PHYSICAL CHEMISTRY

G. M. PANCHENKOV

ON THE DIFFUSION COEFFICIENT IN LIQUIDS

(Presented by Academician A. V. Topchiev on 8 VII 1957)

In our works on the theory of the viscosity of liquids \((^1)\), we proceeded from the idea that, when “coupled” molecules are moved away from one another to a distance greater than the equilibrium distances \(r_0\), corresponding to the minimum of the potential energy, attractive forces act between the molecules. If, in this case, the total kinetic energy of two molecules belonging to two layers of a liquid proves to be less than the binding energy, then, under certain additional conditions, a bond may arise between these molecules, and for a certain time (not more than half the period of oscillation) such a system begins to behave as one “kinetic unit.”

If, however, the total kinetic energy of the pair of molecules under consideration, when they move in opposite directions, is greater than the binding energy, then no bond can arise between the molecules, and the molecules will move independently of one another. Under a certain condition, indicated below, these molecules have the possibility of passing from one layer of the liquid into the nearest one, i.e., diffusion will be observed. Thus, the average kinetic energy per molecule capable of diffusion will be

\[ \varepsilon_{\mathrm{d}} > \frac{1}{2}\varepsilon_0, \tag{1} \]

where \(\varepsilon_0\) is the energy of one bond of liquid molecules.

As was shown in our works \((^1)\),

\[ \varepsilon_0 = 2\lambda_{i0}/\gamma, \tag{2} \]

where \(\lambda_{i0}\) is the internal latent heat of evaporation of the liquid in vacuum at absolute zero; \(\gamma\) is the coordination number of the liquid.

Since the average distance between liquid molecules is smaller than their linear dimensions, one more condition is necessary for a molecule to pass from one layer to another: it is necessary that two nearby molecules of the layer into which the given molecule passes be at a distance permitting the penetration of the new molecule, i.e., at a distance greater than the equilibrium one.

Let us denote the probability of such a favorable arrangement of molecules by \(W\). From the considerations discussed, it follows that the process of diffusion in liquids differs from the process of diffusion in gases in that, in gases at low pressures, all molecules take part in the diffusion process, whereas in liquids only those do whose energy is greater than \(\varepsilon_{\mathrm{d}}\) and opposite which a free place is formed. For the process of “self-diffusion” in gases, kinetic theory leads to the following

equation for the diffusion coefficient:

\[ D=\frac{1}{8}\pi\lambda\bar{c}, \tag{3} \]

where \(\lambda\) is the distance over which mass transfer occurs in one elementary act, and \(\bar{c}\) is the mean velocity of the molecules.

For liquids, in accordance with what was said above, one must write

\[ D=2Wr_0\bar{c}, \tag{4} \]

where \(r_0\) is half the equilibrium distance between the centers of the molecules, which, for simplicity, we shall regard as spherical. Since real molecules are not spherical, in the indicated method of calculation we replace them by spheres with an effective radius, which behave in the same way as the real molecules. The coefficient \(\pi/8\) has been omitted, because the motion of the molecules must be considered only in two directions (toward the next layer and away from it). The quantity \(\lambda\) in liquids cannot exceed the mean distance between the centers of the molecules. Transitions over a greater distance have a very small probability. The quantity \(r_0\) is readily determined as

\[ r_0=\sqrt[3]{\frac{3v_m}{4\pi N_0}}, \tag{5} \]

where \(v_m\) is the volume of a gram-molecule of the liquid, and \(N_0\) is Avogadro’s number. The mean velocity is

\[ \bar{c}=\int_{c_{\mathrm{d}}}^{\infty} c\,\frac{dN}{N}, \tag{6} \]

where \(c_{\mathrm{d}}\) is the velocity of molecules whose kinetic energy is equal to \(\varepsilon_{\mathrm{d}}\). Substituting into expression (6) the value of \(dN/N\) from Maxwell’s velocity distribution law, integrating, and passing from velocities to energies, we obtain

\[ \bar{c}=\frac{2}{\sqrt{\pi}}\sqrt{\frac{2RT}{M}}\,e^{-\varepsilon_0/2RT} \left[\frac{\varepsilon_0}{2RT}+1\right]. \tag{7} \]

Since the process of molecules moving away from one another, and thereby freeing space for the diffusing molecule, is analogous to the process of evaporation, the quantity \(W\) may be written in the form

\[ W=e^{-2\lambda_i T/\gamma RT}, \tag{8} \]

where \(\lambda_iT\) is the internal latent molecular heat of evaporation at the given temperature.*

Substituting (5), (7), and (8) into (4), we obtain the final equation

\[ D=4\,\frac{\sqrt[3]{3}\sqrt{2R}}{\sqrt[3]{4\pi N_0}\sqrt{\pi}}\, \frac{v_m^{1/3}T^{1/2}}{M^{1/2}}\, e^{-2\lambda_i T/\gamma RT}\, e^{-\varepsilon_0/2RT} \left[\frac{\varepsilon_0}{2RT}+1\right]. \tag{9} \]

Since expression (4) is written for the process of “self-diffusion,” all quantities characterizing the properties of the molecules (\(v_m\), \(M\), \(\lambda_iT\), \(\varepsilon_0\)) must be taken as averages for the substances composing such a system.

* Obviously, this expression is valid only under the condition that the attractive forces decrease sharply with distance, i.e., when they can be neglected at distances between molecules equal to their diameter. Otherwise the exponent will include the quantity \(\lambda<\lambda_iT\).

Expression (9) gives the dependence of the diffusion coefficient on temperature, concentration, and pressure, since the quantities \(v_m\), \(\lambda_i T\) are functions of temperature, concentration, and pressure, while \(\varepsilon_0\) is a function of concentration and pressure.

Under the condition that only neighboring molecules interact and that their relative concentrations around any one of them are identical, for a solution consisting of two components \((^2)\):

\[ \varepsilon_0=\varepsilon_{011}c_1^2+2\varepsilon_{012}c_1c_2+\varepsilon_{022}c_2^2, \tag{10} \]

where \(\varepsilon_{0ij}\) is the binding energy between molecules of the \(i\)-th and \(j\)-th kinds, and \(c_1\) and \(c_2\) are the mole fractions of the two components in the solution. Depending on the relation among the quantities \(\varepsilon_{011}\), \(\varepsilon_{022}\), and \(\varepsilon_{012}\), the diffusion coefficient, with changing concentration, will vary either linearly or pass through a maximum, a minimum, or a point of inflection. The quantity \(\varepsilon\) can readily be found from the temperature dependence of the density and viscosity of liquids \((^1)\).

The quantity \(\lambda_iT\) for a given system of two liquids must be calculated from an analogous equation.

How well formula (9) agrees with experiment can be seen from Table 1.

The diffusion coefficients of \(o\)-xylene in \(n\)-xylene were measured in our laboratory by the diffraction micromethod by T. S. Makareva and V. V. Erchenkov. The values of the binding energies \(\varepsilon_0\) needed for the calculations, found from the temperature dependence of the density and viscosity of liquids \((^6)\), proved to be as follows:

For calculating the diffusion coefficients it is necessary to know the coordination number \(\gamma\) for the given system. This quantity was calculated from equation (9), using the diffusion coefficient for a temperature of 15 or 20°.

As is seen from Table 1, the calculated diffusion coefficients for the systems \(o\)-xylene — \(n\)-xylene and benzene — monodeuterobenzene, with increasing temperature, regularly all exceed the experimentally found values. This is explained, on the one hand, by inaccuracies in determining the latent heats of evaporation at different temperatu-

Table 1

Calculated and experimental values of diffusion coefficients and coordination numbers of liquids

pores, and on the other hand, by the fact that as the temperature rises the coordination numbers of liquids decrease. A small change in the coordination number leads to a significant change in the calculated diffusion coefficient. The values of the coordination numbers calculated from formula (9) using experimentally found values of the diffusion coefficients are given in Table 1.

As is seen from the data presented above, the coordination numbers for the systems o-xylol—n-xylol and C₆H₆—C₆H₅D decrease only slightly with temperature; however, if for calculations of the diffusion coefficients one uses a coordination number found for some one temperature, this leads to discrepancies between the calculated diffusion coefficients and the experimental data. In the case of the system C₂H₅OH—C₂H₅OD the coordination number practically does not change with temperature, and if any of its values in the given temperature interval is used, the calculated values of the diffusion coefficients are in good agreement with the experimental values.

Calculations of the same type were carried out by us for a number of other systems and likewise showed good agreement between theory and experiment.

The quantity

\[ -\,2\lambda_i\tau/\gamma T=\Delta S, \tag{11} \]

where \(\Delta S\) is the change in entropy upon bond formation, calculated per 1 mole. The temperature dependence of this quantity is determined by the expression

\[ \frac{\partial \Delta S}{\partial T} = \frac{2}{\gamma}\, \frac{\Delta C_{\mathrm{равн}}}{T}, \tag{12} \]

where \(\Delta C_{\mathrm{равн}}\) is the difference between the heat capacities of the liquid and the vapor that are in equilibrium. If, approximately, it is assumed that the quantity \(\Delta C_{\mathrm{равн}}\) does not depend on temperature, then

\[ \Delta S=\alpha \ln T+\beta. \tag{13} \]

where \(\alpha\) and \(\beta\) are constants.

Substituting (11), with (14) taken into account, into (9), we obtain

\[ D = 4\, \frac{\sqrt[3]{3}\sqrt{2R}} {\sqrt[3]{4\pi N_0}\sqrt{\pi}}\, \frac{v_m^{1/3}T^{1/2}}{M^{1/2}}\, e^{\beta l/R} \left[ \frac{\varepsilon_0}{2RT}+1 \right] e^{(\alpha T\ln T-\varepsilon_0)/RT}. \tag{14} \]

According to calculations, for many liquids

\[ v_m^{1/3}T^{1/2} \left[ \frac{\varepsilon_0}{2RT}+1 \right] \simeq \mathrm{const}. \tag{15} \]

Therefore expression (15) may be written in the following form

\[ D=Ae^{(\alpha T\ln T-\varepsilon_0)/RT}, \tag{16} \]

where

\[ A= \frac{ 4\sqrt[3]{3}\sqrt{2R}\,v_m^{1/3}T^{1/2} }{ \sqrt[3]{4\pi N_0}\sqrt{\pi} } \left[ \frac{\varepsilon_0}{2RT}+1 \right] e^{\beta l/R}. \]

Equation (16), as calculations show, agrees well with experimental data.

If expression (16) is logarithmized, we obtain

\[ \ln D=\ln A+\frac{\alpha \ln T}{R}-\frac{\varepsilon_0}{RT}, \tag{17} \]

from which it is evident that over a broad temperature interval \(\ln D\) is not linear with respect to the variable \(1/T\). In narrow temperature intervals it may be consid-

state that $\ln T = a - b/T$, where $a$ and $b$ are constant coefficients, and, consequently, the dependence of $\ln D$ on $1/T$ in narrow temperature intervals will be linear.

Moscow State University
named after M. V. Lomonosov

Received
25 VI 1957

REFERENCES

1. G. M. Panchenkov, ZhFKh, 24, 1390 (1950); DAN, 50, 329 (1945); 51, 361, 453 (1946); ZhFKh, 20, 811, 1011 (1946); 21, 187 (1947); Theory of the Viscosity of Liquids, 1947; DAN, 63, 701 (1948).
2. G. M. Panchenkov, DAN, 51, 453 (1946); ZhFKh, 20, 1011 (1946); Theory of the Viscosity of Liquids, 1947.
3. K. Graupner, E. R. S. Winer, J. Chem. Soc., 1952, 1145.
4. Physicochemical Properties of Individual Hydrocarbons, vol. 5, Moscow, 1954.
5. Technical Encyclopedia. Handbook of Physical, Chemical, and Technical Quantities, 7, Moscow, 1937.
6. G. M. Panchenkov, ZhFKh, 24, 1390 (1950).