VIII

Surface Tension

The study of a phenomenon such as surface tension by the application of a broad general theory encompassing a variety of associated phenomena has a significant advantage over a more restricted approach to the subject because an understanding of the interrelations between these various phenomena usually carries with it a very definite indication of the nature and magnitude of the factors governing the particular physical process under consideration. In this case we find from theory that surface tension and viscosity are very similar properties of matter; in fact it can be said that each is essentially the complement of the other. The findings with respect to viscosity which were described in paper VII can therefore be transferred en bloc to surface tension with appropriate modifications to compensate for the existing differences between the two phenomena.

As pointed out in the first paper of this liquid series, development of the consequences of the fundamental postulates on which this work is based indicates that physical state is a property of the individual molecule determined by its thermal energy level. In the solid state the inward-directed cohesive forces exceed the outward-directed thermal forces in all dimensions and the molecule therefore maintains a constant spatial relationship with its neighbors. In the gaseous state the net force is outward" permitting each molecule to break all ties and pursue its own independent course. The liquid state is the intermediate situation in which the molecule has sufficient thermal energy to escape from the solid state limitations in at least one dimension but not enough to break loose in all dimensions. Fluidity a measure of mobility, is a reflection of the amount of progress toward the gaseous state and it necessarily has zero value at a temperature analogous to the melting point, rising in linear relation to the temperature from there on.

Surface tension, a measure of the inter-molecular attraction, is obviously the in-verse property, a reflection of the degree to which the substance still retains its solid properties. It follows that the surface tension is zero at a temperature analogous to the critical temperature, rising in linear relation to the decrease in temperature below this point.

As in the case of fluidity these relations apply specifically to the surface tension per molecule. Since surface tension is customarily measured in terms of force per unit of length, we multiply the conventional measurement by the cube root of the density, obtaining the quantity gd^1/3, which is, proportional to the surface tension per molecule and corresponds to the quantity fd^2/3 used in the fluidity study.

By analogy with the findings concerning fluidity we can deduce that the controlling temperature, at which the surface tension is zero can be evaluated from equation 9, paper VI, in the same manner as T_c and T_f, using an appropriate value of the mass increment. The unit value of the surface tension is reached at the point where the difference T_r - T is equal to the liquid base temperature, which has been defined as the value obtained from equation 9 with the mass increment equal to zero. From the previously developed natural units3 we then compute the numerical coefficient of the surface tension equation as 58.314, which gives us

The resemblance to equation 13, the fluidity equation, is obvious.

It was brought out in the fluidity discussion that the mass increment effective in the fluidity relations is determined entirely by the end groups of the molecule, the interior groups having no effect except in the smallest molecules, where they may cause a modification of the increments applicable to the end groups by increasing the separation between the ends of the molecules. Surface tension, however, is a property of the solid-state type and in the solid state each structural group acts independently. This means that from a surface tension standpoint each structural group is an end group and is a potential contributor to the mass increment. Instead of settling down to a constant series level after the first few added CH₂ groups in the manner of the mass increments applicable to fluidity, the increments applicable to surface tension continue to increase (or decrease) as the molecule becomes larger.

In the normal paraffins, for example, methane has the same mass increment, for surface tension as for fluidity and for the critical temperature. But whereas the increment for the critical temperature remains constant at -6 for the entire normal paraffin series, and the increment for fluidity remains constant at zero after dropping to this point as a result of the first two CH₂ additions the negative increment for surface tension rises continuously as the chain lengthens, at the rate of two units per added CH₂ group as far as octane and four units per CH₂ group beyond this point. The 1-olefin pattern is almost identical but two units lower.

Table VIII-1 shows the liquid base temperatures, the mass increments, and the corresponding values of T_r for several common organic series. Table VIII-2 compares the experimental surface tensions with the results obtained from equation 15 using selected values from the preceding table and similar data on other representative organic compounds. The densities used are those given in the reports of the surface tension measurements, except where accurate theoretical values are available from the volumetric work described in previous papers of this series. These density values have not been listed in the table, as they are readily available from other sources.

The great majority of organic liquids and many of the common inorganic liquids follow this pattern without deviation and to simplify this initial presentation the discussion has been confined to these "regular" liquids. The compounds with special behavior characteristics including those chronic dissenters water and the alcohols, will be covered in a later publication.

Since the accuracy of the surface tension measurements is considerably greater than that of the measurements of viscosity we can draw some more definite conclusions from the comparisons of calculated and experimental values than were possible in the preceding paper on viscosity. To those who are interested in this situation primarily from the standpoint of liquid theory these questions as to the exact degree of correlation have little meaning. It is obvious that we cannot prove the validity of the theory by such a comparison; all that we can do is to show that the differences between the calculated and experimental values are within the uncertainty limits of the latter, which indicates that the theory could be correct. Actual evidence of its validity will have to be derived from other sources, the nature and significance of which will be discussed in the next paper in this series. In addition to its theoretical aspects however, the development of an accurate method of calculating the magnitudes of the various physical properties of liquids has a considerable practical value and those who are interested principally in the practical utilization of the figures are likely to give considerable weight to the actual degree of correlation between theory and experiment. Some discussion of this matter is therefore in order.

The important point that should be recognized in this connection is that a closer agreement is not necessarily a better agreement. Everyone realizes that if the differences between the calculated and experimental values are greater than the experimental uncertainty the theoretical results cannot be correct " but it is not always appreciated that too close an agreement has exactly the same meaning. The results obtained from a theoretically correct method of calculation will, of course, show very close agreement with the experimental values in many, cases but if the agreement is consistently much closer than the margin of experimental uncertainty it is clear that method of calculation gives us nothing more than a restatement of the experimental results.

This emphasizes the importance of a realistic evaluation of the magnitude of the experimental uncertainty. For this purpose the most reliable guide is the extent of agreement between competent observers. The investigators' own estimates of uncertainty usually have to be taken with a grain of salt. As B. N. da C. Andrade puts it, the last significant figure reported is often nothing more than an "expression of a genial optimism". Timmermans³⁶ has made a close study of this subject and states that it is difficult to measure surface tension with an accuracy greater than about 0.2 percent. If the surface tension is in the neighborhood of 25 dynes/cm this amounts to approximately .05 dynes/cm. Examination of his selections of "precision" data discloses that in a very large percentage of those cases where two or more values are given for the same temperature the difference between the minimum and maximum is .10 dynes/cm or more. In the normal paraffin hydrocarbons, which are as Timmermans says "a first rate physico-chemical material" and have been extensively studied, the average difference between his 20º values is .033 dynes/cm where only two values are given, but when we look at heptane and octane, which are represented by 4 and 5 values respectively, we find that the difference between the high and low values in both cases is .09 dynes/cm. Since there is no reason to believe that the measurements are any more difficult on these two compounds than on others in the series it is evident that the law figure of .033 dynes/cm is merely a result of selection and that the true uncertainty in the most accurate measurements of surface tension is at least .09 dynes/cm.

This conclusion is corroborated by the fact that the differences between the "best" experimental values for other organic compounds are much higher. The average difference between the 20º values selected by Timmermans for the branched paraffins is .122 dynes/cm, and in other organic families we find deviations up to .45 dynes/am. Since these are "precision" measurements in the most favorable temperature range and have undergone a selection process which has eliminated the most discordant values, it is obvious that the uncertainty in the general run of experimental work is substantially higher.

Let us now see whether the differences between the experimental results and the calculated values (which are the correct values if the theory is valid) are consistent with these experimental uncertainties. Again we will look first at the experimental results which are presumably the most accurate. A recent study by Jasper, Kerr and Gregorich⁴⁴ covered the normal paraffins from pentane to octadecane and from 0º to 80º C. If ire average the differences between the results of this investigation and the calculated values for each compound and then give these averages equal weight in computing an overall average difference we arrive at a figure of .074 dynes/cm. A similar calculation for the 1-olefins using data of the same investigators gives us an average deviation of .110 dynes/cm. The API values for the normal paraffins (propane and beyond) which cover a larger number of compounds and extend to higher and lower temperatures where measurement is more difficult show an average deviation of .149 dynes/cm. The monumental work of Vogel and collaborators produces results such as the following: normal paraffins .118, acetylenes .128, alkyl benzenes .097, propyl esters .134, ethyl diesters .128.

Although uncertainty is a very difficult thing to measure, the foregoing figures leave little doubt but that the average differences between the experimental results and the values calculated by the methods of this paper are consistent with realistic estimates of the uncertainties. The calculated values are therefore within the area in which they should fall if they are correct. The next paper in this series will utilize other methods to show that these values probably are correct.

The particular merit of the new development from the standpoint of practical utilization rests upon this point since it means that the new methods are able to distinguish correct from incorrect values. This is not true of any of the mathematical expressions previously developed for the representation of liquid properties. Such expressions smooth out the experimental values and eliminate minor irregularities but the general degree of accuracy of the results obtained from them is completely determined by the accuracy of the experimental data from which their constants were derived. The equations developed in this work, on the contrary, are primarily theoretical. They are not entirely independent of the experimental results but they utilize these data only for the purpose of finding the general area within which the correct values are located and the specific magnitudes of these correct values are then precisely determined by the equations. The accuracy or lack of accuracy of the experimental work has no bearing on this latter operation.

To illustrate this point let us look at some very interesting and significant data on the normal paraffins from C₁₁ to C₁₆ inclusive. When equation 15 was first formulated several years ago the results obtained from calculations on this group of compounds were checked against the API values, which are a composite of experimental data selected from the literature. At 20º C, which is probably the temperature most favorable for accurate observation, the differences between the calculated and experimental values ranged from .02 to .04 dynes/cm, with an average of .027, If we accept Timmemans' estimate of the maximum accuracy normally obtainable at the present time arid take .05 dynes/cm as the dividing line between correct and incorrect values, the calculations indicate that all of the API values at 20º are correct. When we examine the results at other temperatures, however, we find that only 7 out of a total of 44 measurements in the range from 0º to 80º f all within the limits which we have taken as defining the correct values .9 and the average deviation is .088 dynes/cm. This comparison therefore leads to the conclusion that most of the API values for these compounds at temperatures other than 20º C are incorrect.

Ordinarily the only support which is available for such conclusions as the foregoing comes from the general considerations -which indicate that the results produced by the new system are correct and if these conclusions had been announced at the time they were originally reached they undoubtedly would have been challenged, but the subsequent work of Jasper. et al, fully substantiates the accuracy of the calculations and the validity of the conclusions reached therefrom, Almost 80 percent of the individual, measurements on these six compounds reported by the Jasper group are within .05 dynes/cm, of the calculated values and the average deviation is only .35 dynes/cm. On the other hand the differences between the Jasper and API values average .10 dynes/cm with individual discrepancies as high as .30 dynes/cm. This recent work thus provides a very striking confirmation of the points which ' have been brought out in the foregoing discussion and, together with the theoretical evidence of the validity of the calculated results which will be discussed in paper number nine, justifies the assertion that wherever a homologous series of compounds is long enough or closely enough related to another series to enable establishing the values of the mass increments on a firm basis the calculations from equation 15 produce the correct values of the surface tension and if there is any lack of agreement with the experimental results the error is in the experimental work.

Although the true situation with respect to viscosity is obscured to some extent because of the greater experimental uncertainties in the measurement of that property, it is quite apparent that the theoretical development in this work unites the surface tension and viscosity phenomena and the conclusions with respect to surface tension are therefore valid for viscosity as well.

This ability of the new equations to discriminate between correct and incorrect values has some important implications with respect to their applicability. Obviously such expressions have a potential field of usefulness considerably broader than that of any expressions which merely enable interpolation and a limited amount of extrapolation of the experimental data.

REFERENCES

3. Larson, D. B., The Structure of the Physical Universes, published by the author. 755 N. E. Royal Court, Portland 12, Oregon. 1959, Appendix A.

36. Historians, Jeans Physico-chemical Constants of Pure Organic Compounds, Elsevier Publishing Co, Amsterdam, 1950.

38. American Petroleum Institute Research Project 44, Selected Values, Carnegie Press, Pittsburgh 1953,

44. Jasper, J. J., Kerr, E. R., and Gregorian, F. J., Am. Chem. Soc., 75-5252.

45. Jasper, J. J., and Kerr, E. R., Ibid., 76-2659.

46. Grzeskowiak, R., Jeffery, G. H., and Vogel, A. I., J. Chem. Soc., 1960-4719.

47. Vogel, A. I., Ibid., 1948-644.

48. Vogel, A. I., Ibid., 1948-624.

49. Owen, K., Quayle, O. R., and Clegg, W. J., J. Am. Chem. Soc., 64-1294.