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Comments Applicability of the Kelvin Equation to Vapor/Liquid Systems in Porous Media Introduction Vapor adsorption/desorption data on porous solids, if obtained from an experiment carried out in the relative pressure range above about 0.5, are often interpreted with the aid of an expression involving the curvature of the vapor/liquid interfaces. This expression, first derived in approximate form in 1870, is known as the Kelvin equation.' An exact form of the equation can also be derived by introducing the Gibbsian concept of equal chemical potentials in the vapor and liquid phases.2 This leads to very small corrections, at least at high relative pressures. As is well-known, the approximate form of the equation has been widely and successfully applied to the study of pore size distribution^.^,^ For example, this approach has proven to be useful in two recent studies of the desorption of water vapor from samples of rock formations which function as reservoirs for petroleum and natural gas.6v6 On the other hand, it should be acknowledged that vapor adsorption/desorption techniques are not yet widely used in the study of porous solids of geological interest. This may in part be due to persistent doubts, particularly outside the field of surface chemistry, as to the applicability of the Kelvin equation. These doubts in turn appear to be supported by experimental studies carried out and reported by Shereshevsky and co-workers over a number of years.'-" References to Shereshevsky's work are often found in the older literature, and the work is mentioned both in recent reviews12J3and in various treatises.'*-17 These citations in the literature, however, usually do not emphasize the fact that the work has clearly and unequivocally been refuted by the experiments of other workers.lSm (1) Thomson, W. (Lord Kelvin) Proc. R. Soc. Edinburgh 1870, 7,63; Philos. Mag. 1871, Ser. 4, 42,448. (2) Melrose, J. C. AIChE J. 1966, 12, 986; J . Colloid Interface Sci. 1972, 38,312. (3) Emmett, P. H.Chem. Rev. 1948, 43, 69. (4) Sing, K. S.W.; Everett, D. H.; Haul,R. A. W.; Moscou, L.; Pierotti, R. A.; Rouquerol, J.; Siemieniewska, T. Pure Appl. Chem. 1985,57,603. ( 5 ) Ward, J. S.;Morrow, N. R. SPE Form. E d . 1987,2, 345. (6).Melrose, J. C. In Proceedings, IUPAC Symposium on the Characterization of Porous Solids; Unger, K., Behrens, D., Eds.; Elsevier: Amsterdam, 1988. (7) Shereshevsky, J. L. J. A m . Chem. Soc. 1928, 50, 2966. (8)Shereshevsky, J. L. J. Am. Chem. SOC.1928, 50, 2980. (9) Shereshevsky,J. L.; Carter, C. P. J.Am. Chem. SOC.1950,72,3682. (10) Folman, M.; Shereshevsky, J. L. J. Phys. Chem. 1955,59, 607. (11) Coleburn, N.L.; Shereshevsky,J. L. J. Colloid Interface Sci. 1972, 38,84. (12) Skinner, L.M.; Sambles, J. R. Aerosol. Sci. 1972, 3, 199. (13) Fisher, L. R.;Israelachvili, J. N . J. Colloid Interface Sci. 1981, 80,528. (14) Rideal, E. K.A n Introduction t o Surface Chemistry, 2nd ed.; Cambridge University Press: Cambridge, 1930; Chapter I, p 43. (15) Bikerman, J. J. Surface Chemistry for Industrial Research; Academic: New York, 1947; Chapter I, p 58. (16) Partington, J. R.An Advanced Treatise on Physical Chemistry; Longmans, Green: London, 1950; Vol. 2, p 367. (17) Adamson, A. W. Physical Chemistry of Surfaces, 3rd ed.; Wiley: New York, 1976; Chapter 11, p 52, Chapter VI, p 311. (18)Woodland, D.J.; Mack, E., Jr. J . Am. Chem. SOC.1933,55, 3149.
These citations also do not point out what appears to be the most probable flaw in Shereshevsky's experiments. The purpose of this communication is, first, to present comments on the actual experimentalevidence upon which Shereshevsky based his conclusions. Comments on the historical background relating to the Shereshesky work will also be presented. The experimental work of others, which refutes Shereshevsky's work, will then be reviewed. Finally, previous comments on the experimentalwork of both Shereshevsky and his critics will be briefly examined. Shereshevsky's Early Experiments In discussing the actual experiments reported by Shereshevsky, we may first of all note that the original work of 1928'~~ involved cylindrical capillaries having a very limited size range. Radii varied from 1to 4 pm (soft glass) and from 1 to 6 pm (fused quartz). The liquids used in the experiments were water (soft glass and fused quartz) and toluene (fused quartz only). In the case of the water/quartz system' the relative water vapor pressure in a given capillary was not determined in a direct manner by equilibration, through the vapor space, with an electrolyte solution of known vapor pressure. Instead, measurements were made of the varying rate at which water evaporated from the capillary and presumably condensed in a reservoir containing a KCl solution of varying concentration. Relative vapor pressures of these KC1 solutions were determined indirectly, at various times, from concentration measurements. These relative vapor pressures varied from 0.9861 to 0.9938 for an experiment using a fused quartz capillary of 2.07-pm radius (Table VI1 of ref 7). The relative vapor pressure within the capillary was then found by plotting the observed evaporation rates at various times against the corresponding vapor pressures of the KC1 solution. Such a plot was then linearly extrapolated to zero rate, and the relative vapor pressure of the KC1 solution at zero rate was taken as the equilibrium relative vapor pressure within the capillary. Specific details of the evaporation rate measurements were not disclosed. Neither was any justification given for the linear extrapolation of the evaporation rate data. For the fused quartz capillary of 2.07-pm radius, the equilibrium relative vapor pressure determined by this means was found to be 0.9956. The corresponding value calculated from the Kelvin equation for this value of the meniscus radius of curvature was 0.9995. This discrepancy was taken as experimental proof that the water surface tension was enhanced by a factor of 8.5. The work with soft glass capillaries was even less direct but was said to indicate surface tension enhancement factors as high as 23. As noted by Shere~hevsky,~ the experiments using soft glass capillaries may also have been flawed by the dissolution of glass from the walls of the capillaries. The results for toluene8 were limited to qualitative observations of whether condensation or evaporation occurred in a series of capillaries of varying radius. Nevertheless, it was asserted with some confidence (19) Cohan, L. H.;Meyer, G. E. J. Am. Chem. SOC.1940, 62, 2715. 1963,59, 846. (20) Cross, N. L.; Picknett, R. G. Trans Faraday SOC.
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that the “classical theory as applied to vapor pressure in capillaries is not correct.’’ It should be pointed out that Shereshevsky7i8did not initially choose to interpret his data on the basis of a reduction in the effective meniscus radius by a wetting film,as is now common practice? If such an interpretation is attempted,21it is found that Shereshevsky’sdata for the 2.07-pm-radius capillary imply a water film thickness of 1.83 pm, or about 6.6 X lo3 molecular layers.
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predict films a t least lo2 and possibly lo3 times thinner than those inferred2’ from the Shereshevsky experiments.
Shereshevsky’s Later Experiments Some years after his original work, Shereshevsky, first with Carterg in 1950 and then with Folmanlo in 1955 and with Coleburn” in 1972, published new experimental work which was said to corroborate the 1928 work. For the water vapor/liquid systemg two fused quartz and four pyrex capillaries were used. For the toluene and isopropyl Historical Background alcohol vapor/liquid systems,1° eight different pyrex capillaries were used. In these experiments the capillary tubes At the time of Shereshevsky’s early work (1928), a were actually conical in shape. The radii in the vicinity number of authorities were convinced that solid surfaces of the vapor/liquid meniscus ranged from 3.2 to 9.3 pm could exert, through a process of multiple polarizations, (water), from 0.85 to 1.7 pm (toluene), and from 0.95 to or “relayed forces”,an attraction persisting over a distance 2.6 pm (isopropyl alcohol). of some 103-104 molecular layers. For example, Hardy22i23 In the case of the water vapor/liquid systemga reservoir termed this process “diachysis” and referred to the macontaining a KC1 solution of known concentration was croscopic force which resulted from this process as the again used. Evaporation of water from a given capillary “Leslie pressure”. This designation obviously was intended was allowed to continue until an apparent equilibrium was to recognize the early nineteenth century speculations of reached. This seems to have taken only some 2-3 h (Table L e ~ l i e regarding ~ ~ p ~ ~ the nature of wetting films. It is of I, ref 9). No attempt was made to check what was assumed interest that this terminology has now been superseded to be the equilibrium meniscus position by approaching by the more familiar term, disjoining pressure, introduced this position from the condensation side. by Deryagin and Obukhov26in 1936. An extensive review An important feature of the study was the use of a of experimental work purporting to confirm the relayed number of KC1 solutions of varying concentration in the force concept, with particular emphasis on the contribureservoir. Ten different concentrations ranging from 0.025 tions of M ~ B a i n , was ~ ~ ppublished ~~ by Henniker21 as late to 0.80 m were used. Thus, the relative pressure range as 1949. from 0.9991 to 0.9745 was covered in a systematic way. In contrast to the position taken by the Hardy-McBain school in the 1920s, other workers, following L a n g m ~ i r , ~ ~ Over this entire range of relative pressures, good agreement between the observed meniscus radii in the fused quartz believed that surface forces were of comparatively short and pyrex capillaries was found. Although these observed range and that wetting films could at most be a few moradii differed markedly from those predicted by the Kelvin lecular diameters in thickness. A resolution of the conflict equation, they did vary systematically with the KC1 conbetween the two extreme points of view was then achieved, centration in the reservoir. at least in part, about a decade after Shereshevsky’s 1928 Another significant feature of the observations should work. This resolution occurred as the result of extremely also be noted. It was statedg that capillaries similar to precise measurements indicating an anomaly in the relative those used in the equilibration experiments, when partially surface tension versus concentration relationship for very filled with water under atmospheric conditions, “showed dilute electrolyte solutions. no noticeable deformation of the menisci in the direction These measurements, due to Jones and Ray,30and the of high curvature.” It was also statedgthat “The radii of associated anomaly, called the Jones-Ray effect, were inthe menisci appeared normal and equal to those of the terpreted by L a n g m ~ i r ~inl ,1938. ~ ~ This interpretation capillaries a t these levels.” These observations, if valid, was based on the now well-known concept that overlapping would seem to rule out the possibility of wetting films some electrical double layers, arising at similarly charged in1-2 pm in thickness existing in capillaries which were only terfaces, produce a force tending to thicken the film be3-10 pm in radius. tween such interfaces. Specifically, it was shown by Notwithstanding these latter observations, the results Langmuir that an aqueous film with a thickness of about presented by Shereshevskyand Carter were taken by imply 50 nm would exist above the water vapor/liquid meniscus that the vapor pressure lowering in capillaries 3-10 pm in in a silica capillary tube of 136-pm radius, as used by Jones radius is from 7 to 80 times that predicted by the Kelvin and Ray. The dependence of this thickness on the elece q ~ a t i o n .No ~ concern was expressed that equilibrium trolyte concentration corresponds closely to the observed between water in the capillaries and water in the reservoir anomaly. Applying Langmuir’s theory to capillaries of KC1 solution may not have been established. much smaller radius, as used by Shereshevsky, would obIf the Shereshevsky and Carter data are used to infer viously yield considerably smaller values of the film values for the thickness of the wetting films in these exthickness. Thus, electrical double-layer theory would periments, it is found that these values range from 3 to 8 pm, even larger than those inferred from the earlier (21)Henniker, J. C. Reu. Mod. Phys. 1949,21,322. experiments. As would be expected, these apparent (22) Hardy, W.B. J. Chem. SOC.1925,1207. thicknesses also vary systematically with the electrolyte (23)Hardy, W.B. Philos. Trans. R. SOC.1931,A230, 1. (24) Leslie, J. Philos. Mag. 1802,14, 193. concentration of the reservoir. This variation is the same (25)Leslie, J. Elements of Natural Philosophy; Oliver and Boyd: for both the fused quartz and pyrex capillaries. Edinburgh, 1829; Chapter 111, p 353. It is of course conceivable that the cause of the very (26) Derjaguin, B. V.;Obukhov, E. V.Acta Physicochim. URSS 1936, 5. significant disagreement with the Kelvin equation pre- , 1. (27)McBain, J. W.The Sorption of Gases and Vapors by Solids; dictions is contamination of the presumably pure water Routledge: London, 1932;Chapter XVI. in the capillaries. However, calculations show that the (28)McBain, J. W. Colloid Science; D.C. Heath Boston, 1950; concentration of the contaminating component would have Chapter 4. (29)Langmuir, I. Phys. Reu. 1915,6, 79. to be very nearly that of the particular electrolyte solution (30)Jones, G.;Ray, W. A. J. Am. Chem. SOC.1937,59,187. used in the reservoir. It seems highly improbable that (31)Langmuir, I. Science (Washington D.C.)1938,88, 430. contamination caused by dissolution from the capillary (32)Langmuir, I. J. Chem. Phys. 1938,6,873. ~
292 Langmuir, Vol. 5, No. 1, 1989 walls could (1) vary systematically with the reservoir electrolyte concentration, (2) reach concentrations approaching 0.8 m, and (3) be the same for both fused quartz and pyrex capillaries. On the other hand, it is clear that the process of equilibration in these experiments involves the transport of water, by diffusion in the vapor phase, from a given capillary to the reservoir. Due to the conical shape of the capillaries, the higher the KC1 concentration in the reservoir solution, the greater the mass of water which must be transported. Thus, assuming the experiments were conducted in such a way that the time allowed for reaching an apparent equilibrium was approximately the same for all reservoir concentrations, the degree to which true equilibrium was achieved would be less, the higher the reservoir concentration. It can therefore be concluded that the failure to reach a true equilibrium condition is at least consistent with the dependence of the various inferred properties on the reservoir concentration. That is, failure to reach equilibrium can explain the concentration dependence of both the inferred film thicknesses and the concentration of inferred contaminants, as well as the lack of dependence on the composition of the capillary walls. For the nonaqueous systems studied by Folman and Shereshevsky,lo dibutyl phthalate was used as the nonvolatile component. In this work it was stated that “Equilibrium was approached from two directions, by evaporation from full capillaries and by condensation into empty or partidly filled capillaries.” No indication of the overall times required to reach equilibrium for either process was given, except to note that “The level at which the meniscus remained unchanged for one or two hours was accepted as the equilibrium point.” Neither was any indication given as to the exact sequence of the various evaporation and condensation steps which were followed. In view of the lack of detailed information on these critical procedures, judgement must be suspended. The only conclusion that can be made is that real proof that the data represent true equilibrium states was not presented. Again, notwithstanding the failure to adequately describe critical features of the experiments, the results were interpreted by Folman and ShereshevskylO as indicating an enhancement of surface tension ranging from 2.9 to 5.5 for toluene and from 4.9 to 12.4 for isopropyl alcohol. The alternative thick-film hypothesis21 indicates film thicknesses of 0.8-1.2 pm for toluene and 0.9-1.3 pm for isopropyl alcohol. From the foregoing comments it is clear that Shereshevsky’s first three paper^^-^ are based on experiments which are flawed in an obvious way, i.e., as regards the attainment of true equilibrium conditions. Although the fourth paperlo is not so clearly flawed, insufficient details are given in the paper regarding the processes by which equilibrium was reached. It is not possible, therefore, to state any firm conclusion as to the validity of the results. The final paper” of 1972 reports work very similar in design and scope to that described in the 1955 paperalo It should be stressed that in all of Shereshevsky’s work, if the thick film hypothesis is adopted, the results lead to film thicknesses of the order of 1 pm or more. Experiments Conforming to the Kelvin Equation
Not surprisingly, Shereshevsky’s initial work prompted several other investigators to attempt experiments designed either to disprove his results or to confirm them. However, in no case does any of this work conflict in any way with the classical form of the Kelvin equation.lSm In discussing these experiments we may note, firstly, the work reported by Woodland and Mack18 in 1933. These authors
Comments
also attempted to test the Kelvin equation by experiments involving convex vapor/liquid interfaces, using a form of the Millikan oil-drop experiment. This aspect of their work was reported in the first part of their paper and seems to have received the exclusive attention of later authors. Consequently, the work reported in the second part of their paper, involving capillary rise experiments, appears to have been almost entirely ignored. This work showed, however, that the surface tension of water was not affected in tubes with radii down to 6.7 pm. Detailed criticisms of the experiments reported by Shereshevsky were also presented. These criticisms centered on the failure of Shereshevsky to achieve true equilibrium conditions in his experiments. Later authors have ignored these criticisms. Other workers who attempted to verify Shereshevsky’s results were Cohan and Meyer,lg whose work, published in 1940, was carried out in the same laboratory as the initial experiments of S h e r e s h e v ~ k y . These ~ ~ ~ workers noted that the size of the smallest capillary used by Woodland and Mack was 6.7 pm, whereas Shereshevsky’s work involved capillaries somewhat smaller. They therefore used two capillaries of 2.6 and 14.4 pm for capillary rise experiments with water and four capillaries varying from 1.9 to 10.0 pm for similar experiments with toluene. The size range of the capillaries clearly overlaps that used by Shereshev~ky.~,~ However, again, no indication was found that the surface tension of either water or toluene was enhanced. A final experimental study concerned with the validity of Shereshevsky’s results was reported by Cross and PicknetP in 1963. These authors carried out experiments on the layers or rings of water held by capillary forces between small solid spheres and a flat solid quartz surface. The spheres were beads fused onto the ends of thin quartz fibers. The two radii of curvature characterizing the vaporlliquid meniscus of such a ring are, of course, opposite in sign. However, the mean radii of curvature varied from 2 to 45 pm. This range overlaps that used by Shereshev~ k y . As ~ , in ~ Shereshevsky’s work vapor pressures were controlled by using aqueous KC1 solutions. In these experiments the time allowed for reaching equilibrium was 3-4 days, and equilibrium was approached from both sides. Air was not excluded, but stringent precautions were taken to eliminate sources of contamination. In this work no deviations from the meniscus curvatures predicted by the Kelvin equation were found. No explanation was offered for the highly discordant results reported by Shereshevsky, and the only comment made concerning these results was that “the reason for them remains a mystery.” In view of the very much longer equilibrium times found by Cross and Picknett, it seems very likely, as indicated above, that Shereshevsky’s work was flawed by a failure to reach equilibrium. Discussion of Conflicting Experimental Evidence
In assessing the validity of the various experimental results which have been discussed, it is necessary to consider several points. The first point is concerned with the fact that the capillary rise experiments of Woodland and Mackls and Cohan and Meyerlg did not involve vapor pressure measurements. However, any objection to the relevance of capillary rise experiments can only be valid if there are doubts concerning the thermodynamic relationship between relative vapor pressure and capillary rise. No author has suggested that this might be the case. Nevertheless, the relevance of capillary rise experiments seems to have been overlooked in recent review articles. Thus, Skinner and Sambles12survey the results of a large
Langmuir 1989,5, 293-295 number of investigations but ignore the work of Cohan and Meyerlg and that of Woodland and MacklBon concave menisci, as well as that of Cross and Picknett.20 Similarly, the review by Fisher and I~raelachvili’~ ignores the work of Woodland and Mack, while that of Cohan and Meyer is said not to “bear close examination”. Only the work of Cross and Picknett is said to be “not open to serious criticism”. Unfortunately, Fisher and Israelachvili do not state their reason for objecting to the work of Cohan and Meyer. The second point to be considered is the possible role of contaminants. In this connection Fisher and Israelachvili13suggest that the main problem, in the tests of the Kelvin equation prior to their own,13133was that of “unavoidable accumulation of contaminants”. They thus cast doubt on the use of the Kelvin equation in “practical cases”. They further suggest that Skinner and Sambles12 had reached the same conclusion. It would appear, however, that Skinner and Sambles offered no interpretation of the Shereshevsky results. It should be noted that in their own experiments Fisher and I ~ r a e l a c h v i l i show l ~ ~ ~convincingly ~ that the Kelvin equation is applicable to menisci with radii of curvature in the range from 0.004 to 0.02 pm. Whatever difficulties due to contaminants may have been encountered in these experiments, it is difficult to conceive that such difficulties could account for the results obtained by Shereshevsky. In capillary tubes having radii in the range from 1 to 10 pm any change in the effective tube radius due to the presence of contaminants in the form of an adsorbed film would certainly be negligible. I t is also clear that, in the case of water at least, any conceivable changes in contact angle or vapor/liquid surface tension due to organic contaminants would produce a decrease in the apparent surface tension, rather than a 10-fold or greater increase. If, on the other hand, a water-soluble contaminant were to be responsible for an enhanced lowering of the vapor pressure, the extent of the contamination would have to vary systematically, as pointed out previously, with the electrolyte concentration of the reservoir solution. As has been suggested, this seems highly unlikely. I t is concluded, therefore, that Shereshevsky’s experiments cannot be explained by the formation of an “anomalous component” of a siliceous nature, such as seems to have been involved in studies of the polywater phen~menon.~~ An exception to this conclusion is the case of the initial experiments with soft glass’ mentioned above. In fact, Shereshevsky’s concern over this type of contamination may have contributed to the evident lack of attention to the problem of equilibration. Thus, as has been suggested in the present discussion, the most likely flaw in the Shereshevsky work is the failure to ensure that equilibrium conditions were actually achieved. A further point to be considered relates both to the question of the relevance of capillary rise experiments with water and to that of possible massive contamination by soluble material leached from the solid surfaces in contact with the capillary held liquid. This type of contamination would, of course, have a relatively minor effect in the case of capillary rise, since neither the density nor the surface tension would be markedly affected. Hence, the results of the water capillary rise experiments which have been cited18J9could conceivably lack the relevance which has been indicated. However, if this type of contamination (33) Fisher, L. R.; Israelachvili,J. N. Nature (London) 1979,277,548.
(34)Everett, D.H.; Haynes, J. M.; McElroy, P. J. Sci. B o g . (Oxford) 1971,59, 279.
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were actually to occur in these, qs well as in Shereshevsky’s experiments, it would also be expected to occur in those of Cross and PicknetLm These latter experiments clearly show that massive contamination by soluble material leached from solid surfaces did not occur in this case. The recently reported results for high surface area reservoir rock samples5p6also provide evidence that contamination of this type does not vitiate the applicability of the Kelvin equation. Finally, reference should be made to the adhesion force experiments reported by Fisher and Israela~hvili.~~ This work, an extension of the Kelvin equation s t ~ d i e s l ~ ~ ~ ~ previously referred to, showed that for water the macroscopic value of the surface tension was applicable to the relative pressure range above about 0.95. This range encompasses the relative pressure range involved in the Shereshevsky experiments. Fisher and I~raelachvili~~ also conclude that for relative pressures below 0.99 the wetting film thicknesses were always less than about 1 nm. No evidence for massive contamination by a water-soluble component was found in this work.
Conclusions The experimental work of Shereshevsky relating to the applicability of the Kelvin equation has been directly and unequivocally refuted by the experiments of other investigators. Reviews and comments in the literature concerning Shereshevsky’s work have usually overlooked or downplayed the key experiments refuting this work. Neither the presence of contaminants nor the existence of adsorbed wetting films provides an explanation for the experimental observations reported by Shereshevsky. The most probable flaw in all of Shereshevsky’s work is the failure to ensure the attainment of equilibrium conditions. There exists no reason to doubt the applicability of the Kelvin equation to vapor/liquid systems in capillaries and pores in which the mean radius of curvature of the interface is comparable to those involved in the experiments of Shereshevsky. Acknowledgment. Critical comments on a preliminary draft of this communication by Dr. N. R. Morrow and by the Editor of Langmuir, Professor A. W. Adamson, are gratefully acknowledged. (35)Fisher, L. R.; Israelachvili, J. N. Colloids Surf. 1981,3, 303.
J. C. Melrose Petroleum Engineering Department, Stanford University, Stanford, California 94305 Received June 24, 1988
An Evaluation of Neumann’s “Surface Equation of State”
We have been very interested in the continuing controversy surrounding Neumann’s concept of the “surface equation of state”.l” We present here two unambiguous tests of the theory and then some comments about his thermodynamic analysis. (1) SDelt, J. K.: Neumann, A. W. J. Colloid Interface Sci. 1988,122, 294. (2) Spelt, J. K.; Neumann, A. W. Langrnuir 1987,3, 588. (3)Van Oss, C. J.; Good,R. J.; Chaudhury, M. K. J . Colloid Interface
Sci. 1986,Ill, 378. (4)Spelt, J. K.;Absolom, D. R.; Neumann, A. W. Langrnuir 1986,2, 620.
(5) Neumann, A. W. Adu. Colloid Interface Sci. 1974,4 , 1.
0 1989 American Chemical Society
