W. HERTLAND M. L. HAIR
4676
Hydrogen Bonding between Adsorbed Gases and Surface Hydroxyl Groups on Silica by W. Hertl and M. L. Hair Research and Development Laboratories, Corning Glass Works, Corning, New York
14880 (Received July 12, 1968)
The isosteric heats of adsorption of 23 compounds on the 3750-cmV1surface hydroxyl group on silica have been determined spectroscopically. Comparison of these heats (AH) with the observed frequency shifts (Av) of the hydroxyl band show that the compounds studied can be divided into two groups: (i) those in which AH is constant and in which A v increases with decreasing ionization potential and (ii) those in which AH is a function of Av and of the location of the lone-pair electrons in the adsorbing molecule (AH(kca1) = 0.455(A~cm-~)’/~k ) . The constant k has a value of 3.0 kcal when the lone-pair electrons involved in the hydrogen bonding are in a p orbital and a value of -2.3 koa1 when they are in sp2or spahybrid orbitals. Direct correlations are also observed between the integrated areas of the perturbed hydroxyl bands, the frequency of this band, and the amount of free hydroxyl covered. For compounds in the series in which the frequency shift is a function of the heat of adsorption, a decreasing frequency shift is observed with increasing temperature. Electrostatic interaction probably plays an important part in this hydrogen bonding, but there are other factors which contribute together or individually to the strengths of the H bonds and to the observed frequency shifts.
+
Introduction When physical adsorption takes place on the surface of silica, i t is observed that the band in the infrared spectrum attributed to the freely vibrating hydroxyl group is perturbed to lower frequencies. In many cases the interaction between the adsorbate molecule and the freely vibrating surface hydroxyl group is very specific and adsorption occurs on that site preferentially to other groups on the surface, such as adjacent, hydrogenbonded hydroxy1s.l Previous work on the physical adsorption of molecules on silica has attempted to correlate the observed frequency shift with physical parameters such as polarizability,2 quadrupole moment,a ionization potential14 and heat of adsorption.6 Of these approaches, only the latter shows any success in correlating differing groups of compounds and Kiselev has proposed that the shift in frequency of the free silanol vibration is directly proportional to the heat of adsorption on the hydroxyl groups.6 In the past, these heats have been determined calorimetrically by investigating surfaces both with and without surface silanol groupings. By spectroscopically determining the amount of coverage of the freely vibrating groups on silica, it is possible to obtain the isotherms for adsorption on these groups only, independent of any other adsorption which may take place. From these isotherms, measured at various temperatures, the isosteric heat of adsorption can be calculated. While inherently not SO accurate as calorimetric methods, this method has the distinct advantage that it defines precisely the surface site on which adsorption is taking place. This paper describes the results obtained from measuring the isoThe Journal of Physical Chemistry
steric heats of adsorption of 23 compounds on the freely vibrating hydroxyl group of silica and enables the adsorbate molecules to be separated into three distinct series based on their molecular structure. By the use of an electronic curve resolver, several overlapping bands have been isolated and more accurate data for frequency shifts have been obtained. A complete analysis is given of the perturbed peaks obtained when the following molecules are adsorbed on silica: (CH3)ZCO, CHsCHO, KH3, C6H6, and C6H14. Experimental Section A self-supporting pressed silica disk (Cabot Co., Cab-0-Sill lb0-m2/g surface area) was placed within a cylindrical furnace mounted in a Perkin-Elmer 421 spectrophotometer (the spectral slit width at 3700 cm-I was 1.6 cm-l). The furnace was connected to a conventional glass vacuum rack, and the silica disks were preheated to 800” in air in order to remove all water and hydrogen-bonded hydroxyl groups. To obtain the adsorption isotherm, various pressures of the desired reagent were admitted to the cell, and the change in peak intensity of the band due to the freely vibrating hydroxyl group was measured. The ratio of the peak intensity of this band in the presence (1) M. L. Hair, “Infrared Spectroscopy in Surface Chemistry,” Marcel Dekker, Inc., New York, N. Y . , 1967. (2) R. S. McDonald, J . Amer. Chem. SOC.,79,850 (1957). (3) G. J. C. Fronsdorff and G. L. Kington, Trans. Faraday SOC.,5 5 , 1173 (1959). (4) M. R. Basila, J . Chem. Phys., 3 5 , 1151 (1961); M. R. Basila, E. L. Saier, and L. R. Cousins, J . Amer. Chem. Soc., 87, 1665 (1965). (5) G. A. Galkin, A. V. Kiselev, and V. I. Lygin, Russ. J . Phys. Chem., 41, 20 (1967).
ADSORBED GASESAND SURFACE HYDROXYL GROUPSON SILICA
4677
Table I : Isosteric Heats of Adsorption on Silica, Observed Hydroxyl Band Frequency Shifts, and Ionization Potentials
- -Hadso
Compd
a
(isosteric), kcal/rnol
1. 2. 3. 4. 5. 6. 7. 9. 10.
SiCla CHC13 CClr CHaSiCla (CH3)2SiC12 (CH3)3SiC1
11. 12. 13. 14. 15. 16.
CH3Si(OCH& (CH3)2Si(OCH& (CzH6)zO (CH&SiOCHa NOC1 NHa
6 . 0 f0 . 1 6 . 7 -I: 0 . 3 7.9 f 0 . 2 8 . 3 f0 . 1 3.7 8 . 9 f0 . 2
17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.
C6H6N (pyridine) (CzH6hN n-Pentane CH&1 n-Heptane Cyclohexane Benzene C&I Toluene Xylene n-Hexane
10.8 f 1 . 0 11.0 f 1 . 2 5 . 7 f0 . 4
cs2
CH3CHO (CHa)zCO
5 . 4 i0 . 3 5 . 6 f0 . 3 6 . 0 f0 . 5 6.5 f1.0 7 . 8 f0 . 8 8 . 7 i0 . 4 6 . 9 f0 . 3 10.5 i1 . 0 12 f 1 , 5
... 6 . 4 i0 . 6 5.6 f1 . 0 6.0 f1.0
...
6 . 2 i 0.3
...
Ionization potential, eV
A v 9 cm-1
25 f 5 45 =k 4 45 f 3 57 90 135 f 10 60 f 5 280 f 10 395 f 15 (at 25’) 345 i 20 (at 100’) 363 f 7 403 8 460 f 20 477 i 8 178, 195 580 (at 90’) 675 (at 25’) 765 f 50 975 f 50 30 25 45 45 120 f 10 128 f 20 127 155 f 5 30
11.6 11.42 11.47
...
...
10.4
... 9.69
...
... 9.6
10.16 9.7 7.50 10.58 10.7 10.34 10 06 9.2 9.1 I
8.81 8.45-8.60 10.43
The error limits for the heats of adsorption are estimated.
of the gas to the peak intensity under vacuum gives the fraction of hydroxyl groups remaining uncovered (1 e). The quantity 0 is the fraction of hydroxyl groups covered a t any given pressure of added gas. With each reagent this procedure was repeated at three or more temperatures. A new value of the peak intensity under vacuum was obtained a t each temperature, since the peak intensity decreases slightly with increasing temperature. The spectrum of the perturbed hydroxyl band was also recorded. I n the case of reagents where the adsorption bands due to the gas phase overlapped the band due to the perturbed hydroxyl band, a compensating cell was mounted in the reference beam in order to cancel out the absorption due to the gas phase.
Results Heat of Adsorption. From the isotherms obtained at several temperatures, the heat of adsorption was calculated as follows. The pressures required to give exactly the same degree of physical coverage a t various temperatures were determined to give the adsorption isostere. A plot of log P(O = constant) vs. 1/T was then made. The slope of the resultant straight line is proportional to the isosteric heat of adsorption. For each compound, plots were made a t various coverages
and, in all cases, were found to be linear and parallel up to a coverage of about 0 = 0.5 or 0.6. Beyond 0 = 0.6 deviations occurred. These could be due either to mutual repulsion of the adsorbed molecules or to errors in the calculation, since at high coverages the isotherm is almost flat and it is difficult to obtain the precise pressure for a given e. It should be noted here that 0 refers to coverage of the surface hydroxyl group and that the enthalpy change measured is the isosteric heat of adsorption of the molecule o n the freely vibrating hydroxyl groups. The measured enthalpy changes are listed in Table I, together with the observed frequency shifts and literature values for the ionization potentials of the compounds. In obtaining these ionization potentials from the literature, no attempt has been made to distinguish which were the “best ” values. Frequency Shift and Intensity. The perturbed peaks obtained when physical adsorption takes place on silica were resolved on an electronic curve resolver (Du Pont Model 310 curve resolver). Two typical analyses are shown in Figures 1 and 2 (benzene and acetone). Both the observed band and the individually resolved bands are shown. Also shown in the figures are the gas-phase spectra taken without the silica present. It should be Volume 78, Number 13 December 1868
4678
W. HERTLAND M. L. HAIR BENZENE- GAS PHASE
I
ONLY
fi
5
PERTURBED OH PEAK WHEN BENZENE IS ADSORBED
B Figure 1. Resolution of perturbed hydroxyl band due to benzene adsorption.
ACETONE
-
GAS PHASE ONLY
e to
Figure 2. Resolution of perturbed hydroxyl band due to acetone adsorption.
noted that overtone bands occur in the gas-phase spectrum in the same frequency region where the perturbed hydroxyl group band appears. The net effect of this superimposition is to create an initial observation that the frequency shift changes with increased coverage. With the exception of the benzene adsorption, this appears to be untrue. In any accurate analysis of the perturbed bands, therefore, these gas-phase overtones (and the overtones from the gaseous molecules which have adsorbed on the silica) must be subtracted from the perturbed peak. From each spectrum the following information was obtained: (a) the shape of the peak due solely to the perturbed hydroxyl, (b) the integrated area under this peak, and (c) the position of the peak ( i e . , the frequency). Except for the aromatic compounds, all the perturbed peaks resolved were found to consist of a single, symmetrical Gaussian band. In the case of the aromatic hydrocarbons, the Gaussian band was skewed.
Discussion Band Xhape. Gas-phase infrared absorption bands are normally Lorentzian in shape. Examination of an expanded spectrum of the band at 3750 cm-’, attributed to the freely vibrating surface silanol group, shows that it can be satisfactorily accounted for by the superimposition of two symmetrical Lorentsian peaks whose maxima are separated by less than 5 cm-’. After physical adsorption and interaction between the adThe Journal of Physical Chemistry
sorbate and the hydroxyl group, the perturbed peaks are all fitted by a single Gaussian peak. This is taken as an indication that the adsorbed molecules show a distribution of energies; i.e., there is a Boltsmann distribution, suggesting that the thermal kinetic energy modulates the precise adsorption interaction energy. The perturbed peaks, with the exception of those due to aromatic hydrocarbons, are symmetrical in shape. Frequency Shifts. The adsorption studied here takes place on a freely vibrating surface hydroxyl group and is thus a form of hydrogen bonding. It is of some interest then to comment briefly on the available theories of hydrogen bonding. Pimentel and RllcLellnn6 have reviewed earlier spectroscopic studies of hydrogen bonding, most of which dealt with hydrogen bonding in solutions. It was generally noted that the 0-H stretching mode was shifted to lower frequencies, the half-width of the perturbed peak was broadened, and the integrated intensity of the perturbed peak was many times larger than that of the unperturbed peak. Correlations have been found between the observed hydroxyl frequency shift and various other parameters, such as the band half-width, the integrated intensity, Hammett’s u function, the length of the H-B bond, the heat of formation of the hydrogen bond, and others.6 These correlations only hold, however, within restricted classes of compounds. Of particular interest is the approximately linear relationship which is observed between Av and AH-a relationship that is only approximately true out to about Av = 500 cm-l, after which gross deviations are observed. In a study of adduct formation with the hydroxyl group of phenol, Purcell and Drago’ have correctly pointed out that the enthalpy change for the interaction of phenol with a Lewis base consists of two contributions. These are (i) the change in the phenol 0-H bond energy and (ii) the bond energy involved in forming the new hydrogen bond between the phenol and the base; i.e.
AH
=
+ EH-B
6Eo-~
Unfortunately, H-B vibrations are normally observed below 200 cm-’ and are not readily susceptible to observation with solids. It is now believed that hydrogen bonding is due primarily to electrostatic interaction. The electrostatic model, however, does not explain the following two important points:8 (i) the increase in the intensity of the ir absorption band, which is many fold in excess of that explainable by the electrostatic model, and (ii) the absence of a correlation between hydrogen-bond strengths and molecular dipole moments. The expla(6) G. C. Pimentel and A. L. McLellan, “The Hydrogen Bond,” W. H. Freeman and Co., San Francisco, Calif., 1960. (7) K. F. Purcell and R. S. Drago, J . Amer. Chem. Soc., 89, 2874 (1967). (8) C. A. Coulson, quoted in ref 6, p 233.
ADSORBED GASESAND SURFACE HYDROXYL GROUPSON SILICA nations offered for these behaviors are lumped together by attributing some covalent character to the H bond. I n brief, one ma,y say that there is no one theory of H bonding which explains all the experimental phenomena. Basila has studied the physical adsorption of methylbenzenes and chLoromethanes on silica. He considered the hydrogen bonding on the freely vibrating groups to be a special case of charge-transfer interaction and the observed frequency shift to be a measure of the strength of the interaction. In this case, the strength of the interaction of a series of donor molecules with a given acceptor should be related to the ionization potential of the donors. Within any one homologous series (the benzenes or the chloromethanes), he found a nonlinear correlation between the ionization potential of the donor and the frequency shift of the hydroxyl group, but no relationship was found between different series of compounds. More recently, Kiselev, et a1.,6 have studied the relationship between the observed frequency shifts of the hydroxyl groups on silica and both the ionization potentials and the heats of adsorption of the adsorbates. From their data they concluded that the ionization potential cannot serve as a measure of the effect of electron-donor properties in the specific interactions of molecules diff ering markedly in electronic structure. By calorimetrically measuring the heat of adsorption (Q) of various molecules on hydroxylated and dehydroxylated silica, they concluded that the differential heat between these two quantities was the heat of specific adsorption on the hydroxyl groups. A reasonably linear plot was obtained of AQ vs. AJJfor shifts up to 500 cm-1. The heats of adsorption and AV values obtained in this work are given in Table I. These show that, within a given homologous series, there is a monotonic rise in the frequency shift with a rise in the heat of adsorption (compounds 1-6, 9-10, 11-14, and 15-18). With the hydrocarbons and methyl halides (compounds 19-26), no such relationship holds. The heats of adsorption are all the same within experimental error, although the frequency shifts do vary over a wide range. Thus, in this series, there is a contribution to the frequency shift which does not show up in the heat of adsorption. Comparing the values of the ionization potentials of the nonhydrocarbon compounds with the frequency shifts or heats of adsorption shows that there is no correlation between these parameters. However, for the hydrocarbons, where the heat of adsorption is approximately constant, there is a correlation between the frequency shift and the ionization potential. This is shown in Figure 3. The adsorbates used in this study can be immediately separated into two classes of compounds: (i) those in which the frequency shift increases with increasing heat of adsorption and (ii) those in which the heat of adsorp-
4679
I 86
I 9.0
I
I
96
m.0
I 10.5
1f.V)
Figure 3. Comparison of frequency shifts and ionization potentials for the series of hydrocarbon adsorbates on silica.
0
I lo
m
I
I
20
30
Figure 4. Isosteric heats of adsorption of compounds adsorbed on silica as a function of ( A V ) ’ ’ ~ .
tion is approximately constant and thus independent of the frequency shift. In Figure 4,the heats of adsorption of compounds 118 are plotted as a function of ( A v ) ” ~ . The plot gives two good linear relationships (upper-curve compounds 1-10 and lower-curve compounds 11-18). Since the adsorbing sites (the OH groups) are the same for all the measurements, it seems likely that this difference can be traced to electronic differences in the adsorbing molecules. One striking difference between the molecules in the upper and lower curves is apparent. In the upper set, the lone-pair electrons are all located in p orbitals, whereas in the lower set, the lone-pair electrons are all located in hybrid orbitals (sp2 or spa). Thus, for a given heat of adsorption, a low shift is observed when p-orbital electrons are available for H bonding and a large shift is observed when hybrid orbital electrons are used for H bonding. “The most noticeable distinction between a hybrid orbital and its component s, p, , . . orbitals is that the hybrid no longer possesses central symmetry. . .the Volume 7.9, Number 13 December 1068
W. HERTLAND M. L. HAIR
4680 centre of mean position of a hybrid may be a t some distance from the In hybrid orbitals, then, the atomic dipole can be quite large and, in the case of carbon, can contribute as much as 2.2 D. As suggested above, the theory of hydrogen bonding is incomplete. However, if we assume the hydrogen bond is principally due io electrostatic interaction, then the process taking place is as follows. As the lone-pair electron orbital of the adsorbing molecule approaches the surface hydroxyl group, the electrostatic interaction distorts the hydrogen 1s orbital. This causes the 0-H stretching vibration frequency to be lowered. The amount of electrostatic interaction will determine the magnitude of the frequency shift and will be greater in the case of a hybrid orbital than in the case of a p orbital. The over-all heat of adsorption which is measured will be determined by the amount of this electrostatic interaction plus contributions due to other interactions with the hydroxyl groups. The factor causing the observed frequency shift is thus only one of several effects which contribute to the over-all heat of adsorption. This must be so, since the magnitude of the shift in the case of the hydrocarbons is independent of the over-all heat of adsorption. On the basis of these results, it appears that adsorbates on silica can be divided into two groups : (i) those in which the heat of adsorption is constant (6.0 zt 0.4 kcal/mol) and in which the frequency shift can be given in terms of the ionization potential Av(cm-’)
=
10.9 - I(eV) 0.015
=
0.455(Av~m-’)”~4-(constant)
0
5 AREA OF
(2)
ao’c
IO
15
PERTURBED OH
20
25
PEAK
Figure 5. (Top) effectof coverage (e) on the frequency of the perturbed hydroxyl group during hexane adsorption; (bottom) effect of coverage on the area of the perturbed hydroxyl peak.
I
I
m AREA
When the electrons responsible for the H bonding are located in p orbitals, the constant is equal to 3.0; when the electrons are located in sp2 or sp3 hybrid orbitals, the constant is equal to -2.3. For compounds in the first series the dependence of the shift on ionization potential indicates a chargetransfer interaction, as proposed by B a ~ i l a . However, ~ in the second series, the lack of this dependence makes a charge-transfer mechanism unlikely, and, in this case, the shift must be attributed mainly to electrostatic interaction. However, since there is a nonzero intercept, there must be other contributions to Av. The choice of a relationship based on (A,)’’’ is difficult to justify on theoretical grounds. It should be pointed out, however, that the linear relationships which hnve been predicted and then observed, in solution chemistry, are useful only when Av < 500 cm-l. Above this value, gross deviations occur which cannot The Journal of Physical Chemistry
Q
A S0.C
0
and (ii) those in which the heat of adsorption is a function of the frequency shift and of the location of the lone-pair electrons responsible for the H bonding. These heats are given by the equation Ahir(ltca1)
‘eHI4
OF
PERTURBED
100 OH
PEAK
Figure 6. (Top) effect of coverage (e) on the frequency of the perturbed hydroxyl group during acetaldehyde adsorption; (bottom) effect of coverage on the area of the perturbed hydroxyl peak.
be understood in terms of our present knowledge of hydrogen bonding and which are usually ignored. Intensity, Frequency, and Temperature. The results of measurements made on the perturbed OH bands following adsorption of hexane, acetaldehyde, ammonia, and benzene are shown in Figures 5-8. The lower part of each figure shows the area of the perturbed peak as a function of the fractih of freely vibrating hydroxyl groups consumed when adsorption takes place. Except for benzene the plots are linear and pass through the origin. The upper plots of Figures 5, 6, and 8 show the frequency of the perturbed peak as various fractions of the free hydroxyl group are consumed. With the possible exception of benzene, the position of the perturbed peak does not change with coverage at any given temperature. However, in the case of the nonhydrocarbon compounds, there is a (9) C. A. Coulson, “Valence,” Oxford University Press, London, 1968, p 207.
ADSORBED GASESAND SURFACE HYDROXYL GROUPSON SILICA
4681 IT
0
c z
J
I BO
BO
I
100
TEMPERATURE 1.C)
3.018
/
G 0
eo-
I z
E 8 0
g
0.4 o'6/
AREA
OF
PERTURBEO
OH
I
'gH6
AREA
;O OF
;O PERTURBED
&%-"."
PEAK
Figure 7. (Top) ammonia adsorption position of the perturbed hydroxyl peak of various temperatures; (bottom) effect of coverage on the area of the perturbed hydroxyl peak.
10
lo-
I8 0
40 MI
50 PEAK
60
Figure 8. (Top) effect of coverage (e) on the frequency of the perturbed hydroxyl group during benzene adsorption ; (bottom) effect of coverage on the area of the perturbed hydroxyl peak.
change in the frequency shift a t different temperatures. This is best illustrated in Figure 7 (top) for ammonia, where the frequency of the perturbed OH band is plotted for various temperatures ; the frequency increases by about 100 cm-l over the range from 30 to 90". In the case of liydrocarbon compounds no significant shift was noted with change in temperature. With the exception of the aromatic hydrocarbons, the perturbed bandis were all symmetrical and Gaussian. The measured areas (normalized for the amount of free OH consumed) are plotted in Figure 9 against the observed frequency shift. It is secn that the intensities increase linearly with increasing wave number shift out to about 700 cm-l, beyond which deviations from the straight line occur. The linear relationship between Av and the integrated absorption intensity has been observed in studies of hydrogen bonding in solution. The existence of this linear relationship in this work demonstrates a confidence in the use of the electronic
curve resolver in analyzing the data obtained. Measurements made before resolving the spectra gave a random scatter instead of the expected relationship. It should be noted that this simple linear relationship was not observed with benzene, which throughout these experiments gave anomalous results. With increasing temperature, acetone, acetaldehyde, and ammonia all cause a decreasing frequency shift for the perturbed hydroxyl band. These compounds fall in the group in which the heat of adsorption can be described in terms of the observed frequency shift. Thus the heat of physical adsorption falls off with increasing temperature. This effect is very often noted in physical adsorption. The magnitude of the change in the observed shift and the change in heat to which this corresponds can be calculated from eq 2 , which was determined from the average AH-Av data of a large number of compounds. These results are given in Table 11. By using the observed frequency shift at various temperatures one has a very sensitive estimate of this change in heat of adsorption. I n the case of the compounds in which the frequency shift is a function of the ionization potential and independent of the heat of adsorption (benzene and hexane), the shift is independent of temperature in the range reported. Table I1
Compd
Change in shift, cm-1 (temp range, "C)
Change in heat, koa1
Acetone Acetaldehyde Ammonia
50 (25-100) 15 (30-90) 100 (25-90)
0.6 0.2 0.9
Except for benzene, the observed shifts are independent of the degree of physical coverage, showing that over the range of coverages studied there is not appreciable lateral interaction. Benzene shows anomalous behavior compared with Volume 7.9, Number 1.9 December 1968
4682
NOTES
the other compounds. Thus the linear relationship between the amount of OH consumed and the area of the perturbed peak is not obeyed, and the frequency of the perturbed OH peak decreases with increasing surface c ~ v e r a g e . ~This increase in frequency shift with increasing coverage is somewhat strange. Since benzene belongs to the series of compounds in which the shifts are independent of the heat of adsorption, it seems unreasonable to attribute this t o changes in the heat of adsorption due to coverage. Moreover, the shift is in the opposite direction. If, however, it can be attributed to a change in the ionization potential, this would indicate that the ionization potential of adsorbed benzene decreases by about 0.4 eV as its coverage of the surface hydroxyl group increases.
Acknowledgment. The authors wish to acknowledge the able assistance of Miss Ethel Herritt in the experimental work.
Addendum An anonymous reviewer has suggested that ketone adducts of Lewis acids have C-0-X bond angles near
120" (where X is the first atom of the Lewis acid). In view of this, he suggests that the lone pairs in C=O groups are sp2hybrids rather than pure p states. This view can be reconciled with the two lines in Figure 4 by assuming that in compounds 7-10 it is the double bond that is involved in hydrogen bonding rather than the lone-pair electrons on oxygen or sulfur. The referee suggests that the work of Fritzsche'o on phenol bonding to ketones in CCl, solution supports this supposition. This comment has obvious validity. However, the authors would like to emphasize the incomplete theoretical state of hydrogen bonding and the difficulties of explaining hydrogen-bonded interactions in terms of Lewis acid adducts where strong orbital overlap occurs. Both Coulsonll and Ballhausen and Gray12in their discussions of carbonyl groups and formaldehyde, respectively, refer to the existence of pure p states on the carbonyl oxygen atom. (10) H. Fritzsche, -4eta Chim. Acad. Sci. Hung.,40, 31 (1964). (11) Reference 8, p 185. (12) C. J. Ballhausen and H. B. Gray,"Molecular Orbital Theory," W. A. Benjamin, Inc., New York, N. Y., 1964, p 85.
NOTES
The volume fraction may then be expressed by (b = cVh, where c is the concentration (fM)and Vh is the molar volume of the solute including attached solvent ( M - l ) . Thus eq 1can be modified to
A Proposed Viscosity-Concentration Equation beyond Einstein's Region
by S. P. Moulik
17/70 = 1
Department of Physical Chemistry, JadavpuT University, Calcutta, India (Receised M a y 22, 196'8)
Rigid particles suspended in a continuous medium will interfere with the stream lines of flow pattern and the suspension should have a higher viscosity than the medium, From hydrodynamic considerations Einstein' was the first to treat this problem in the case of rigid spheres suspended in a continuum. His results for dilute suspensions may be expressed by 11/70 =
1
+ 2.w
(1)
where 7 is the absolute viscosity of the suspension, that of the medium, and (b the volume fraction occupied by the particles. In the above equation the volume fraction # must be taken to include the volume of any solvent immobilized on the surface of the solute particle. The Journal of Physical Chemistry
+ 2.5cVh
=
1
+ KC
(2)
where 2.5Vh = K = constant. This viscosity relation of Einstein is restricted to the dilute domain of solute concentration and consequently neglects the solvent-solute interaction. Jones and Dole2 formulated an empirical equation which can account for both solute-solute electrostatic interaction and solventsolute interaction. According t o RIerker and Scott3 it is the difference between the microscopic and macroscopic viscosities, depending on the solvent-solute interaction, that determines the various values of Einstein's constant for different solute-solvent systems. To deal with viscosity of concentrated solutions, Suryanarayana and Venkatesan4empirically formulated (1) A. Einstein, Ann. Physik, 19, 289 (1906); 34, 591 (1911). (2) G. Jones and M. Dole, J . Amer. Chem. Soc., 51, 2950 (1929). (3) R. L. Merker and M. J. Scott, J . Colloid Sci., 19, 245 (1964).
