CHAPTER 7

Temperature Relations

As explained in introducing the comparisons of the theoretical specific heats with experimental results, the curves in Figure 5 to13 verify only the specific heat pattern, the temperature scale of each curve being adjusted to the empirical results. In order to complete the definition of the curves we will now turn our attention to the temperature relations.

All of the distinctive properties of the different kinds of matter are determined by the rotational displacements of the atoms of which these substances are composed, and by the way in which the displacements enter into the various physical phenomena. As stated in Volume I,

The behavior characteristics, or properties, of the elements are functions of their respective displacements. Some are related to the total net effective displacement… some are related to the electric displacement, others to the magnetic displacement, while still others follow a more complex pattern. For instance, valence, or chemical combining power, is determined by either the electric displacement or one of the magnetic displacements, while the inter-atomic distance is affected by both the electric and magnetic displacement, but in different ways.

The great variety of physical phenomena, and the many different ways in which different substances participate in these phenomena result from the extension of this “more complex pattern” of behavior to a still greater degree of complexity. One of these more complex patterns was examined in Chapter 4, where we found that the response of the solid structure to compression is related to the cross-section against which the pressure is exerted. The numerical magnitude involved in this relation is determined by the product of the effective cross-sectional factors, together with the number of rotational units that participate in the action, a magnitude that determines the force per unit of the cross-section. Inasmuch as one of the dimensions of the cross-section may take either the effective magnetic displacement, represented by the symbol b in the earlier discussion, or the electric displacement, represented by the symbol c, two new symbols were introduced for purposes of the compressibility chapter: the symbol z to represent the second displacement entering into the cross-section (either b or c), and the symbol y to represent the number of effective rotational units (related to the third of the displacements). The a-b-c factors were thus represented in the form a-z-y.

The values of these factors relative to the positions of the elements in the periodic table follow the same general pattern in application to specific heat as in compressibility, and most of the individual values are either close to those applying to compressibility or systematically related to those values. We will therefore retain the a-z-y symbols as a means of emphasizing the similarity. But the nature of the thermal relations is quite different from that of the relations that apply to compressibility. The temperature is not related to a cross-section; it is determined by the total effective rotation. Consequently, instead of the product, azy, of the effective rotational factors, the numerical magnitude defining the temperature scale of the thermal relations is the scalar sum, a+z+y, of these rotational values.

This kind of a quantity is quite foreign to conventional physics. The scalar aspect of vectorial motion is recognized; that is, speed is distinguished from velocity. But orthodox physical thought does not recognize the existence of motion that is inherently scalar. In the universe of motion defined by the postulates of the Reciprocal System of theory, on the other hand, all of the basic motions are inherently scalar. Vectorial motions can exist only as additions to certain kinds of combinations of the basic scalar motions.

Scalar motion in one dimension, when seen in the context of a stationary spatial reference system, has many properties in common with vectorial motion. This no doubt accounts for the failure of previous investigators to recognize its existence. But when motion extends into more than one dimension there are major differences in the way these two types of motion present themselves (or do not present themselves) to observation. Any number of separate vectorial motions of a point can be combined into a single resultant, and the position of the point at any specified time can be represented in a spatial system of reference. This is a necessary consequence of the fact that vectorial motion is motion relative to that system of reference. But scalar motions cannot be combined vectorially. The resultant of scalar motion in more than one dimension is a scalar sum, and it cannot be identified with any one point in spatial coordinates. Such motion is therefore incapable of representation in a spatial reference system of the conventional type. It does not follow, however, that inability to represent this motion within the context of the severely limited kind of reference system that we are accustomed to use means that such motion is non-existent. To be sure, our direct perception of physical events is limited to those that can be represented in this type of a reference system, but Nature is not under any obligation to stay within the perceptive capabilities of the human race.

As pointed out in Chapter 3, Volume I, where the subject of reference systems was discussed at length, there are many aspects of physical existence (that is, many motions, combinations of motions, or relations between motions) that cannot be represented in any single reference system. This is not, in itself, a new, or unorthodox conclusion. Most modern physicists, including all of the leading theorists, have realized that they cannot accommodate all of present-day physical knowledge within the limitations of fixed spatial reference systems. But their response has been the drastic step of cutting loose from physical reality, and building their fundamental theories in a shadow realm where they are free from the constraints of the real world. Heisenberg states their position explicitly. “The idea of an objective real world whose smallest parts exists objectively in the same sense as stones and trees exist, independently of whether or not we observe them… is impossible,”⁸ he says. In the strange half-world of modern physical theory the only realities are mathematical symbols. Even the atom itself is “in a way only a symbol,”⁹ Heisenberg tells us. Nor is it required that symbols be logically related or understandable. Nature, these front rank theorists contend, is inherently ambiguous and subject to uncertainties of a fundamental and inescapable nature. “The world is not intrinsically reasonable or understandable,” Bridgman explains, “It acquires these properties in ever-increasing degree as we ascend from the realm of the very little to the realm of everyday things.”¹⁰

What the Reciprocal System of theory has done in this area is to show that once the true status of the physical universe as a universe of motion is recognized, and the properties of space and time are defined accordingly, there is no need for the retreat from reality, or for the attempt to blame Nature for the prevailing inability to understand the basic relations. The existence of phenomena not capable of representation in a spatial reference system is a fact that we must come to terms with, but the contribution of the Reciprocal System has been to show that the phenomena outside the scope of the conventional spatial reference systems can be described and evaluated in terms of the same real entities that exist within the reference system. The scalar sum of the magnitudes of motions in different dimensions, the quantity that we will now use in analyzing the temperature relations, is an item of this nature. It is just as real as any other physical quantity, and its components, the motions in the individual dimensions, are motions of the same nature as those one-dimensional scalar motions that are capable of representation in the spatial reference systems, even though the scalar sum cannot be so represented in any manner accessible to our direct perception.

In the theoretical minimum situation, where the effective thermal factors are 1-0-0, and the scalar sum of these factors is one unit, the temperature of the initial negative level is one unit out of the total of 128 that corresponds to the full 510.7 degrees temperature unit of the condensed states. But since the thermal motion is effective in only one direction, the ratio becomes 1/256, and the zero point temperature, T₀, the temperature at which the thermal motion counterbalances the negative initial level of vibration, is 1.995° K. For a substance with thermal factors a, z, and y, and the normal 2/9 initial specific heat level, we then have:

T0 = 1.995 (a+z+y) degrees K

(7-1)

This value completes the definition of the specific heat curves by defining the temperature scales. It will be more convenient, however, to work with another of the fixed points on the curves, the first transition point, T₁. As this is the unit specific heat level on the initial linear section of the curve, while T0 is 2/9 unit above the initial point, the temperature of the first transition point is:

T1 = 8.98 (a+z+y) degrees K

(7-2)

Thermal factors of the elements for which reliable specific heat patterns are available, and the corresponding theoretical first transition temperatures (T₁) are listed in Table 22, together with the T₁ values derived from curves of the type illustrated in Figures 5 to13, in which the temperature scale is empirical. In effect, this is a comparison between theoretical and experimental values of the temperature scales of the specific heat curves. The experimental values are subject to some uncertainty, as they have been obtained by inspection from graphs in which the linear portions of the curves were also drawn from visual inspection. Greater accuracy could be attained by using more sophisticated techniques, but the time and effort required for this refinement did not appear to be justified for the purposes of this initial investigation of the subject.

The compressibility factors derived in Chapter 4, with a few values restated in different, but equivalent, terms, are shown in the table for comparison with the corresponding thermal factors. The principal determinants of the compressibility values, aside from the effect of the pressure level itself (including the internal pressure), were found to be the magnitude and sign (positive or negative) of the displacement in the electric dimension. The rotational group to which the element belongs (determined by the magnetic displacements) is much less significant. In the thermal situation the rotational group becomes the dominant influence. The elements of Group 3B (magnetic displacements 3-3), midway in the group order, generally have thermal factors close to the compression values. In half of the 3B elements included in the table the deviation is no more than one unit. But in each direction from this central group there is a systematic deviation from the compressibility values, upward in the lower groups and downward in the higher groups. Every element above number 42, molybdenum, that is included in the table, with one exception, has thermal factors either equal to or less than the corresponding compressibility factors. Every element below molybdenum, with three exceptions (two of which are alkali metals), has thermal factors that are either equal to or greater than the corresponding compressibility factors.

It was noted in Chapter 4 that in compression the lowest electropositive elements do not take the minimum 1-1-1 factors of their electronegative counterparts, but have a = 4 in all of the elements of this class investigated by Bridgman. The reason for this difference in behavior is not yet known (although it is no doubt connected with the all-positive nature of the rotational displacement of these elements), but it is even more pronounced in the thermal factors. Except for the alkali metals above sodium, which, as noted above, have thermal factors even lower than the compressibility values, the lower electropositive elements not only maintain the 6-unit minimum (4-1-1 or equivalent) but raise the effective magnitudes of their thermal factors still farther by omitting the n = 1 section of the specific heat curve based on equation 5-6, and going immediately to n = 2, which increases the temperature scale by a factor of 8. This pattern is followed by boron and carbon, and in part, by beryllium. The corresponding members of the next higher group, magnesium, aluminum, and silicon, also have n = 2 from the start of the thermal motion, but here the second unit is one-dimensional rather than three-dimensional. Beryllium combines the two patterns. It has the same thermal factors as lithium, but a dimensional multiplier halfway between those of lithium and boron, the two adjoining elements.

The option of one dimension or three dimensions is open whenever motion advances from one unit to two units, but not under any other conditions. Three-dimensional motion of one displacement unit is meaningless, as 1³ = 1. After two units there is no option, as there cannot be more than two units in linear succession, for reasons that were discussed in Volume I. But two-unit motion can be either one-

Table 22: Effective Rotational Factors

Factors					T1		Factors				T1
	Comp	Therm.	n	Tot.	Calc.	Obs.		Comp.	Therm.	Tot.	Calc.	Obs.
Li	4-1-1	4-2-1	2	14	126	131	Y	4-2-1	4-3-1	8	72	71
		4-1-1	2	12	108	110	Zr	4-8-1	4-4-1	9	81	84
Be	4-4-1	4-2-1	2	14	314	323	Mo	4-8-2	4-8-2	14	126	129
			8	56	314	323			4-6-2	12	108	107
		4-1-1	2	12	269	267	Ru	4-8-2	4-8-2	14	126	128
			8	48	269	267			4-6-2	12	108	107
B		4-1-1	8	48	431	420	Rh	4-8-2	4-8-1	13	117	117
C-d	4-6-1	4-4-1	8	72	647	635			4-6-1	11	99	95
C-g	4-2-1	4-3-1	8	64	575	578	Pd	4-6-2	4-4-2	10	90	91
Na	4-1-1	4-1-1		6	54	52			4-4-1	9	81	78
Mg	4-4-1	4-1-1	2	12	108	109	Ag	4-4-2	4-3-1	8	72	72
		3-1-1	2	10	90	91	Cd	4-4-1	2-2-1	5	45	46
Al	4-5-1	4-2-1	2	14	126	131	In	4-4-1	4-6-2	12	108	105
		4-1-1	2	12	108	112	Sn	4-4-1	4-2-1	7	63	66
Si	4-4-1	4-6-2	2	24	216	220			4-1-1	6	54	57
P-r		4-6-2	2	24	216	207	Sb	4-4-1	4-3-1	8	72	68
P-w		4-2-1		7	63	66	Te	4-3-1	4-2-1	7	63	61
S	4-1-1	4-4-1		9	81	84	I		2-2-1	5	45	44
Cl		4-2-1		7	63	62	Xe		1-1-0	2	18	19
Ar		1-1-1		3	27	28	Cs	4-1-1	1-1-0	2	18	17
K	4-1-1	2-1-1		4	36	32	Ba	4-2-1	2-1-1	4	36	34
Ca	4-3-1	4-3-1		8	72	76	La	4-4-1	2-2-1	5	45	42
Sc		4-6-1		11	99	103	Pr	4-4-1	1-1-1	3	27	27
		4-5-1		10	90	88	Nd	4-4-1	1-1-1	3	27	31
Ti	4-8-1	4-8-2		14	126	124	Sm	4-4-1	2-1-1	4	36	36
V	4-8-1	4-8-3		15	135	133	Eu	4-4-1	2-1-1	4	36	33
		4-6-2		12	108	107	Gd	4-4-1	2-2-1	5	45	48
Cr	4-8-1					162	Tb	4-4-1	2-2-1	5	45	44
		4-8-2		14	126	128	Dy	4-4-1	2-2-1	5	45	41
Mn	4-8-1	4-8-1		13	117	115	Ho	4-4-1	2-1-1	4	36	33
		4-5-1		10	90	92	Er	4-4-1	1-1-1	3	27	28
Fe	4-8-1	4-8-4		16	144	142	Tm	4-4-1	1-1-1	3	27	29
		4-6-2		12	108	108	Yb	4-2-1	2-1-1	4	36	37
Co	4-8-1	4-8-2		14	126	126	Hf		4-3-1	8	72	71
		4-6-1		11	99	100	Ta	4-8-2	4-3-1	8	72	74
Ni	4-8-1	4-8-2		14	126	131	W	4-8-3	4-6-2	12	108	108
		4-6-1		11	99	97	Re		4-4-2	10	90	93
Cu	4-6-1	4-6-2		12	108	108			4-4-1	9	81	78
Zn	4-4-1	4-3-1		8	72	73	Ir	4-8-3	4-6-1	11	99	98
Ga		2-1-1		4	36	36			4-5-1	10	90	88
Ge	4-4-1	4-8-1		13	117	119	Pt	4-8-2	4-3-1	8	72	76
As	4-4-1	4-6-2		12	108	106	Au	4-6-2	4-1-1	6	54	57
Se	4-1-1	4-3-1		8	72	75	Hg		2-1-1	4	36	32
Br		4-2-1		6	56	54	Tl	4-4-1	2-1-1	4	36	34
Kr		1-1-0		2	18	20	Pb	4-4-1	2-1-1	4	36	33
Rb	4-1-1	1-1-0		2	18	20	Bi	4-3-1	2-2-1	5	45	44

dimensional or three-dimensional. At the point where the advance from one to two units takes place, the motion is therefore able to take the dimensions that are best suited to the existing situation. A one-dimensional increase in the value of n results in increasing the temperature scale by a factor of 2 rather than 8. The alkali metals, which diverge from the normal electropositive behavior in a number of respects because of their low electric displacement, follow the same pattern as the elements listed in the preceding paragraph, but one step lower, as indicated in the following comparison:

Group	Alkalis	Other Positive
1B	n = 2	n = 8
2A	4-x-x	n = 2
2B	1-1-x	4-x-x

As we found in the specific heat investigation, the electronegative elements below displacement 7 have a half-size initial negative specific heat level: 1/9 unit instead of the normal 2/9. It might be expected that this would result in a net effective specific heat of 8/9 unit or 2 2/3 R, at the transition point instead of the 7/9 unit (2 1/3 R) that exists when the initial negative level is 2/9 unit. But it is quite clear from the measured specific heat values that this is not true. The first transition point in the specific heat curves of the electronegative elements is 2 1/3 R just as it is in the curves with the 2/9 unit (2/3 R) negative initial level. Apparently the restriction that prevents the existence of the more negative initial level in the specific heat of these elements is gradually eliminated as the temperature rises, so that at the transition point the effective negative component of the specific heat is the normal 2/9 unit.

The thermal factors of the higher inert gases, krypton and xenon, which have no rotation in the electric dimension, are 1-0-0 rather than 1-1-1, as in compressibility. This is a peculiarity of the mathematics, and has no physical significance. In both cases the meaning of the symbols is that the effective magnitude is determined entirely by the factors a and z. In multiplication this requires a unit value in the y position, whereas in addition a zero is required for the same purpose. But this equivalence of the 1-1-1 compressibility and 1-1-0 thermal factors does not mean that 1-1-1 thermal and 1-1-0 thermal are equivalent. The 1-1-1 thermal combination is the minimum for a substance with effective rotational displacement in all three dimensions. Where the thermal factors drop to 1-1-0, as indicated for rubidium and cesium, there is no effective displacement in the electric dimension, and the thermal motion is following the inert gas pattern. Such behavior is uncommon, but it is not without precedent in other properties. We found in Chapter 1, for instance, that a number of elements, including the halogens, the elements corresponding to the alkalis on the opposite side of the inert gases, have inter-atomic distances in one or two dimensions that are similarly based on magnetic rotation only.

Since the empirical values listed in Table 22 are subject to a considerable degree of uncertainty, small differences between them and the calculated values have no significance. In some cases, however, the discrepancy is large enough to be real, and further study of the thermal relations of these elements will be required. Only one of the experimental values shown in the table, one of those applicable to chromium, is too far from any theoretical temperature to be incapable of explanation on the basis of the theoretical information now available.

As brought out in the discussion of the general pattern of the specific heat curves in Chapter 5, in many substances there is a change in the temperature scale of the curve at the first transition point (T₁), as a result of which the first and second segments of the curve do not intersect at the 2^1/3 R end point of the lower segment of the curve in the normal manner. This change in scale is due to a transition to the second set of thermal factors given, for the elements in which it occurs, in Table 22. With the benefit of the information that we have developed regarding the factors that determine the temperature scale we can now examine the quantitative aspects of these changes.

As an example, let us look at the specific heat curve of molybdenum, Figure 9, which, as previously noted, also applies to ruthenium. The thermal factors applicable to these elements at low temperatures are 4-8-2, identical with the compressibility factors. The first transition point, specific heat 4.63, is reached at 126° K on the basis of these factors. The corresponding empirical temperatures, determined by inspection of the trend of the experimental values of the specific heats, are 129 for molybdenum and 128 for ruthenium, well within the range of uncertainty of the techniques employed in estimating the empirical values. If the thermal factors remained constant, as they do in the “regular” pattern followed by such elements as silver, Figure 5, there should be a transition to n = 2 at this 126° K temperature, and the specific heat above this point would follow the extension of a line from the initial level of 3.89 to 4.63 at 126° K. But instead of continuing on the 4-8-2 basis, the thermal factors decrease to 4-6-2 at the transition point. These factors correspond to a transition temperature of 108° K. The specific heat of the molecule therefore undergoes an isothermal increase at 126° K to the extension of a line from the initial level of 3.89 to 4.63 at 108° K, and follows this line at higher temperatures. The effect of the isothermal increase in the specific heat of the individual molecules is, of course, spread out over a substantial temperature range in application to a solid aggregate by the distribution of molecular velocities.

The temperature of the subsequent transition points and the end points of the various segments of the specific heat curves can be calculated from the temperatures of the first transition points by applying the relative values listed in Chapter 5 to the appropriate values of T1. An approximate agreement between the empirical data and the higher transition points thus calculated is indicated, but the angles at which the upper segments of the curves intersect are too small to permit any close empirical definition of the temperature of intersection. The only one of the end points that has any real significance is the end point of the last segment of the curve applicable to the substance under consideration. This is the temperature limit of the solid. Any further addition of heat initiates the transition to the liquid state.

Inasmuch as it is the individual molecule that reaches its thermal limit at the solid end point, it is the individual molecule that makes the transition to the liquid state. Physical state is thus basically a property of the individual molecule rather than a property of the aggregate, as seen in conventional physical theory. The state of the aggregate is merely a reflection of the state of the majority of its constituents. Recognition of this fact some forty years ago, in the early stages of the investigation that led to the results now being reported, was a major step in the clarification of physical fundamentals that ultimately opened the door to the formulation of a general physical theory.

The liquid state has long been an enigma to conventional physics. As expressed by V. F. Weisskopf, “A liquid is a highly complex phenomenon in which the molecules stay together yet move along each other. It is by no means obvious why such a strange object should exist.”¹¹ Weisskopf goes on to speculate as to what the outcome would be if physicists knew the fundamental principles on which atomic structure is based, as present-day theory sees them, but “had never had occasion to see structures in nature.” He doubts if these theorists would ever be able to predict the existence of liquids.

In the Reciprocal System of theory, on the other hand, the liquid state is a necessity, an intermediate condition that must necessarily exist between the solid and gaseous states. When the thermal motion of a molecule reaches equality with the inward progression of the natural reference system in one dimension of the region outside unit distance, the cohesive force in that dimension is eliminated. The molecule is then free to move in that dimension, while it is held in a fixed position, or a fixed average position, in the other dimensions by the cohesive forces that are still operative. The temperature at which the freedom in one dimension is reached is the melting point of the aggregate, because any additional thermal energy supplied to the aggregate is absorbed in changing the state of additional molecules until the remaining content of solid molecules reaches the percentage that can be accommodated within the liquid aggregate.

These remaining solid molecules are gradually converted to the liquid state in a temperature range above the melting point. Thus the liquid aggregate in this range contains a percentage of solid molecules, while the solid aggregate in a similar temperature range below the melting point contains a percentage of liquid molecules. The presence of these “foreign” molecules has a significant effect on the physical properties of matter in both of these temperature ranges, an effect which, as we will see in the subsequent discussion of the liquid state, can be evaluated accurately by using probability relations to determine the exact proportions in which molecules of the two states exist at each temperature.

While the end point of the solid state is the temperature at which the intermolecular forces reach an equilibrium at the unit level, arrival at this end point does not mean automatic entry into the liquid state. It merely means that the cohesive forces of the solid are no longer operative in all three dimensions, and therefore do not prevent the free movement in one dimension of space that is the distinguishing characteristic of the liquid state. The significant point here is that a liquid molecule is limited to certain specific temperatures. A liquid aggregate can take any temperature within the liquid range, but only because the aggregate temperature is an average of a large number of the restricted individual values.

This same restriction to one of a limited set of values also applies to the temperature of the solid molecule, but in the vicinity of the melting point the solid is at a high time region temperature level, where the proportionate change from one possible value, n units, to the next, n + 1 units, is small. The motion of the liquid state, on the other hand, is in the region outside unit space, and is equivalent to gas motion in the one dimension in which the thermal energy exceeds the solid state limit. As we saw in Chapter 5, temperatures in the vicinity of the melting point are very low on the scale applicable to this outside region, and the proportionate change from n to n + 1 is large. The intervals between the possible temperatures of liquid molecules are therefore large enough to be significant.

Because of the limitation of the liquid temperatures to specific values, the temperature at which a molecule qualifies as a liquid is not the end point temperature of the solid state, but a higher value that includes the increment necessary to bring the end point temperature up to the next available liquid level. This makes it impossible to calculate melting points from solid state theory alone. Such calculations will have to wait until the relevant liquid theory is developed in a subsequent volume in this series, or elsewhere. But the temperature increment beyond the solid end point is small compared to the end point temperature itself, and the end point is not much below the melting point. A few comparisons of end point and melting point temperatures will therefore serve to confirm, in a general way, the theoretical deductions as to the relation between these two magnitudes.

There is a considerable degree of uncertainty in the experimental results at the high temperatures reached by the melting points of many of the elements, and there are also some theoretical aspects of the thermal situation in the vicinity of the melting point that have not yet been fully explored. The examples for discussion in this initial approach to the subject have been selected form among those in which these uncertain elements are at a minimum. First, let us look at element number 19, potassium. This element has a specific heat curve of the type identified by the notation n = 3 in Figure 4. Its thermal factors are 2-1-1, and it maintains the same factors throughout the entire solid range. As indicated in Chapter 5, the end point temperature of this type of curve is 9.32 times the temperature of the first transition point. This leads to an end point temperature of 336° K. The measured melting point is 337° K. In this case, then, the solid end point and the melting point happen to coincide within the limits of accuracy of the investigation.

Chlorine, an element only two steps lower in the atomic series than potassium, but a member of the next lower group, has the lower type of specific heat curve, with n = 2. The end point temperature of this curve is 4.56 on the relative scale where the first transition point is unity. The thermal factors that determine the transition point, and are applicable to the first segment of the curve, are 4-2-1, but if these factors are applied to the end point they lead to an impossibly high temperature. It is thus apparent that the factors applicable to the second segment of the curve are lower than those applicable to the first segment, in line with the previously noted tendency toward a decrease in the thermal factors with increasing temperature. The indicated factors applicable to the end point in this case are the same 2-1-1 combination that we found in potassium. They correspond to an end point temperature of 164° K, just below the melting point at 170° K, as the theory requires.

Next we look at two curves of the n = 4 type, the end point of which is at a relative temperature of 17.87. On the basis of thermal factors 4-6-1, the absolute temperature of the end point is 1765° K, which is consistent with the melting points of both cobalt (1768) and iron (1808). Here, too, the indicated factors at the end point are lower than those applicable to the first segment of the specific heat curve, but in this case there is independent evidence of the decrease. Cobalt, which has the factors 4-8-2 in the first segment is already down to 4-6-1 at the second transition point, while iron, the initial factors of which are also 4-8-2, has reached 4-6-2 at this point, with two more segments of the curve in which to make the additional reduction.

Compounds of elements about group 1B, or having a significant content of such elements, follow one or another of the Type 1 patterns that have been illustrated by examples from the elements. The hydrocarbons and other compounds of the lower group elements have specific heat curves of type 2 (Figure 3) in which the end point is at a relative temperature of 1.80. As an example of this class we can take ethylene. the thermal factors of these lower group compounds are limited to 1-1-1, 2-1-1, and the combination value 1½-1-1. As we found in Volume I, however, the two groups of atoms of which ethylene and similar compounds are composed are inside one time region unit of distance. They therefore act jointly in thermal interchange rather than acting independently in the manner of two inorganic radicals, such as those in NH₄NO₃. Each group contributes to the thermal factors of the molecule, and the value applicable to the molecule as a whole is the sum of the two components. Ethylene uses the 1-1-1 and 1½-1-1 combinations. A difference of this kind between the two halves of an organic molecule is quite common, and no doubt reflects the lack of symmetry between the positive and negative components that was the subject of comment in the discussion of organic structure. the combined factors amount to a total of 6½ units. This corresponds to a transition point at 58° K, which agrees with the empirical curve, and an end point at 104° K, coincident with the observed melting point.

The joint action of the two ends of an organic molecule that combines their thermal factors in the temperature determination is maintained when additional structural units are introduced between the end groups. As brought out in Chapter 6, such an extension of the organic type of structure into chains or rings also results in the activation of additional thermal motions of an independent nature within the molecules. The general nature of this internal motion was explained in the previous discussion. The same considerations apply to the transition point temperature, except that the internal motion is independent of the molecular motion in vectorial direction as well as in scalar direction. It is therefore distributed three-dimensionally, and the fraction in the direction of the molecular motion is 1/8 rather than ½. Each unit of internal motion thus adds 1/8 of 8.98 degrees, or 1.12 degrees K to the transition point temperature. With the benefit of this information we are now able to compute the temperatures corresponding to the specific heats of the paraffin hydrocarbons of Table 21. These values are shown in Table 23.

Table 23: Temperatures of Critical Points—Paraffin Hydrocarbons

Thermal Factors
	Trans. Point	Total		End point	Total
Propane	1-1-1	1-1-1	6	1-1-1	1-1-0	5
Butane	1-1-1	1-1-½	5½	2-1-1	1½-1-1	7½
Pentane	1½-1-1	1½-1-1	7	2-1-1	2-1-1	8
Hexane and above	2-1-1	1½-1-1	7½	2-1-1	2-1-1	8
Temperatures
	Internal Units	T1	End Point Factors		End Point	Melting Point
			Internal	Total
Propane	0	54		5	81	85
Butane	0	50	1	8½	137	138
Pentane	2	65	1	9	145	143
Hexane	3	71	3	11	178	179
Heptane	4	72	3	11	178	182
Octane	5	73	5	13	210	216
Nonane	6	74	5	13	210	220
Decane	7	75	7	15	242	243
Hendecane	8	76	7	15	242	247
Dodecane	9	77	8	16	259	263
Tridecane	10	79	8	16	259	268
Tetradecane	11	80	9	17	275	279
Pentadecane	12	81	9	17	275	283
Hexadecane	13	82	10	18	291	291

The first section of this table traces the gradual increase in the thermal factors as the molecule makes the transition from a simple combination of two structural groups, with properties that are similar to those of inorganic binary compounds, except for the joint thermal action due to the short inter-group distance, to a long-chain organic structure. The increase in the factors follows a fairly regular course in this range except in the case of butane. If the experimental values of the specific heat of this compound are accurate, its transition point factors drop back from the total of 6 that applies to propane to 5½, whereas they would be expected to advance to 6½. The reason for this anomaly is unknown. At the C₆ compound, hexane, the transition to the long-chain status is complete, and the thermal factors of the higher compounds as far as hexadecane (C₁₆), the limit of the present study, are the same as those of hexane.

In the second section of the table the transition point temperatures are calculated on the basis of 8.98 degrees K per molecular thermal factor, as shown in the upper section of the table, plus 1.12 degrees per effective unit of internal motion. The number of internal motions shown in Column 1 for each compound is taken from Table 21.

Columns 3 and 4 are the values entering into the calculation of the solid end point, Column 5. As the table indicates, some of the internal motions that exist in the molecule at the transition temperature are inactive at the end point. However, the active internal motion components are thermally equivalent to the molecular motions at this point, rather than having only 1/8 of the molecular magnitude as they do at T₁. This is a result of the general principle that the state of least energy takes precedence (in a low energy environment) in cases where alternatives exist. Below the transition point the internal thermal motions are necessarily one-dimensional. Above T₁ they are free to take either the one-dimensional or three-dimensional status. The energy at any given temperature above T₁ is less on the three-dimensional basis. This transition therefore takes place as soon as it can, which is at T₁. At the melting point the energy requirement is greater after the transition to the liquid state. Consequently, this transition does not take place until it must because there is no alternative. A return to one-dimensional internal thermal motion is an available alternative that will delay the transition. This motion therefore gradually reverts back to the one-dimensional status, reducing the energy requirement, and the solid end point is not reached until all effective thermal factors are at the 8.98 temperature level. The end point temperature of Column 5 is then 8.98 × 1.80 = 16.164 times the total number of thermal factors shown in Column 4.

The calculated transition points are all in agreement with the empirical curves within the margin of uncertainty in the location of these curves. As can be seen by comparing the calculated solid end points with the melting points listed in the last column, the end point values are also within the range of deviation that is theoretically explainable on the basis of discrete values of the liquid temperatures. It is quite possible that there is some “fine structure” involved in the thermal relations of solid matter that has not been covered in this first systematic theoretical treatment of the subject. Aside from this possibility, it should be clear from the contents of this and the two preceding chapters that the theory derived by development of the consequences of the postulates of the Reciprocal System is a correct representation of the general aspects of the thermal behavior of matter.