \(70^\circ\)—3

To a greater extent these assumptions and calculations should be justified for such forms of association in which the dipoles are arranged antiparallel to one another, which leads to a decrease in the total dipole moment, and also for molecules with a long hydrocarbon chain or at a high degree of molecular association, when the influence of the dipole moment on the intermolecular interaction and the heat of activation of viscous flow is weakened.

The molecular weight of associated particles \(M_{\mathrm{acc}}\) at \(25^\circ\) is determined graphically (Fig. 1) or by the formula

\[ M_{\mathrm{acc}}=\frac{E_{\mathrm{act}}}{18.4}, \tag{2} \]

where \(E_{\mathrm{act}}\) is the heat of activation of viscous flow of the associated liquid at \(25^\circ\). The degree of association of molecules \(f_{\mathrm{acc}}\) is determined from the relation

\[ f_{\mathrm{acc}}=\frac{M_{\mathrm{acc}}}{M_{\mathrm{form}}}, \tag{3} \]

where \(M_{\mathrm{form}}\) is the formula molecular weight of the monomer.

The results of calculating the degree of association of alcohols, carboxylic acids of normal structure, and some other liquids from the data of [5, 6] are given in Table 2. The heats of activation of viscous flow of the liquids were determined graphically by the formula

\[ E_{\mathrm{act}}=4575\,\operatorname{tg}\Theta, \tag{4} \]

where \(\operatorname{tg}\Theta\) is the angular coefficient of the straight line plotted in the coordinates \(\lg \eta\)—\(1000/T\). The dependence of the degree of association of alcohols and acids on

the number of carbon atoms in the monomer is presented in Fig. 2, whence it follows that the degree of association of alcohols whose molecules contain 1–3 carbon atoms is close to 4; for alcohols with 3–6 carbon atoms in the molecule the degree of association is close to 3; the degree of association of other alcohols is less than 3. These data agree well with the results of studies of the association of lower alcohols by spectral methods [7, 8]. There are data [8] that aggregates of methyl, ethyl, and

Table 2

For calculating the average degree of association of liquid molecules from the temperature dependence of viscosity

Note: In view of the assumptions adopted concerning the structure of the molecules, the calculation of \(f\) is approximate in character.

propyl alcohols at a temperature of 0–20° contain on average 4.5–5 molecules each. Calculation of the degree of association of alcohols by other approximate methods, using the formulas of Longinescu, Eötvös, Ramsay, and Shields, gives values 1.5–2 times smaller.

Formic (\(f \approx 4.2\)) and, to a small degree, acetic acids (\(f \approx 2.5\)) in the liquid state are capable of polymeric association,

other carboxylic acids are associated predominantly into dimers ($f \simeq 1.8$) (Table 2, Fig. 2). The results obtained agree with known literature data on the association of carboxylic acids [1].

The ability of aldehyde groups to form H-bonds remains unclear to this day [1]. Calculation of the degree of association of aldehydes from the heat of activation of viscous flow gives the following results (Table 2): $f = 2$ for acetic aldehyde and $f = 1.5$ for propionic aldehyde.

The calculated values of the degree of association of some other liquids are, respectively: ethylene glycol $\sim 10$, glycerol $\sim 9$, water — 11.6. The latter calculations may prove to be more approximate, since when the number of H-bonds is equal to three or more, association of molecules tends to extend throughout the entire mass of the liquid.

Fig. 2. Dependence of the average degree of association of alcohols (1) and carboxylic acids (2) of normal structure on the number of carbon atoms in their molecules at 25°

When two chemically noninteracting components with a straight chain of molecules are mixed (the molecular weight of the mixture $M_{\mathrm{mix}}$ is additive when the composition is expressed in relative mole units), whose viscosities differ by no more than a factor of 6, the heat of activation of viscous flow is an additive quantity. It can be shown that the specific heat of activation of viscous flow of such a mixture and of its constituent components is the same. This circumstance was used to determine the average molecular weight of particles in solutions of associated liquids, taking into account the assumptions adopted earlier for formula (2), using as an example water (B)—methanol (A) mixtures according to the data of G. M. Panchenkov [2]. At a content of 27.3 mol.% methanol in the mixture, the value of $E$ is equal to 5.1 kcal/mole, which corresponds to a molecular weight of the particles equal to 277. Taking into account that a methanol molecule is most strongly solvated by three water molecules [8], it should be assumed that the methanol hydrate has the form $(AB_3)_3$ with a molecular weight equal to 258.

The degrees of association of molecules, calculated by an approximate method from the temperature dependence of viscosity, were used to estimate the strength of H-bonds from the heats of evaporation of associated liquids.

The experimentally determined heat of evaporation of associated liquids (in the absence of molecular association in the vapor) $\Delta H_{\mathrm{evap}}^{\mathrm{assoc}}$ includes the total energy of rupture of H-bonds $\sum \Delta H_{\mathrm{H}}$ and the heat of evaporation of the liquid consisting of monomeric molecules (the hypothetical state of the liquid in the absence of H-bonds) $\Delta H_{\mathrm{evap}}^{\mathrm{hyp}}$.

$$ \Delta H_{\mathrm{evap}}^{\mathrm{assoc}} = \sum \Delta H_{\mathrm{H}} + \Delta H_{\mathrm{evap}}^{\mathrm{hyp}} . \tag{5} $$

To determine the latter quantity empirically, the influence of molecular weight \(M\) on \(\Delta H_{\text{vap}}\) of nonassociated liquids at \(25^\circ\) was considered. Table 3 gives the specific heats of vaporization of liquids whose molecules do not have a branched chain and do not contain heavy atoms. The constancy of the specific heat of vaporization of the liquids determines the desired relation:

Table 3

Heats of vaporization of certain practically nonassociated liquids

\[ \Delta H_{\text{vap.}}^{\text{hyp.}} = 87,2\,M_{\text{form.}}. \tag{6} \]

To calculate the strength of the \(H\)-bond (kcal/mole per 1 \(H\)-bond), \(\Delta H_{\text{H}}\), from \(\sum \Delta H_{\text{H}}\), it is necessary, in addition to the quantity \(f\), to know the form of molecular association and the coordination number of the liquid with respect to the \(H\)-bond, \(\gamma_{\text{H}}\), determined by spectral methods. The strength of the \(H\)-bond is determined by the formula

\[ \Delta H_{\text{H}} = \frac{\sum \Delta H_{\text{H}}} {\dfrac{\gamma_{\text{H}}}{2} \left( \sum_m a_m + \sum_n b_n \frac{f_n - 1}{f_n} \right)} , \tag{7} \]

where \(a_m, b_n\) are the fractions of molecules participating in cyclic and noncyclic associations of degree \(m\) and \(n\), respectively. Experimentally, the average degree of association of molecules is determined:

\[ f = \frac{1} {\displaystyle \sum_m \frac{a_m}{f_m} + \sum_n \frac{b_n}{f_n}} . \tag{8} \]

Possible cases of assumptions in calculating the number of \(H\)-bonds per molecule are: 1) noncyclic association predominates, \(f=f_n\); 2) cyclic association predominates, \(f=f_m\); 3) one cyclic and one noncyclic form of association of different degrees, \(f_m \ne f_n\).

Taking into account two forms of molecular association, relations (7, 8) take the form

\[ \Delta H_{\mathrm{n}}= \frac{\Sigma \Delta H_{\mathrm{n}}} {\dfrac{\gamma_{\mathrm{n}}}{2}\left[x+(1-x)\dfrac{f_n-1}{f_n}\right]}, \tag{9} \]

\[ f=\frac{1}{\dfrac{x}{f_m}+\dfrac{1-x}{f_n}}, \tag{10} \]

where \(x=a_m\). If \(\Delta H_{\mathrm{n}}\) is determined by some other method, the content of the individual forms of molecular association in the structure of the liquid can be calculated as follows. From relations (9, 10), eliminating \(x\), we obtain

\[ f_n=\frac{1-f_m[1/f-(1-K)]}{1-K}, \tag{11} \]

where \(K=\dfrac{\Sigma \Delta H_{\mathrm{n}}}{\Delta H_{\mathrm{n}}\cdot \gamma_{\mathrm{n}}/2}\). To obtain an unambiguous result \(f_n\), it is necessary to know the number of molecules in the cyclic form of association, \(f_m\). The existence of cyclic forms of association with an even number of molecules, 2, 4, 6, has been proven. The results of calculating \(\Sigma \Delta H_{\mathrm{n}}\) for certain liquids by formulas (5, 6) are given in Table 4. The dependence of \(\Delta H_{\mathrm{vap}}^{\mathrm{ass}}\) of alcohols on their molecular weight is shown in Fig. 3. For four of the eight alcohols with 1–8 carbon atoms in the molecule, \(\Sigma \Delta H\) is about \(6.13\ \mathrm{kcal/mol}\); for the remaining alcohols it is somewhat smaller. The smallest value of \(\Sigma \Delta H_{\mathrm{n}}\) is observed for butanol—\(4.62\ \mathrm{kcal/mol}\). The coordination number \(\gamma_{\mathrm{n}}\) for alcohols is taken to be 2 [8]. If it is taken into account that the lower alcohols have a cyclic form of molecular association [1], then \(\Delta H_{\mathrm{n}}\) for methanol, ethanol, and propanol is about \(6.13\ \mathrm{kcal/mol}\), which is close to the result obtained by other methods and accepted in the literature (\(6.2\ \mathrm{kcal/mol}\)) [9].

Fig. 3. Dependence of the external heat of vaporization of alcohols of normal structure (1) and of a hypothetical liquid (2) on molecular weight at \(25^\circ\).

If one assumes that \(\Delta H_{\mathrm{n}}\) is approximately the same in all alcohols [9], then for \(\Sigma \Delta H_{\mathrm{n}}<6.2\ \mathrm{kcal/mol}\) it is necessary to admit the existence, in the structure of alcohols, along with cyclic forms of molecular association, of linear forms whose energy of breaking \(H\)-bonds, calculated per molecule, is smaller. Calculation of the content of linear forms of association \((1-x)\) in butanol by formulas (9, 10), assuming \(f_m=2, 4, 6\), gives 90, 80, 72%, respectively, which indicates the predominance of linear forms of association of butanol molecules. For the higher alcohols (hexanol, octanol), \(\Sigma \Delta H_{\mathrm{n}}\) corresponds to the strength of the \(H\)-bond in alcohols. Taking into account that their degree of association is less than 3, it may be assumed that they are associated predominantly in the form of cyclic dimers.

The energy of rupture of \(H\)-bonds during evaporation of aldehydes is considerably less than that of alcohols (Table 4). Taking into account the activity of the hydrogen atom of the aldehyde group, the tendency of aldehydes toward cyclic polymerization, and the small molecular size, the structure of the dimer \((f=2)\) of acetaldehyde may be represented as

Table 4

Total energy of rupture of hydrogen bonds during evaporation of liquids practically unassociated in the vapor

Note: Reference data on the heat of evaporation of amyl alcohol are contradictory [5, 6].

\[ \mathrm{H_3C{-}C\!\begin{array}{c} \diagup\ \mathrm{H\cdots O}\backslash \\ \diagdown\ \mathrm{O\cdots H}\diagup \end{array}\!C{-}CH_3}. \]

In this case \((Y_{\mathrm{h}}=2)\), per 1 molecule of aldehyde there is 1 \(H\)-bond, and \(\Delta H_{\mathrm{h}}=\sum \Delta H_{\mathrm{h}}=2.56\) kcal/mol per 1 \(H\)-bond. The strength of the \(H\)-bond in aldehydes is unknown. However, the strength of the \(H\)-bond in the complex of acetone with chloroform is known,

\[ \begin{array}{c} & \mathrm{CH_3}\\ & |\\ \mathrm{Cl_3C{-}H\cdots O{=}C} \, ,\\ & |\\ & \mathrm{CH_3} \end{array} \]

equal to 2.5 kcal/mol [1]. In essence, in both cases under consideration one and the same type of \(H\)-bond is observed,

\[ {>} \mathrm{C{-}H\cdots O{=}C} {<}, \]

and the close values of \(\Delta H_{\mathrm{h}}\) in both complexes confirm the correctness of the calculations.

The energy of rupture of the \(H\)-bonds of water is greater than the strength of any type of \(H\)-bond in liquids (Table 4). The strength of the \(H\)-bond in water, assuming the absence of cyclic forms of molecular association \((Y_{\mathrm{h}}=4\ [8])\), is 4.9 kcal/mol per 1 \(H\)-bond. The strength of the \(H\)-bonds of water is estimated in the literature as 5–6 kcal/mol [8], and that of ice as 4.5 kcal/mol per 1 \(H\)-bond [1].

Conclusions

1. It has been shown that, in the theoretical equation for the viscosity of liquids, associated molecules must be regarded as independent particles in the process of viscous flow of the liquid.

2. It has been established that, for hydrocarbons of normal structure and their derivatives that do not contain heavy atoms, in the absence of molecular association, the specific heat of activation of viscous flow and the specific heat of evaporation at \(25^\circ\) are, respectively, 18.4 and 87.2 cal/g.

3. Approximate methods are proposed for calculating the degree of association of molecules from the heat of activation of viscous flow, and the total energy of rupture of \(H\)-bonds from the heats of evaporation of associated liquids.

4. On the basis of the theoretical equation, the possibility has been shown of determining the strength of an \(H\)-bond (in kcal/mole per one \(H\)-bond) from these data, provided that additional information is available on the coordination number of the liquid with respect to the \(H\)-bond and on the forms of molecular association.

5. The results of calculating the degree of association and the strength of the \(H\)-bond by the proposed method agree well with literature data obtained mainly by spectral methods.

Literature

1. J. Pimentel, O. McClellan. The Hydrogen Bond, translated from English, Mir Publishers, Moscow, 1964.

2. G. M. Panchenkov. Theory of the Viscosity of Liquids. Gostoptekhizdat, Moscow, 1947.

3. A. A. Tager. Physical Chemistry of Polymers. Goskhimizdat, Moscow, 1963.

4. M. I. Shakhparonov. Methods for Studying the Thermal Motion of Molecules. Moscow State University Press, Moscow, 1963.

5. Chemist’s Handbook, Vol. 1. Goskhimizdat, Moscow, 1963.

6. A. Weissberger, E. Proskauer, J. Riddick, E. Toops. Organic Solvents, translated from English, IL, Moscow, 1958.

7. Z. S. Drutman. Scientific Reports of Higher Education Institutions. Chemistry and Chemical Technology. Soviet Science Publishing House, Moscow, 3, 513, 1958.

8. A. I. Altsybeeva, V. P. Belousov, A. G. Morachevskii. Chemistry and Thermodynamics of Solutions. Collection of articles. Leningrad State University Press, Leningrad, 1964, p. 145.

9. T. I. Temnikova. Course in the Theoretical Foundations of Organic Chemistry. Goskhimizdat, Leningrad, 1962.