VIIViscosity and FluidityOne of the most useful findings that has resulted from the development of the general theory of physical state on which this present work is based is the theoretical deduction that there is a linear relation between temperature and each of the temperaturedependent properties of the pure liquid molecule (the molecule which is individually at a temperature somewhere between the melting point and the critical temperature). An approximate linearity has long been recognized in some of these properties, such as surface tension, for example. The present series of papers will not only show that the linearity in these instances is exact rather than approximate but will also demonstrate that a linear relationship exists in the case of those other liquid properties in which such a relation has not hitherto been recognized. The previous papers in this series have already established the linear nature of the liquid volumetemperature relationship. The liquid aggregate follows a nonlinear relation but these papers show that the deviation of the aggregate volume from linearity is not a true liquid characteristic, It is due to the presence of nonliquid molecules in the aggregate: solid molecules at the lower temperatures and critical molecules (see paper II for an explanation of this term) at relatively high temperatures. When the proportion of these foreign molecules is calculated and their volume is subtracted from the volume of the aggregate we find that the remainder, the pure liquid volume, follows the exact linear relation required by theory. The situation with respect to viscosity is quite similar. Here again we find that the observed values show no indication of a linear relation to temperature but when the effect of the various modifying influences is evaluated and removed the true molecular relationship is seen to be very definitely linear. The underlying theory indicates., however,, that the theoretically significant property is fluidity rather than viscosity and the linear relation is therefore between temperature and fluidity. All of the calculations will be made on the fluidity basis and for convenience in setting up the tabular comparisons of experimental results and calculated values the experimental viscosities will be converted to fluidities. It is evident that In the tipper part of the liquid temperature range where the number of molecules that have individually reached the critical temperature represents a significant proportion of the total number of molecules the fluidity of the liquid aggregate exceeds the true liquid fluidity because of the greater fluidity of the more mobile critical molecules., just as the liquid aggregate volume in this same region exceeds the volume of the true liquid. In this temperature range,, therefore., the additional fluidity due to the presence of the critical molecules will have to be evaluated and added to the true liquid fluidity in order to arrive at figures comparable with the experimental results. The experimental measurements Of Viscosity have., however., been confined almost entirely to the lower temperature range where this critical component Is negligible and it seems advisable to limit this present discussion to this range in which the results of the calculations can be verified by comparison with experimental values, leaving the relatively unexplored high temperature range for later treatment. According to the basic theory a solid molecule has no fluidity under any circumstances. At first glance this may seem to conflict with the observed fact that many solids do have a certain amount of plasticity, which is a low degree of fluidity. It has previously been explained, however, that solids at temperatures in the vicinity of the melting point, the same temperature range in which plasticity is found, contain a significant proportion of liquid molecules. The observed fluidity or plasticity of the solid aggregate is not actually a solid property; it is a property of the liquid molecules existing within the solid aggregate. The same theoretical considerations, which exclude the possibility of any fluid properties for the solid molecule also, indicate that the liquid molecule has zero fluidity below the liquid temperature, which corresponds to the melting point of the solid. The first requirement in setting up the mathematical relations for calculating fluidity is therefore to establish this liquid equivalent of the melting point: a quantity which we will designate T_{0}. Preliminary consideration of the theoretical background would seem to indicate that this temperature T_{0} should be onethird of the critical temperature. More detailed study discloses that this basic relation is modified in two respects. First, we find that the effective value of the mass increment applicable to fluidity is not necessarily the same as that applicable to the critical temperature and we therefore replace the critical temperature with a temperature T_{f} computed from equation 9, paper VI, in the same manner as the critical temperature but utilizing the modified mass increment. Second, we find that T_{0} is related to the 3/2 power of T_{f} rather than to T_{f}. On this basis we may set up the expression
As stated in the introductory paragraph of this paper, the linear relation between temperature and the various liquid properties applies specifically to the liquid molecule. In the case of properties, which are measured in such a way that the number of molecules is variable, it is therefore necessary to introduce a density term into the mathematical expression of the property. Viscosity is measured in terms of force per unit area and to put the corresponding fluidity on a basis proportional to the force per molecule we multiply the experimental value by the 2/3 power of the density. The product fd^{2/3} thus obtained in the quantity, which varies linearly with the temperature. For convenience the fluidity will be expressed in reciprocal centipoises and the symbol f will refer to the fluidity in terms of these units. From further theoretical considerations we next deduce that in organic compounds and others of similar general characteristics the linear increase in the quantity fd^{2/3} takes place at such a rate that this quantity reaches the equivalent of one natural unit at 2T_{0}. The value of the natural units previously developed^{3} enables us to compute the unit of fd^{2/3} in terms of reciprocal centipoises times (g/cm^{3})^{3/2} as 4.116. The temperaturefluidity equation is then
and we may express the theoretical fluidity of the pure liquid at any temperature T as
In comparing the values calculated from equation 13 with experimental results we note that the experimental fluidities diverge from the linear relation in the vicinity of T_{0} and approach zero by a gradual curve rather than showing a sharp cutoff at T_{0}. It is obvious that this is another result of the distribution of molecular velocities due to the operation of the probability principles. When the temperature of the liquid aggregate is T_{0} the temperatures of the individual molecules (their thermal energy levels) arc distributed over a range of values above and below T_{0}, that temperature being merely the average of the individual temperatures. Those molecules with individual temperatures above T_{0} have the positive fluidities which correspond to the prevailing temperatures but there is no such thing as negative fluidity hence this fluidity contributed by the molecules above T_{0} cannot be offset by the molecules below T_{0} and the total of the positive fluidities manifests itself as the fluidity of the liquid aggregate. Below T_{0} this aggregate fluidity follows a probability curve as the number of molecules above T_{0} gradually decreases. As explained in paper II, a determination of the exact probability relations governing such phenomena as this is beyond the scope of a broad general study of this kind and wherever probability is involved it has been considered sufficient for present purposes to establish a reasonably close correlation between the experimental values and some form of the socalled "normal" probability function' thus verifying the theoretical conclusion that the relation is one of the probability type without necessarily implying that the normal function is the exact probability expression applicable to the particular case. In the present instance the contribution of each temperature T shove T_{0} to the fluidity deviation depends not only on the number of molecules at this temperature but also on the difference x between T and T_{0}. The probabilities are therefore computed from the integral of x fx dx rather than the integral of fx dx, which was, used in most of the previous probability applications. The probability unit in which x is expressed has been taken as 510n/72 degrees and the numerical coefficient as 6.116n/18. For all of the compounds included in the tables accompanying this paper the value of n is 4. From the foregoing it will be noted that in the class of compounds covered by the tables the fluidity at temperatures below T_{0} is independent of the composition of the liquid and is a function only of the temperatures difference T_{0}T. In the range immediately above T_{0} the probability increment adds to the normal fluidity corresponding to the temperature T. Aside from the effect of the density, a quantity which can be evaluated by methods independent of the viscosity, the entire fluidity pattern of these normal organic liquids through the full temperature range of the pure liquid is completely defined by a single numerical value: the zero point temperature, T_{0}. This in turn can be computed from the molecular mass and the appropriate mass increment. As in the case of the critical temperature the whole fluidity question therefore reduces to a matter of determining the mass increment. Here again we find it necessary, at the present stage of the development, to utilize series relationships as the means of obtaining the increments. The general series pattern is the same as that of the values applicable to the critical temperature, all members of each series beyond the first three or four having the same mess increment, filth slightly higher values for the lower compounds. These series values of the increment are usually close to those previously established for the critical temperature. The paraffin increments, for example, shift about 6 or 8 units in the positive direction, the normal paraffins going from 6 (critical temperature basis) to 0 (fluidity basis), the 2methyl and 3methyl paraffins from 10 to 4, and the 2,2dimenthyl paraffins from 14 to 6. At this point it may be appropriate to comment briefly on the nature of this quantity which has been called the mass increment. The use of this term is not intended to imply that the actual mass of the molecule is in any way variable. We find that in many physical properties, including fluidity, the relation between mass and the property in question is subject to modification in such a manner that if the normal effect of the mass m of a particular atom is xm the modified value is nxm. The average coefficient applicable to the molecule as a whole is the weighted average of the x and nx values applicable to the individual atoms. It would be entirely possible, and perhaps more logical, to use average numerical coefficients of this nature in conjunction with the actual molecular mass rather than to throw the adjustment into the mass term by treating the difference between nm and m as an increment adding to the effective mass, but the latter has the very great advantage of bringing out clearly and distinctly the regularities and series relationships in the various properties which would be completely obscured by the alternate method of treatment. Table VII1 shows the mass increments and the corresponding values of T_{0} for a number of common organic series. In Table VII2 representative values of T_{0} from the preceding tabulation together with the appropriate values for some additional compounds are applied to the calculation of fluidities at specific temperatures. Column 2 in this table, headed P, shows the fluidity increment due to the probability distribution, if any significant increment is present. Column 3 is the sum of this increment and the molecular fluidity, fd^{2/3}, computed from equation 12. Dividing by the 2/3 power of the density we then obtain the theoretical fluidity (column 4) corresponding to the measured fluidity in column 5. Most of the calculated fluidities listed in Table VII2 agree with the experimental values within one or two percent. This degree of correlation is found in at least a substantial part of the experimental temperature range of practically all of the hundreds of liquids that have been studied in the course of this work, but at other temperatures the divergence frequently amounts to five percent or more. The significance of such discrepancies depends, of course, on the degree of accuracy that can be attributed to the experimental results, Timmermans^{36} tells us that it is difficult to measure viscosity with an accuracy of better than one percent and he includes among his selections of "precision" data some values from different sources which differ by three to six or eight percent. (See for example the values listed for pentane, ethyl alcohol, ethyl, ether, and methyl butyrate.) A general comparison of the results obtained on identical substances by different observers indicates that the accuracy of the normal run of experimental work is considerably lower than that of Timmermans' selected values, as might be expected. The differences between the experimental fluidities and the values calculated from equation 13 are therefore clearly within the limits of accuracy of the experimental work, In this connection it should be remembered that agreement with the experimental results is not an and in itself here as it is in the case of those studies which are made specifically for the purpose of devising mathematical expressions that fit the experimental data. An expression, which is provided with a sufficient number of adjustable constants, can be made to fit any set of experimental results, right or wrong, as long as it is not irregular. The aim of this present works on the other hand, is to ascertain the correct values and the calculations produce a rigid pattern with no latitude for adjustment. If this pattern agrees with the general pattern of experimental values and a reasonably large proportion of the figures agree closely (within the one percent deviation suggested by Timmermans, let us say) then individual discrepancies are not significant, even if they are numerous. In a situation where even the "percision" measurements often differ widely it is obvious that the general run of experimental work must have produced many inaccurate values and these will necessarily show up as deviations from the theoretical fluidities. In addition to the previously mentioned alternate values of the factor n in the expressions from which the probability units are calculated, there are a few other possible variations in the structural factors involved in the fluidity and a number of substances, notably rater and the alcohols, follow a somewhat modified pattern. This initial presentation of the fluidity relations has, however, been confined to those substances, which can be considered regular, and discussion of the liquids with special behavior characteristics will be deferred to a later publication. REFERENCES 3. Larson, D, B., The Structure of the Physical Universe, published by the author, 755 N. E. Royal Court, Portland 12, Oregon, 1959. Appendix A, 36. Timmermans, Jean, Physicochemical Constants of Pure Organic Compounds, Elsevier Publishing Co., Amsterdam, 1950. 38. American Petroleum institute Research Project 44, Selected Values, Carnegie Press, Pittsburgh, 1953. 39. Bingham. E, C., and Spooner, L. W., Physics, 4387. 40. Friend, J. N., and Hargreaves, W. D., Phil. Mag., 35619. 41. Friend and Hargreaves, ibid., 35136. 42. Friend and Hargreares, Ibid., 36731. 43. Lacher, J. R., Walden, C. H., and Park, J. D., J. Am. Chem. Soc., 713026.
