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The Current State of the Theory of Viscosity
B. V. Bak, Tomsk
1. Introduction
Up to now theory has still not uncovered the true mechanism of the viscosity of liquids. Alongside other questions—such, for example, as specific heat, specific volume and their temperature behavior—viscosity and its dependence on temperature constitute an as yet unresolved part of the general problem that may be called the problem of the liquid state.
The intermediate position occupied by a liquid between the solid and gaseous states, whose theory has been developed much more fully, permits, generally speaking, an approach to the liquid state from two sides: from the solid body and from the gas. The circumstance that the transition from the liquid state to the gaseous one takes place continuously, whereas the transition from the solid to the liquid state is connected with a sudden “collapse” of the lattice, the loosening of which with increasing temperature is very small and becomes catastrophic at the melting point, seems, as it were, to incline one to the view that a liquid should stand closer to the gaseous aggregate state than to the solid one. The transfer of the ideas of the kinetic theory of gases to liquids in order to explain one or another of their properties, and the good agreement with experimental data*, obtained as a result of this transfer, also compel one to accept the existence of an extremely close connection between these two states and, naturally, to try to find the correct approach to the liquid by proceeding from the gas.
On the other hand, it seems beyond doubt that the intermolecular forces in the solid and liquid states cannot differ too greatly from one another, as is shown, for example, by Lindemann’s theory of melting.^1 The density of a liquid also differs little from the density of a solid, and the concept of a free path, extremely essential for the kinetic theory of gases, essentially does not exist for liquids: the large distances between gas molecules make it permissible to neglect intermolecular forces, whose intense fields completely alter the motions of the particles of a liquid. These same forces lead to the formation within the liquid
* See, for example, the work of Goldhammer discussed below.
molecular groupings that give the liquid a quasi-crystalline character. The difference from a solid here can be seen only in the instability of these groupings, both in the sense of their total number and in the sense of the size of each such group. The presence of such groupings was shown, for example, by Stewart² by means of X-ray diffraction.
All these circumstances permit an approach to the liquid from the solid state, with the transfer to the liquid of the concepts about the molecular state that are usual for solids (for example, oscillations of particles about an equilibrium position, intermolecular forces, etc.).
The existence of these two opposite, but in principle possible, paths toward an understanding of the liquid state was also reflected in attempts to solve the problem of viscosity. The essential difference between the viscosity of liquids and that of gases, consisting in the opposite course of the temperature variation of viscosity, compels one to assume fundamentally different mechanisms of this phenomenon for the two states; however, the possibility of transferring the concepts of the kinetic theory of gases, taking into account, of course, the special features of the liquid state, is nevertheless not excluded.
The viscosity of a liquid, as indeed its other properties, is very substantially influenced by the association of molecules. Liquids possessing strongly associated molecules usually do not obey the quantitative regularities valid for non-associated liquids. Thus, for example, from the experimental data on the basis of which Batschinski derived his well-known viscosity formula, it is evident that deviation from linearity in the system whose coordinates are specific volume and fluidity is characteristic, for example, of water and alcohols, of which it is well known that they are strongly associated liquids.
Extremely interesting is the fact that some associated liquids nevertheless satisfy the quantitative regularities following from one or another theory of viscosity, whereas other, likewise associated, liquids do not conform to these regularities. This makes one reflect on the very nature of association and, in particular, on the arrangement of dipoles within associated molecules, and also on the change in the degree of association over the temperature interval corresponding to the existence of the liquid phase.
It is quite natural that a theory capable of giving, in good agreement with experiment, a quantitative expression for the change of viscosity with temperature and pressure, both for non-associated and for associated liquids, on the basis of some conception of the mechanism of this phenomenon, will inevitably prove to be an essential, if not decisive, factor in the understanding of the liquid state in general.
The literature on this question has been enriched over the last several years by a series of works in which theories of viscosity are proposed that give a qualitative and quantitative picture of this phenomenon in a more...
...or less good agreement with experimental data. These include first of all the works of Herz and Kudar³, Andrade⁴, and also the work of A. Goldhammer⁵. In addition, the works of M. Wohler⁶ are of known interest, indicating a connection between molecular structure and viscosity, although, following F. Miles⁷, they must be approached very critically.
A review of all this material is the aim of the present article.
2. The Theory of Herz and Kudar³
Of all the existing mathematical expressions for the value of the coefficient of internal friction and for its temperature dependence, the best agreement with experimental data is given by Bachinskii’s empirical formula
\[ \eta=\frac{c}{v-\omega}, \tag{1} \]
where \(\eta\) is the coefficient of internal friction, \(v\) is the specific volume, and \(c\) and \(\omega\) are constants characteristic of the liquid. Here \(\omega\) is the specific volume occupied by the molecules upon transition to the solid state (then \(v=\omega\) and \(\eta=\infty\)), i.e. it is a quantity proportional to the proper volume of the molecules and equal for many substances to \(\omega=0.31\,v_{\mathrm{crit}}=b\) of the van der Waals equation. The second constant \(c\), according to Bachinskii, is equal to
\[ 3.58\cdot10^{-5}\frac{T_k^{1/2}v_k^{1/3}}{M^{1/6}} . \]
The difference \(v-\omega\) represents the “free” volume.
Strictly speaking, the proper volume of the molecules of a liquid should decrease with increasing temperature, since the greater kinetic energy of the colliding molecules should lead to a deeper penetration of one molecule into the sphere of another. Thus the increase of \(v-\omega\) with temperature should occur more rapidly than the increase of the specific volume observed experimentally. On the other hand, the constant \(c\) in the numerator of formula (1), written in the form
\[ c=3.58\cdot10^{-5}\frac{v^{1/3}T^{1/2}}{M^{1/6}}, \]
i.e. without the index \(k\), corresponding to critical values, should also increase with increasing temperature. The good agreement of formula (1) (in which the temperature and volume refer to the critical point and \(\omega\) is likewise taken as constant) with experimental data leads to the conclusion that the increase of the numerator compensates for the additional increase of the denominator due to the temperature dependence, as a result of which the change of the coefficient of internal friction with temperature proves to depend only on the observed change of the specific volume.
If in Bachinskii’s equation one substitutes the relations proposed by Lorentz,
\[ \frac{T_k}{T_e}=2.273 \quad \text{and} \quad \frac{v_k}{v_e}=3.120, \]
where the indices \(k\) and \(e\) refer respectively to the critical point and the melting point, then Bachinskii’s formula can be rewritten as:
\[ \eta=\frac{1}{v-\omega}\cdot 7.90\cdot 10^{-5}\frac{T_e^{\frac12}v_e^{\frac13}}{M^{\frac16}} . \tag{2} \]
Einstein’s diffusion formula
\[ \eta=\frac{kT}{4\pi rD} \tag{3} \]
gives good agreement with experiment only for the case of monatomic particles in the liquid state. In formula (3), \(D\) is the diffusion coefficient, \(r\) is the radius of the particle, \(k\) is Boltzmann’s constant, and \(T\) is the absolute temperature.
By a series of transformations formula (3) can be reduced to
\[ \eta=\frac{1}{v-\omega}\cdot 4.18\cdot 10^{-5}\frac{T_e^{\frac12}v_e^{\frac13}}{M^{\frac16}} . \tag{4} \]
From a comparison of formulas (3) and (4) it is seen that they differ by a numerical factor close to 2, and that, thus, Einstein’s formula can also be applied for transmitting the temperature dependence of the viscosity of polyatomic molecules, but in this case the right-hand side of the formula must be corrected by this factor.
What, then, does Einstein’s formula fail to take into account that is characteristic precisely of polyatomic molecules?
The authors of the theory believe that this discrepancy may be attributed to the circumstance that, in the case of liquids containing polyatomic molecules, not only the displacement of the molecules takes place, but also their rotation about certain axes, the position of which depends on the spatial arrangement of the atoms within the molecule, i.e., on its structure. Naturally, such rotation cannot occur in monatomic particles, for which formula (4) proves to be valid.
The essence of the theory, therefore, reduces to taking into account those forces by which this additional rotation of the molecule is caused, and to a quantitative transfer of those changes that are introduced by this circumstance into the viscosity of liquids.
Since the mean energy of displacement is equal to \({}^{3}/_{2}\,kT\), whereas for rotation, to which two degrees of freedom correspond, it is \(kT\), the energy expended in one way or another will be converted into the energy of displacement and rotation in the ratio \({}^{3}/_{2}\).
*
To the velocity \(u\), caused by the expended energy, there will correspond the kinetic energy \(\frac{mu^2}{2}\), and the additional energy of rotation will be
\[ \frac{\theta \alpha^2}{2}=\frac{2}{3}\frac{mu^2}{2}=\frac{mu^2}{3}, \tag{5} \]
where \(\theta\) is the moment of inertia and \(\alpha\) is the angular velocity.
For this rotation there must act a moment, produced by a couple of forces, so that, according to hydrodynamic data, it must be true that
\[ 2rK=8\pi\eta r^3\alpha, \]
where \(r\) is the radius of the particle.
Hence
\[ K=4\pi\eta r^2\alpha. \]
For a sphere with a uniform distribution of densities, the moment of inertia is
\[ \theta=\frac{2}{5}mr^2, \]
so that, taking a spherical form for the molecules and substituting the value of \(\theta\) just obtained into formula (5), we obtain
\[ \alpha=\sqrt{\frac{5}{3}}\,\frac{u}{r}, \]
whence
\[ K=5.16\,\pi\eta u. \tag{6} \]
Comparison of expression (6) for the force causing the rotation of molecules with Stokes’ formula
\[ K=6\pi\eta ru \tag{7} \]
for the force causing the displacement of molecules shows that these forces are almost equal to each other, differing by only a very small numerical factor.
Thus, the force causing the motion of the molecules of a liquid, in the case of polyatomic molecules, must be almost twice as great as in the case of monatomic molecules, where rotation does not occur.
This circumstance can also explain the inapplicability of Einstein’s formula to polyatomic molecules.
If the numerical coefficient of formula (4) is multiplied by the ratio obtained from (6) and (7), \(\frac{11.16}{6}=1.86\), then one obtains \(7.77\), which differs very little from \(7.90\), the coefficient of Batschinski’s empirical formula (2).
Thus, from this theory it follows that the empirically determined coefficient of internal friction is the sum of two components—translational and rotational:
\[ \eta_{\mathrm{emp}}=\eta_{\mathrm{trans}}+\eta_{\mathrm{rot}} =\eta_{\mathrm{trans}}\left(1+\frac{\eta_{\mathrm{rot}}}{\eta_{\mathrm{trans}}}\right). \tag{8} \]
The component \(\eta_{\mathrm{trans}}\) is conveyed sufficiently well by formulas that do not take molecular rotation into account, while the ratio \(\dfrac{\eta_{\mathrm{rot}}}{\eta_{\mathrm{trans}}}\) can be expressed by the ratio of the corresponding forces, so that
\[ \frac{\eta_{\mathrm{rot}}}{\eta_{\mathrm{trans}}} = \frac{K_{\mathrm{rot}}}{K_{\mathrm{trans}}}. \]
\(K_{\mathrm{trans}}\) can be obtained from the velocity of the particles relative to the medium. Substituting the value of this relative velocity \(\dfrac{du_x}{dz} r\) into formula (7), we obtain
\[ K_{\mathrm{trans}} = 6\pi \eta r^2 \frac{du_x}{dz}. \tag{9} \]
\(K_{\mathrm{rot}}\) can be determined from the moment of rotation, taking into account the structure of the molecule. If it is assumed that the molecules are not deformed under this rotation, then only the tangential component of the moment of rotation may be considered, without the radial one. Then \(K_{\mathrm{rot}}\), as the algebraic sum of all tangential forces of rotation, will be equal to
\[ K_{\mathrm{rot}} = \dot{\alpha} \sum m_i r_i^2, \]
where \(\dot{\alpha}\) is the angular acceleration, and \(r_i\) is the distance of the mass points \(m_i\) from the axis of rotation.
The moment of rotation
\[ M = \dot{\alpha} \sum m_i r_i, \]
where \(\sum\) denotes the moment of inertia.
Since
\[ M = \delta K_{\mathrm{rot}}, \]
where \(\delta\) is the arm, then
\[ \delta = \frac{\sum m_i r_i^3}{\sum m_i} = \frac{\tau^2}{s}. \]
Here
\[ \tau = \frac{\sum m_i r_i^2}{\sum m_i r_i} \]
is the radius of inertia, and
\[ s = \frac{\sum m_i r_i}{\sum m_i}. \]
Here one may confine oneself to an examination of the application of the above arguments to the case of a spherical distribution of masses in the molecule.
If the \(x\)-axis corresponds to the direction of the flow, and the \(z\)-axis to the velocity gradient, then the rotation of a material point of the molecule \(P\) will occur about the \(y\)-axis. The velocity of this material point in the direction of the \(x\)-axis will be
\[ \rho \frac{du_x}{dz} \sin^2 \varepsilon. \]
On the other hand, this same velocity must be equal to
\[ \frac{\rho u}{\sin \varepsilon}, \]
whence
\[ u = \frac{du_x}{dz}\sin^2 \varepsilon . \]
The angular velocity \(\alpha\) is obtained from the mean value of the tangential components
\[ \alpha = \frac{1}{\pi}\int_0^\pi \frac{du_x}{dz}\sin^2 \varepsilon\, d\varepsilon = \frac{1}{2}\frac{du_x}{dz}. \]
Then the hydrodynamic moment of rotation, equal to \(7\pi\eta r^3\alpha\), will be
\[ M = 4\pi\eta r^3\frac{du_x}{dz}. \]
If we assume that the entire mass is distributed over a spherical surface of radius \(\rho\), then the equalities
\[ \tau^2 = \frac{2}{3}\rho^2 \quad \text{and} \quad s = \frac{\pi}{4}\rho, \]
will be valid, whence
\[ K_{\mathrm{rot}}=\frac{M}{\delta} = \frac{3}{2}\pi^2\eta r^2\frac{r}{\rho}\frac{du_x}{dz}, \tag{10} \]
and by dividing (10) by (9) we obtain
\[ \frac{K_{\mathrm{rot}}}{K_{\mathrm{trans}}}=\frac{\pi}{4}\frac{r}{\rho}. \]
Thus, for the viscosity of liquids with a spherical distribution of mass, the formula is derived
\[ \eta = \eta_{\mathrm{trans}}\left(1+\frac{\pi}{4}\frac{r}{\rho}\right). \tag{11} \]
For liquids with monatomic molecules the factor in parentheses is absent.
By analogous calculations, taking into account the possible axes of rotation, expressions may also be obtained for molecules of another shape. Thus, for example, for dumbbell-shaped molecules one obtains the expression
\[ \eta = \eta_{\mathrm{trans}}\left(1+\frac{8}{9}\frac{r}{\rho}\right), \]
where \(\rho\) denotes one half of the internuclear distance.
This theory does not take into account the influence on the viscosity of the dipole moment of the molecules and, as the authors indicate, is applicable to nonpolar liquids. The influence of polarity, however, may be expressed not only in the formation of associated aggregates, but also in a definite orientation of the molecules relative to one another. Both of these factors undoubtedly influence internal friction.
It is interesting that the calculation of the coefficient of viscosity by the formulas given, for some dipolar liquids, gives good agreement with experiment. Since this is valid only for some associated liquids, the authors
It is assumed that in these cases the formation of bimolecular groups evidently takes place. Then the dipole moments of each half of such a double molecule may be directed antiparallel, and, if the dipole action of neighboring molecules is neglected, the total dipole moment may prove to be equal to zero. If, however, the particles of the liquid are situated sufficiently densely, the influence of neighboring molecules can no longer be neglected, and then the antiparallelism of the dipoles in the double molecules will be disturbed, leading to a dipole moment different from zero.
In the latter case, obviously, one cannot expect agreement between the values calculated from the formulas and the experimental values. The same discrepancy may be expected in the case of liquid metals and molten electrolytes. The authors consider that liquid metals are either monatomic or consist of molecules that do not represent sufficiently strong combinations of atoms, so that rotation does not arise. The same applies also to molten electrolytes, where it may be considered that the ion pair does not form a strong compound. However, the value of the molecular weight in the viscosity formula must be taken equal to that in the liquid state, i.e., for Pb, for example, the atomic weight must be quadrupled, for Bi—doubled, etc. In the case of molten salts, the weight of the ion pair must be taken.
Proceeding from these considerations, all liquids may be divided into four types: to the first will belong nonconducting monatomic liquids; here, obviously, Einstein’s formula should be applicable; to the second type belong nonpolar polyatomic liquids; here Einstein’s formula must be multiplied on the right-hand side by the factor ∼ 2; to the third type—dipolar liquids and molten electrolytes, and to the fourth—liquid metals. In both of the latter cases the formula for nonrotating molecules is applicable, taking into account the molecular weight in the liquid state.
It seems to us interesting to dwell on one circumstance about which the authors say nothing, but which follows logically from all that has been said above.
If Einstein’s formula gives a good account of the change in the coefficient of internal friction for nonrotating molecules, and the coefficient 4.18 enters into the value of the constant \(c\), then it is quite natural to conclude that Batschinski’s formula with the coefficient 7.90 for \(c\) should not give agreement with experiment precisely for liquids with monatomic particles. For them the slope angle in the axes \(\dfrac{1}{\eta}\) and \(\upsilon\) (Batschinski’s formula may be rewritten as
\[ \upsilon = \omega + \frac{c}{\eta} \]
) must be different from that which follows from Batschinski’s formula. In the cited paper by Batschinski all the experimental results given refer to polyatomic liquids, and the sole case of a monatomic liquid—the case of mercury—shows a deviation from a rectilinear dependence.
The essential difference, in the sense of the theory mentioned, between the association of polyatomic molecules and the association of monatomic ones consists in the fact that, if one does not speak of a change in volume, in the first case the conditions for molecular rotation are only disturbed, whereas in the second case this rotation first arises.
The very fact of the association of polyatomic molecules does not yet introduce any substantial change into the temperature course of the dependence of fluidity on specific volume. Indeed, for example, acetic acid, for which many authors assume bimolecular association, and it may be considered that the bond in these pairs is so strong that, in the temperature interval corresponding to the liquid phase, the degree of association does not change appreciably, gives a very good straight line in the coordinate system mentioned.^8 Thus it is not the fact of association itself, but its change with temperature, that causes the deviation from the rectilinear dependence of \(\dfrac{1}{\eta}\) on \(v\) for associated liquids.
The dissociation of associated polyatomic molecules should not, obviously, cause a noticeable change in the numerical coefficient at \(c\), since rotation occurs in this case both for the associated and for the non-associated molecules. The concavity of the curve toward the fluidity axis may be explained by the different temperature course of the specific and proper volumes. For all liquids the volume of the associated aggregate is less than the sum of the volumes of the particles entering into it. With increasing temperature the specific volume \(v\) increases, and, owing to the dissociation of the associated molecules, \(\omega\) also increases. At lower temperatures the difference \(v-\omega\) will increase more slowly than at higher ones, where the growth of \(\omega\) is slowed owing to the decrease in the degree of association. This accelerated growth of \(v-\omega\) leads to a relatively more rapid growth of fluidity, and the curves prove to be concave toward the abscissa axis.
For water, whose anomaly, as is commonly believed, consists precisely in the fact that the volume of the associated aggregate is greater than the sum of the volumes of its constituent particles, one must expect the opposite course of the curve, which is also observed experimentally.^8
In a similar way one may explain the anomalous behavior of the viscosity of water with change in pressure. Water is the only liquid whose viscosity decreases with increasing pressure.^9 This occurs at temperatures up to \(30^\circ\mathrm{C}\) and pressures up to \(1000\) atm. At about 30 degrees the viscosity of water, up to a pressure of 400 atm, does not change its value, and, finally, above \(30^\circ\) the viscosity exhibits normal behavior, i.e. a steady increase with pressure. Applying the same method of reasoning, one may conclude that with increasing pressure at low temperatures the volume \(v\) decreases, but \(\omega\) decreases still more in absolute magnitude owing to the dissociation of associated molecules, which at low temperatures is fairly considerable. This causes an increase in \(v-\omega\) (despite the fall of \(v\)), and \(\eta\) decreases. At \(30^\circ\) the decrease of \(v\) is exactly equal to the decrease of \(\omega\) with pressure, since
heating water to \(30^\circ\) led to partial dissociation of the associated molecules, owing to which the decrease of \(\omega\) with pressure is already smaller. Finally, at still higher temperatures the change of \(\omega\) with pressure is insignificant, and the increase of \(\eta\) occurs at the expense of a decrease of \(v\).
In the case of mercury mentioned above, association should, in the sense of the theory, lead to the appearance of a new factor—the rotation of molecules. Owing to this, the numerical coefficient at \(c\), and hence also the angle of inclination, should increase with decreasing temperature, i.e. the curve should be concave toward the axis of flow; in Batschinski’s work, however, it is convex toward this axis. One must suppose either that the experimental data used by Batschinski are incorrect (there are a number of indications of this), or that at low temperatures some new factor appears, overriding the influence of association and causing the straight line to curve in the opposite direction. The authors, however, believe that this deviation may be ascribed to association, although all the arguments given above concerning the change in the angle of inclination follow directly from their theory.
3. Andrade’s Theory⁴
Basing himself on the opposite temperature behavior of the viscosity of liquids and gases, and also on the numerous unsuccessful attempts to explain the mechanism of internal friction by transferring to liquids the concepts of the kinetic theory of gases, the author comes to the conclusion that the problem of viscosity, like the problem of the liquid state in general, will most probably be solved from the point of view of the solid state. In the fact that the distance between particles differs little in the solid and liquid states, that in both cases the molecules are in the field of considerable intermolecular forces, and that liquids, especially at low temperatures close to the freezing point, have a quasi-crystalline structure, the author sees grounds for this assertion.
One of the basic premises of the theory is the transfer of momentum from layer to layer; but, in contrast to gases, there is no transition of molecules from one layer to another, but only a temporary connection of molecules at the boundary of layers, so that, in a first approximation, one may consider that the molecules remain in their own layers, while the equilibrium position of the oscillating molecules is displaced only slightly. That this displacement is indeed small may be inferred from the small rate of diffusion in liquids.
The second basic premise of the theory is the assumption that the frequency of oscillations of molecules about the equilibrium position for a liquid differs little from the frequency of such oscillation for the solid state. These two states differ from one another not in the frequency of the oscillations, but in their amplitude. The amplitude of oscillations in the solid state is small in comparison with the distances between particles, whereas in a liquid it is so large that a collision occurs at each extreme deviation. In addition,
the equilibrium position, which in the solid state is fixed, while in the liquid it is slightly displaced. The theory, however, assumes that the oscillation of particles about the displaced equilibrium position does not differ in frequency from the oscillation in the immobile case.
The third basic premise consists in the necessity of assuming a certain potential energy for the transfer of momentum upon collision. In this case, obviously, the number of molecules with a given potential energy will be determined by Boltzmann’s law of exponential distribution, and the number of cases favorable to the transfer of momentum will decrease with increasing temperature.
The temporary combination of molecules at the boundary of layers is connected with the orientation of molecules relative to one another, and this orientation must, obviously, be disrupted by thermal oscillations. Thus a lowering of the temperature will favor this orientation.
The local electrostatic fields, which are also assumed to exist in the liquid, may be regarded as directed in such a way that they too favor the orientation of molecules. The quasi-crystalline character of the liquid, especially at low temperatures, can precisely be explained by the presence of small groups of molecules formed as a result of this tendency toward orientation. It may be assumed that a large part of the volume of the liquid is occupied by these groupings, the number of which, as well as the size of each, changes continuously in the liquid and depends, generally speaking, on the potential energy. An increase in temperature disrupts the tendency toward the grouping of molecules, since thermal oscillations affect the degree of orientation of the molecules.
That, as a first approximation, the molecules may be taken as remaining in their own layer can be shown in the case of liquid lead, the self-diffusion of which can be observed thanks to the presence in it of “marked” molecules of the radioactive isotope thorium B. The diffusion coefficient \(D\) in this case is equal to \(2.4\ \text{cm}^2/\text{day}\). Since
\[ \frac{\overline{x^2}}{\tau}=2D, \]
where \(\overline{x^2}\) is the mean value of the square of the displacement during time \(\tau\), then
\[ 2D=\frac{4.4}{86\,400}\ \frac{\text{cm}^2}{\text{sec}}=5.1\cdot 10^{-5}\ \frac{\text{cm}^2}{\text{sec}}. \]
The mean distance between particles may be obtained for lead from
\[ \bar{x}=\sqrt[3]{\frac{M}{Nd}}=3.1\cdot 10^{-8}, \]
where \(M\) is the atomic weight, \(d\) the density, and \(N\) Avogadro’s number.
Then the time \(\tau\) necessary to traverse this distance will be
\[ \tau=\frac{\bar{x}^{2}}{2D}=1.89\cdot 10^{-11}. \]
On the basis of the second main premise, one may take for the frequency of oscillations of the molecules of a liquid its value for the solid state, which, for the case of lead under consideration, is equal to \(2\cdot 10^{12}\); this can be established by various methods\({}^{10}\). For the time between two collisions, taking into account that they occur at each extreme displacement, one can obtain the value \(2.5\cdot 10^{-13}\) sec. During the time \(\tau=1.89\cdot 10^{-11}\), consequently, 76 collisions occur.
Such a large number of collisions, occurring during the time necessary for the displacement of the equilibrium position by a distance equal to the distance between layers, permits one to consider that the molecules remain in their own layer.
If it is assumed that the displacement of molecules takes place along \(n\) identical paths of unknown length \(l\) in any direction, then, with a certain known value of the mean velocity \(\bar{v}\) of such a displacement, the number \(n\) during the time \(\tau\) over the distance \(x\), equal to the distance between particles, can be found. Putting
\[ \frac{1}{2}m\overline{v^{2}}=\frac{3}{2}kT, \]
where \(\overline{v^{2}}\) is the mean value of \(v^{2}\), we obtain
\[ \bar{v}=\frac{2V}{\pi}\sqrt{\overline{v^{2}}}=\frac{4}{\pi}\sqrt{\frac{3}{2}}\sqrt{\frac{kT}{m}}. \]
For a large number of successive paths \(l\) in any direction in three-dimensional space, the probability that the end of the \(n\)-th path lies between \(r\) and \(r+dr\) is determined by Rayleigh’s theorem
\[ \Phi r\,dr=3\sqrt{\frac{6}{\pi}}\,\frac{n}{l^{3}}\,e^{-\frac{3}{2}-\frac{r^{2}}{3nl^{2}}}r^{2}dr. \]
Taking \(nl=d=\bar{v}\tau\), we obtain
\[ \Phi r=3\sqrt{\frac{6}{\pi}}\,\frac{n^{3/2}}{d^{3}}\,e^{-\frac{3r^{2}n}{2d^{2}}}r^{2}. \]
Choosing \(n\) so that the probability of a molecule traversing the distance \(x\) would be greatest, we obtain
\[ \frac{d\Phi r}{dn}=0 \]
and
\[ n=\left(\frac{d}{r}\right)^{2}. \]
Since for lead \(r = 3.1 \cdot 10^{-8}\ \text{cm}\), \(\tau = 1.89 \cdot 10^{-11}\), then, taking \(T = 616^\circ\), we obtain
\[ d = v\tau = 2.95 \cdot 10^{-8}, \]
whence
\[ n = \left(\frac{d}{r}\right)^2 = 90. \]
Another method of calculating the number of paths leads to a value of the same order as the first.
Thus one may assume that the molecules remain in their own layer, and their temporary union at the boundary of the layers does not exceed the interval of time necessary for the transfer of momentum. Viscosity is due to the transfer of momentum from layer to layer.
If \(\sigma\) is the distance between the centers of molecules, the number of which per unit volume is equal to \(n\), then
\[ \sigma = n^{-\frac{1}{3}};\quad \rho = n^{\frac{2}{3}} = \frac{1}{\sigma^2}, \]
where \(\rho\) is the number of molecules per unit surface.
The transfer of momentum through a unit surface in unit time will be
\[ \frac{4}{3}\frac{1}{\sigma^2}\nu m\frac{dv}{dx}\sigma = \eta \frac{dv}{dx} \]
or
\[ \eta = \frac{4}{3}\nu\frac{m}{\sigma}, \]
where \(\nu\) is the frequency of oscillations.
In accordance with one of the basic assumptions of the theory, let us take for the frequency the value given for a solid by Lindemann’s formula
\[ \nu = c\sqrt{\frac{T_{\text{zam}}}{A V_A^{\frac{2}{3}}}}, \]
where \(A\) is the atomic weight and \(V_A\) the volume at the freezing temperature.
Substituting for \(\sigma\) the value \(\left(\frac{V_A}{N}\right)^{\frac{1}{3}}\) and for \(m\) the value \(\frac{A}{N}\), we obtain for the viscosity near the melting point
\[ \eta = \frac{4}{3}c\frac{(A T_{\text{zam}})^{\frac{1}{2}}}{(N V_A)^{\frac{2}{3}}}, \]
and substituting the values of \(c\) and \(N\),
\[ \eta = 5.1 \cdot 10^{-4} \frac{(A T_{\text{zam}})^{\frac{1}{2}}}{(V_A)^{\frac{2}{3}}}. \tag{12} \]
Formula (12) must be valid for the viscosity near the melting point. Comparison of the values obtained from this formula with experimental data shows more or less good agreement both in the case of metals (mercury, lead, tin, copper) and in the case of polyatomic liquefied gases (halogens, oxygen, hydrogen).
All the quantities entering into formula (12) are constants, and for this reason it is applicable only for one temperature, close to \(T_{\text{melt}}\). To derive the dependence of \(\eta\) on temperature, it is necessary to use the third basic premise of the theory.
If the frequency of oscillations \(\nu\) is considered independent of temperature, then the influence of the latter will be reflected only in the number of molecules which, at an extreme deviation, possess sufficient potential energy to carry out the transfer of momentum. According to Boltzmann, the ratio of the numbers of molecules possessing this energy at temperatures \(T\) and \(T_1\) will be
\[ e^{\frac{E}{k}\left(\frac{1}{T}-\frac{1}{T_1}\right)} . \]
Then the temperature course of the change of viscosity may be represented as follows:
\[ \frac{\eta_T}{\eta_{T_1}} = e^{\frac{E}{k}\left(\frac{1}{T}-\frac{1}{T_1}\right)} \]
or
\[ \eta_T = A e^{\frac{c}{T}}, \tag{13} \]
The constant \(A\) will correspond to the viscosity at \(T=\infty\).
The formula proposed in this form still contains nothing new, since an exponential dependence of viscosity on temperature had already been proposed earlier by a number of authors, and it was established that the correct temperature course is conveyed by it only approximately. What is essentially new is that, further on, the change of specific volume with temperature is also taken into account, as a result of which the formula obtained is
\[ \eta v^{\frac{1}{3}} = A e^{\frac{c}{vT}}, \tag{14} \]
which is the principal result of the theory.
The fact that \(v\) enters in the power \(\frac{1}{3}\) can be explained by the circumstance that, with increasing temperature, the distance between molecules grows as \(v^{\frac{1}{3}}\), while the number of molecules per unit surface decreases as \(v^{-\frac{2}{3}}\). The appearance of \(v\) in the exponential term is a consequence of the assumption that the potential energy is also a function of the specific volume and, moreover, just as in the van der Waals equation, \(v\) stands in the denominator. Substitution of \(v^2\) leads to worse agreement with the experimental data.
An attempt to take into account the change of \(\nu\) with temperature, which is obtained by the relation proposed by Einstein,
\[ \nu = C v^{\frac{1}{6}} \frac{1}{\sqrt{k}}, \]
where \(k\) is the adiabatic compressibility, leads to
\[ \eta d^{\frac{1}{6}}=\frac{A'}{\sqrt{k}} e^{\frac{c'}{vT}}, \tag{15} \]
since \(A\) in formula (14) is proportional to \(\nu\).
Comparison of formulas (14) and (15) with the experimental data shows that, although both give very good agreement (for more than 100 substances the error does not exceed \(2\%\)), formula (14) nevertheless agrees somewhat better. Evidently there is some cause that brings about compensation of the actual change of \(\nu\) with temperature and leads to the independence of \(\eta\) from the temperature change of \(\nu\).
Using the fact that formula (15) gives sufficiently good agreement with experiment and includes the coefficient of adiabatic compression \(k\), it may be used to transmit the change of viscosity with pressure. The relation between the viscosity at a certain pressure and the viscosity at atmospheric pressure will be transmitted by the formula
\[ \frac{\eta_p}{\eta_1} = \left(\frac{v_1}{v_p}\right)^{\frac{1}{6}} \sqrt{\frac{k_1}{k_p}}\, e^{\frac{c}{T}\left(\frac{1}{v_p}-\frac{1}{v_1}\right)} . \tag{16} \]
The indices \(p\) and \(1\) refer respectively to pressure \(p\) and to atmospheric pressure.
Formula (16) gives agreement with the experimental data to an accuracy of up to \(10\%\), and it must be borne in mind that the experimental data often have to be extrapolated. At pressures above \(2000\) atm the formula ceases to be valid, since, besides considerable changes in the magnitude of \(\nu\) and in the potential energy, the molecules themselves undergo considerable deformation.
If \(E_{\mathrm{mol}}\) is the energy of the field of intermolecular forces, calculated per mole, then it can be shown that
\[ E_{\mathrm{mol}}=\frac{2a_{\mathrm{mol}}}{NMv}, \]
where \(a\) is the constant of the van der Waals equation per mole in CGS units, \(N\) is Avogadro’s number, and \(M\) is the molecular weight. Comparing the exponents in \(e\), which express the potential energy, we obtain
\[ \frac{E}{kT}=\frac{c}{vT}, \]
where \(E\) is the part of the potential energy whose existence is postulated in order to explain the temperature course of the viscosity, as was already noted.
It may therefore be considered that, for all substances, the quantity \(c\) will be proportional to \(\dfrac{2a}{Nmk}\). A check for 36 substances showed that the ratio \(\dfrac{c}{a_0}\left(a_0=\dfrac{2a}{Nmk}\right)\) has, for all of them, a value of about 0.134. A deviation is found in highly symmetrical molecules, for which, evidently, the conditions are least favorable for the orientation of molecules that promotes the formation of molecular groupings. Everything that disrupts the symmetry in the structure of molecules leads to the agreement of \(\dfrac{c}{a_0}\) with 0.134.
A great merit of the theory is that formula (14) proves to be valid, in the main, also for the case of associated liquids, since one of the basic points of the theory—the temporary joining of molecules at the boundary between layers—leads to the formation of association-type bonds between molecules. Even such strongly associated liquids as alcohols obey the formula; their viscosity, in the temperature interval from \(T_{\mathrm{freez}}\) to \(T_{\mathrm{boil}}\), changes by thousands of times. However, associated liquids show a noticeable deviation of the ratio \(\dfrac{c}{a_0}\) from 0.134, i.e., they have an abnormally large constant \(c\).
Since, as already mentioned, an exponential dependence of the coefficient of internal friction on temperature had been proposed on various grounds by other authors, the very fact that such a dependence is obtained from the mechanism proposed by the author does not seem to us a sufficient criterion of the correctness of this mechanism. The correction for the change of specific volume with temperature, which substantially distinguishes the final formula (14), leading to better agreement with experiment, from the intermediate formula (13), which coincides with the formulas of other authors, likewise does not yet testify to the correctness of the mechanism.
It seems to us that a number of other points speak in favor of the basic propositions of the theory, and first of all the fact that the exponent at \(e\) in the proposed formula, for the majority of liquids, turns out to be proportional to the potential energy, which may be regarded as confirmation of the third basic premise.
The circumstance that an attempt to take into account the change in frequency with temperature leads to somewhat poorer agreement with experimental data than is obtained from formula (14), where \(\nu\) enters into the constant \(A\), also confirms one of the basic propositions concerning the independence of \(\nu\) from temperature and the validity of transferring the value of \(\nu\) in the solid state to the liquid, irrespective of the causes determining this constancy.
The deviations of highly symmetrical molecules of the type \(\mathrm{CCl_4}\), benzene, and others from the law \(\dfrac{c}{a_0}=0.134\) may also be regarded as confirmation of the assumption concerning the action of internal electrostatic fields that promote the orientation of molecules, which causes the formation of groups.
Finally, the fact that the transition from formula (14), which conveys the temperature behavior of viscosity, to formula (15), which conveys the behavior of viscosity with pressure, is accomplished only by taking into account the ratio of the coefficients of adiabatic compression for different pressures, also speaks in favor of the present theory.
It must be noted that a mechanism basically analogous to this was proposed as early as 1926 by Ya. Frenkel[^11] in developing A. Ioffe’s ideas on conductivity in crystals. The atoms (or ions) of crystals oscillate about certain equilibrium positions, forming a crystal lattice; moreover, some of the atoms, having broken away from their places, have passed into the “interstitial space.” Such “dissociated” atoms oscillate about new unstable equilibrium positions, forming an additional lattice (Zwischenraumgitter). The displacement of the dissociated atoms from one point of this intermediate lattice to another is what determines conductivity in crystals.
Ya. Frenkel transfers this mechanism also to the liquid state, considering that the latter differs essentially from the crystalline state precisely in that not some of the atoms, but all of them, are in such a dissociated state. Then, of course, there is no normal crystal lattice, but oscillations of the atoms about shifting equilibrium positions are preserved.
The calculation of the coefficient of internal friction carried out on the basis of these ideas led to good agreement with experiment for simple liquids. In this case an exponential dependence on temperature is obtained for \(\eta\).
4. Tammann’s Work[^5]
Batschinski’s formula can no longer be regarded as empirical, since it has been derived by various authors[^12] and by various methods, and the epithet “empirical” can remain attached to it perhaps only in a historical sense.
One possible derivation of this formula from Einstein’s formula may be found in the cited work of Herzog and Kudar, where this derivation is carried out by substituting into Rieke’s diffusion formula
\[ D=\frac{\pi}{8}\lambda w \]
the values of the factors entering into it (the coefficient \(\frac{\pi}{8}\) differs somewhat from the \(1/3\) in Rieke’s formula). The “depth of penetration” in diffusion—a quantity proportional to the mean free path between collisions—is equal to
\[ \lambda=\frac{8}{\pi}\frac{V-b_e}{4\pi r^2 N}, \]
where \(V\) is the volume of a mole and \(b_e\) is its volume at the freezing temperature.
For the velocity \(w\) one may take the value
\[ w=\sqrt{\frac{8kT}{\pi m}}. \]
Then
\[ D=\frac{V-b_e}{4\pi r^2N}\sqrt{\frac{8kT}{\pi m}}. \]
Substituting the \(D\) thus obtained into Einstein’s diffusion formula (3), replacing \(V\) and \(b_e\) by the products \(vM\) and \(\omega M\) (where \(M\) is the molecular weight, \(v\) is the specific volume, and \(\omega\) is the intrinsic volume of a gram of substance), and putting
\[ r=\left(\frac{3}{4}\frac{1}{\pi N}vM\right)^{\frac{1}{3}}, \]
we obtain formula (4).
As has already been indicated, the numerical coefficient of formula (4), equal to 4.18, can be corrected to 7.77 if one adopts the theory of Herzog and Kudar, which takes into account the rotation of polyatomic molecules.
Another derivation from the concepts of the kinetic theory is given in his work by A. Goldhammer. The derivation is based on a comparison of the resistance force experienced by a sphere moving in a certain medium, obtained from various formulae with allowance for the circumstance that, in the case of a liquid, the molecules cannot be regarded as incompressible spheres and that the mean free path must depend on the velocity and volume. The author proposes the dependence
\[ l=l_0[1-f(T)], \tag{17} \]
where \(l_0\) is the mean free path for incompressible molecules, and \(f(T)\) is a decreasing function of temperature, tending to 0 and attaining it at the critical point.
If \(N\) is the number of molecules, \(V\) their volume, and \(S\) the actual volume of all molecules, then
\[ l_0=\frac{V-\alpha S_0}{\sqrt{2}\,4\pi r^2N}, \tag{18} \]
where \(\alpha\) is a numerical coefficient of order 3.
Comparison of the magnitude of the resistance force from Stokes’ formula (7)* with the formula obtained from kinetic considerations:
\[ F=M\frac{\bar{c}}{l}v\frac{m}{M+m}\frac{\frac{4}{3}M+m}{M+\frac{4}{\pi}m}, \tag{19} \]
where \(M\) and \(m\) are the masses of the sphere and of the particles of the medium (in the case under consideration
* Taking into account that, for the case of motion of a particle in a medium of particles of the same order of size, the coefficient in formula (7), owing to the absence of sliding friction, changes from 6 to 4.
\(M=m\), \(\bar c\) and \(\bar v\) are the mean velocities of the particles of the medium and of the sphere, and \(l\) is the free path of the sphere, leads to
\[ \eta=\frac{0.513}{4\pi}\,\frac{m}{r}\,\frac{\bar c}{l}. \tag{20} \]
Substituting for \(l\) its value from (17), and for \(l_0\) its value from (18), i.e., thereby introducing a correction for the peculiarity of the liquid state, which, of course, is not taken into account by formula (19), we obtain
\[ \eta=\frac{0.726Mr\sqrt{\dfrac{8R}{\pi M}}}{V-\alpha S_0}\, \frac{\sqrt{T}}{1-f(T)}. \tag{21} \]
Thus, if
\[ \frac{\sqrt{T}}{1-f(T)}=\mathrm{const}, \]
then formula (21) can be reduced to (1). Using the above-mentioned condition \(f(TK_p)=0\), passing to the specific volume \(v\) and \(S_0\), we obtain
\[ \omega=\alpha S_0 \]
and
\[ c=\frac{0.726r\sqrt{8R}\sqrt{T_k}}{\sqrt{\pi}\sqrt{M}}, \]
where \(\omega\) and \(c\) are the constants of equation (1). Since \(S_0=\dfrac{b}{4}\) (where \(b\) is the constant of the van der Waals equation), putting \(b=\dfrac{v_k}{2}\) (!) and taking \(\alpha=2.5\), we obtain
\[ \frac{\omega}{v_k}=0.31, \]
which corresponds to the value obtained for this ratio by Batschinski. Further, substituting numerical values for the constants and expressing \(r\) through \(S_0\) \((S_0=N_0 4\pi r^3)\), and \(S_0\), in turn, through \(v_k\), we obtain
\[ c=3.87\cdot 10^{-5}\,\frac{T_k^{\frac12}v_k^{\frac13}}{M^{\frac16}}, \]
which is in complete agreement with the value of \(c\) found by Batschinski.
Thus formula (1) has received yet another theoretical derivation.
There is one point in these arguments that we would like to note. The ratio \(\dfrac{\omega}{v_k}\) was checked by Batschinski for a large number of substances, and it was established that the deviation from the value 0.31 generally does not exceed \(2\%\). Such are the experimental data.
In Goldhammer’s work this ratio is equal to 0.31 only in the case where one assumes \(b=\dfrac{v_k}{2}\) and \(\alpha=2.5\). The author himself points out in this connection: “For many substances the constant \(b\) of van der Waals—
Waal’s is equal to \(b=\dfrac{v_k}{2}\), where \(v_k\) is the critical volume.” However, it is difficult to suppose that among the numerous substances for which Batschinski tested the ratio \(\dfrac{\omega}{v_k}\), there were only those for which \(b\) is exactly equal to one half of the critical volume. One might rather expect the more probable value \(b=\dfrac{v_k}{3}\). With regard to \(\alpha\), the author points out that it is in general less than 4; for water it is about 2.8, while the value 2.5 refers to gases that are not very highly compressed. If one adopts as more probable for liquids a value of \(\alpha\), for example 3, then one obtains
\[ \alpha S_0=\frac{3}{4}b=\frac{3v_k}{4\cdot 3}=0.25v_k, \]
which gives a discrepancy of 20% with the value 0.31.
If, however, the author’s accepted value \(\alpha=2.5\) is retained and only for \(b\) one adopts \(\dfrac{v_k}{3}\), then the discrepancy becomes still larger, namely
\[ \alpha S_0=\frac{2.5b}{4}=\frac{2.5}{4}\frac{v_k}{3}=0.21v_k, \]
i.e. it reaches 33%.
The absence of so considerable a deviation of this ratio in Batschinski’s experiments and, on the other hand, the undoubtedly greater probability for \(b=\dfrac{v_k}{3}\) and \(\alpha\sim 3\), contain a certain contradiction.
As the author quite correctly notes in his work, the very good agreement of formula (1) with experiment confronts every theory of viscosity with the necessity of explaining this formula, whatever mechanism may underlie the theory.
The absence of even a mention of Batschinski’s formula in Andrade’s extensive cited work can undoubtedly be noted as a negative fact; conversely, a theory that succeeds in giving an interpretation of this formula will receive additional confirmation of the correctness of its basic starting points.
The derivation of Batschinski’s formula carried out by Hollgammel is extremely important for supporters of the approach to liquids from the side of gases.
5. Viscosity and the Structure of Molecules
It has been noted above more than once that the structure of molecules is substantially reflected in viscosity. Under the influence of this factor, the Herzog and Kudlar theory is encountered; for different distributions of masses in space, it gives various multipliers in parentheses in formula (11). One of the basic propositions of Andrade’s theory—the proportionality of the coefficient \(c\) to the potential energy—undergoes a quantitative violation in the case of symmetrical molecules.
The presence of a regular increase of viscosity as one moves
to more complex members of the homologous series has long been known. The study of the dependence of \(\eta\) on structure is of present interest both along the lines of homologous series and along the lines of particular derivatives of a given substance or of its isomers.
For a whole series of properties one can observe a definite regularity of change in a homologous series; very often the transition from an even to an odd number of C atoms in the chain leads to a different magnitude of change in this property than the transition from an odd to an even number. This can be traced, for example, in the changes of melting temperatures, the difference of which, on passing from an odd to an even number of C atoms, amounts to several tens of degrees, while on passing from an even to an odd number it is only \(4—5^\circ\).
In Uoller’s article\(^{6}\), the ratios of the viscosity near the melting point to the viscosity near the boiling point are compared for members of a homologous series, for various derivatives, for example of benzene, and in general for molecules built symmetrically and unsymmetrically. The comparison was made on a large body of experimental material, and it was established that
\[ r=\frac{\eta_{\text{m.p.}}}{\eta_{\text{b.p.}}} \]
is the greater, the less symmetrical the molecule.
Thus, for molecules of the benzene type, carbon tetrachloride, cyclic compounds, etc., \(r\) does not exceed 4. For less symmetrical molecules, \(r\) is greater than four, reaching several thousands for strongly associated liquids (for isoamyl alcohol \(r = 24\,000\)).
Since \(r\) increases on passing from a more symmetrical isomer to a less symmetrical one, the author sees in the magnitude \(r\) the possibility of determining which isomer is actually in question, something which until now has usually been determined by the method of studying the dipole moment. Thus, the author asserts that the cis form of some substance must have a larger value of \(r\) than the trans form.
A definite connection between the magnitude \(r\) and the structure of molecules is beyond doubt. Equally unjustified, however, is the assumption that the magnitude \(r\) varies in proportion to the difference of the temperatures corresponding to the interval of existence of the liquid phase. A number of examples can show that \(r\) may be large for substances for which the difference \(T_{\text{m.p.}} - T_{\text{b.p.}}\) is small.
The boiling temperature may serve as the temperature at which the viscosities of different substances can be compared, i.e. it may serve as a corresponding temperature. This is evident at least from the fact that the viscosity at the boiling temperature varies, for any substances whatever (even if the most symmetrical are compared with completely unsymmetrical or associated substances), from 0.002 to 0.005, i.e. very slightly and quite independently of the structure of the substance. The question of whether the freezing temperature can serve as such a corresponding temperature, however, raises great doubts. It was already pointed out above that, in a homologous series, the freezing temperature changes differently depending on whether the transition is made from an even to an odd number of C atoms in the chain, or conversely. This makes the dependence completely unquestionable
of freezing temperatures on molecular structure, at least within a homologous series.
For the case of benzene and a series of its substituted derivatives, cited by the author in his work, the following is observed: for toluene, ethylbenzene, and o-, m-, and p-xylene, \(\eta_{\mathrm{boil}}\) is respectively equal to 0.0026, 0.0025, 0.0026, 0.0024, and 0.0024, i.e. it changes practically not at all and is entirely independent of molecular structure. The values of \(\eta_{\mathrm{fr}}\) change from the greatest value, 0.059 for ethylbenzene, to the smallest, 0.007, for symmetrical p-xylene, i.e. by approximately a factor of 8, quite parallel to the change in symmetry. Correspondingly, for ethylbenzene \(r=24\), and for p-xylene \(r=3\), i.e. the ratio of these values of \(r\) gives the same value, 8.
The zigzag curve given by the author in the work in a system of coordinates whose abscissa is the number of C atoms in a series of saturated hydrocarbons and whose ordinate is \(r\), can be obtained in exactly the same form if one plots on the ordinate axis not \(r\), but simply \(\eta_{\mathrm{fr}}\), since for all the members of this series cited by the author the viscosity at the boiling temperature is exactly the same and equals 0.0021. Obviously, the quantities \(r\) for the different homologues of this series will simply be related as \(\eta_{\mathrm{fr}}\).
An extremely important comment on M. Wolmer’s work is made by F. Miles.\(^{7}\)
It was noted above that there is no dependence whatever between \(r\) and the difference \(T_{\mathrm{fr}}-T_{\mathrm{boil}}\), which made it possible for M. Wolmer to compare \(r\) for different substances. However, on the basis of the formula \(\eta=Ae^{B/T}\), which Wolmer used to extrapolate the viscosity data to the melting and boiling points in the form
\[ \ln \eta=A' + \frac{B}{T}, \tag{22} \]
F. Miles showed that the change in \(r\) must depend on the difference
\[ \frac{1}{T_{\mathrm{fr}}}-\frac{1}{T_{\mathrm{boil}}}. \]
Indeed, substituting into equation (22) the indices “fr” and “boil” and subtracting the second equation obtained in this way from the first, we obtain
\[ \ln r=\ln \eta_{\mathrm{fr}}-\ln \eta_{\mathrm{boil}} = B\left(\frac{1}{T_{\mathrm{fr}}}-\frac{1}{T_{\mathrm{boil}}}\right). \tag{23} \]
In the series cited above of benzene and its substituted derivatives, the value \(B\) changes extremely little and has no connection with symmetry. The change in \(r\) may therefore be attributed to the change in the factor in parentheses in formula (23), which for benzene, for example, is equal to \(0.76\cdot 10^{-3}\), and for p-xylene to 1.04. In this same direction \(r\) also increases.
In the homologous series of saturated hydrocarbons, \(B\) shows a uniform increase by approximately the same amount, irrespective of whether the transition is from an even number of C atoms to an un-
…to an even or, conversely, to an odd one, whereas the difference \(\dfrac{1}{T_{\text{zam}}} - \dfrac{1}{T_{\text{kip}}}\) shows a zigzag variation, which is repeated exactly by the quantity \(r\).
Thus the quantity \(r\), by its very definition, already includes a dependence on symmetry, since both \(T_{\text{zam}}\) and \(T_{\text{kip}}\) are directly connected with the structure of the molecules; from this one may conclude that \(T_{\text{zam}}\) cannot serve as the appropriate temperature for comparing viscosities.
It is extremely difficult to give preference to one or another theory of viscosity and to the mechanism underlying it. The decisive word on this question will be obtained when this mechanism is extended to other properties of the liquid and gives, for them as well, good quantitative agreement with experiment. Until this has been done, neither the problem of the liquid state nor the problem of viscosity can be considered solved even in a first approximation.
LITERATURE
- Lindemann, Physik Z. 11, 609, 1911.
- Nannier, Stewart, Trans. Farad. Soc. Discussion on liquid crystals, 1933.
- Herzog u. Kudar, Z. Physik, 80, 217, 1933; 83, 28, 1933; Physik. Z., 11, 437, 1934.
- Andrade, Phil. Mag. 17, 497, 698, 1934.
- Goldhammer, Reports of the Academy of Sciences, No. 7, 1934.
- Waller, Phil. Mag., 18, 505, 574, 1934.
- Miles, Journ. Am. chem. Soc., 57, 698, 1935.
- Batschinski, Z. physik. Chem., 84, 643, 1913.
- Bridgman, Proc. Nat. Acad. Am. 11, 603, 1925.
- Blom, Ann. d. Phys. 42, 1397, 1913.
- Frenkel, Z. Physik, 35, 652, 1926.
- Predwoditelew, Z. Physik, 49, No. 3—4.