VI. Critical Constants

Having completed the coverage of the property of liquid volume we now turn to another important liquid characteristic: the critical temperature, As brought out in the first paper of this series, development of the consequences of the fundamental postulates on which the entire 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. At the lower transition temperature, the melting point, the thermal forces attain equality with the cohesive forces in one dimension and the molecule acquires freedom of movement in this dimension. At the location of the upper transition, the critical temperature, the thermal forces are equal to the cohesive forces in all three dimensions and the molecule is therefore able to escape from all ties with its neighbors.

A complete explanation of the derivation of this and the other theoretical conclusions that will be utilized in setting up the mathematical relations presented in this paper is beyond the scope of the present work, The general theoretical background is described in a preliminary edition of the authors book "The Structure of the Physical Universe' and further details will be added in a more complete edition to follow. This paper, like those which preceded it' will be devoted to demonstrating that the mathematical expressions developed from this new theoretical system are capable of reproducing the experimental results within the probable error of the latter.

It is apparent merely from inspection that the critical temperature is related to the molecular mass and from the basic theory we can obtain the additional information that enables expressing this relation in specific mathematical form. Three items are involved: (1) the direct relationship is not with the critical temperature itself but with a quantity that we will call the liquid base temperature, which is the critical temperature plus half of the 510 degree temperature unit (the same quantity that was utilized in computing the probability unit in paper II), (2) the effective mass is proportional to the fourth power of the base temperature, this relation being analogous to the relation between temperature and radiant energy, and (3) the effectiveness of each mass unit toward increasing the critical temperature varies with the environment and the effective mass added by such a unit is n units, where n may be zero or any small integer. For our purposes it will be convenient to regard the total effective mass as the sum of the actual molecular mass and a positive or negative mass increment, representing the net effect of the environmental factors. Denoting this mass increment by I, we then have the relation

and we may express the critical temperature as

If the critical temperature thus calculated is below 510.2° K a negative correction amounting to 1/36 of the difference between the calculated value and 510.2° is required.

On this basis evaluation of the critical temperature is reduced to a matter of determining the appropriate mass increment. A purely mathematical method of calculating this increment still remains to be worked out but the general principles, which are involved, are quite well defined. In the smaller atoms the atom as a whole takes part in any modification of the effective mass and the mass increment, if any, is n times the atomic number. We find, for example' that the mass increment of the normal straight chain paraffins is -6, which we may interpret as indicating that in each of these compounds the departure from the normal mass is limited to one carbon atom which has a six-unit negative increment. In Table VI-1 the critical temperatures calculated on this basis are compared with the values obtained experimentally. Following the paraffins in this table are the corresponding olefins which likewise take the -6 mass increment except for ethylene which has -8, the 2-methyl paraffins (-10, except 2-methyl propane) and the 2,2-dimethyl paraffins (-14, except 2~2-dimethyl propane).

These hydrocarbon values bring out two of the general factors, which enter into the determination of the mass increments of all of the organic compounds. First, we note that the increment of the CH2 interior groups is zero and the total increment of the molecule is attributable to the end groups of the chain. Since branched chains have more ends they tend toward larger increments unless the direction of the branch increments is opposite to that of the principal chain. Second, we find that in three of these four hydrocarbon series the first compound has a larger increment than those, which follow. In some of the substituted organic series where the initial increment is considerably greater than 6 this deviation at the lower end of the series extends still farther and the normal series level is not reached before the third or fourth compound. We may conclude from this that the addition of CH2 groups between the end groups of the molecule tends to weaken the influences, which are responsible for the existence of the mass increment. Where the original increment is small this weakening is generally a minor reduction completed with the first CH2 addition as in the hydrocarbon examples; where the original increment is large several CH2 additions may be required in order to attain the maximum effect of separation of the end groups.

As these tabulations indicate, the mass increment of the oxygen atoms is normally positive. The same is true of sulfur and nitrogen. The halogens, on the other hand, normally have negative increments. The increments of carbon and hydrogen can be either positive or negative; positive increments predominating in the acetylenes and the ring compounds and negative in the paraffins and olefins. The factors governing the specific values of the mass increments applicable to particular compounds will be discussed in more detail in subsequent papers dealing with viscosity and surface tension, two liquid properties which are functions of certain controlling temperatures not necessarily identical with the critical temperature but closely related to it.

Unfortunately the amount of experimental data available on critical temperatures is very limited and outside of the aliphatic hydrocarbons there are only a few series, which have been studied extensively, enough to enable any significant comparison between calculated and experimental values. Many of the figures in the table are therefor without direct experimental confirmation but an indirect indication of their validity can be obtained from a consideration of other properties related to the critical temperature and this matter will be covered in subsequent papers in this liquid series, particularly in connection with the discussion of boiling points and other vapor pressure phenomena.

If we compare the divergence between the calculated values and those experimental critical temperatures which are available with the divergence between the results obtained by different observers it is evident that the agreement between theory and observation is about as satisfactory as we can expect. There are a few definite conflicts, one of the most obvious of which is in the results on propane. Here the observed value is consistent with a mass increment of -7 rather than -6. This could well be another instance where a relatively small molecule deviates from the regular pattern of the series to which it belongs but there are also indications to the contrary and it appears that this is a question which will require more detailed study than can be justified in this present work.

Experimental values listed in the table have been taken from many sources, with preference being given to the values selected by Timmermana³⁶ and those included in the tables of Kobe and Lynn³⁷. Beyond the range of the experimental work a few of the values obtained by extrapolation processes have been included but the usefulness of these figures is rather limited since their reliability falls off rapidly as the length of the extrapolation increases.

The critical pressure, the vapor pressure at the critical temperature, can be computed by making use of a relation between two hypothetical quantities: the critical liquid volume, which is the volume that would be occupied by the substance in question if it could exist as a pure liquid at the critical temperature, and the critical gas volume, which is the volume that would be occupied if the substance could exist as a pure or "ideal" gas at that temperature. (The observed critical volume is a composite quantity, which has no particular significance in relation to the subjects being covered in this liquid series.) The critical liquid volume can be calculated from equation 3 (paper II), using the appropriate values of V₀ attained by the methods of Paper III. For use in the pressure computations we next adjust this volume to V_L + 3V_H, where V_L is the volume originating below 510° K and V_H is the increment above 510°, if any. The adjusted critical liquid volume, V_CL, thus obtained is related to the critical gas volume by a factor K_C and we may express the critical gas volume as K_CV_CL. Tracing the molar gas volume as 22.4145 x 10³ cm³, the value calculated from the natural units developed in the work on which this liquid study is based, we then have the relation

At the present stage of the development of theory it appears that the factor K_C should be either 5.0 or 5.5 and the value Applicable to each of the series that have been studied thus far does remain constant at one of these two levels, or at the intermediate value 5.25, after the first few members of the series. In some cases these first compounds also take the same factor but more generally there is a positive or negative deviation, which starts at a maximum and decreases toward zero, as the molecule increases in size. Thus we note in Table VI-1 that K_C in the normal paraffin series is methane 5.125, ethane 5.25, propane 5.25, butane and all subsequent compounds 5.50, The acetates have a similar pattern on the positive side of the normal 5.50 value: methyl 5.875, ethyl 5.75, propyl 5.625, butyl et seq., 5.50, and most other organic series approach the normal series value in the same manner. The exact cause of this deviation at the lower ends of the series has not been determined as yet but it is probably another effect of the same influences that modify the mass increment in the smaller molecules.

Preliminary studies of a number of inorganic compounds indicate that the critical constants of these compounds are governed by the same kind of factors that determine the organic constants but for the present we are relying primarily on series relationships for establishing the mass increments and since there are no inorganic series of the necessary length and constancy of structure the present discussion will be confined to the compounds of the organic division.

REFERENCES

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

37. Kobe, K. A. and Lynn, R. E. Jr., Chemical Reviews, 52-117.