CHAPTER 6Specific Heat PatternsFig.5 is a specific heat curve derived from experimental data. The points shown in this graph are the measured values of the specific heat of silver. The accompanying solid lines are the segments of the theoretical fourunit curve of Fig.4 with the temperature scale located empirically. While the curve defined by the plotted points has the same general shape as the theoretical curve, it is quite different in appearance inasmuch as the sharp angles of the theoretical curve have been replaced by smooth and gradual transitions. The explanation of this difference lies in the manner in which the measurements are made. As indicated by equation 58 and the curves in Figs.3 and 4, the specific heat of an individual molecule can be represented by a succession of straight lines. Experimental observations, however, are not made on single molecules, but on aggregates of molecules, and the observed temperature of the aggregate is the average of many different individual molecular temperatures, which are distributed about the average in accordance with the probability relations. Midway between the transition points the relation between temperature and specific heat for most of the individual molecules is such that their specific heats lie on the same straight line in the diagram. The average consequently lies on the same line, and coincides with the true molecular specific heat corresponding to the average temperature. In the neighborhood of a transition point, however, the molecules that are individually at the higher temperatures cannot continue on the same line beyond the 3R limit, and must conform to a lower curve based on a higher number of rotating units. This operates to reduce the specific heat of the aggregate below the true molecular value for the prevailing average temperature. In the silver curve, Fig.5, for example, the true atomic specific heat at 75º K is 4.69. This would also be the average specific heat of the silver aggregate at that temperature if the silver atoms were able to continue vibrating on the basis of one rotating unit up to the point beyond which the probability distribution is negligible. But at a specific heat of 2^{¹/3} R (4.633) the vibration changes to the twounit basis. Those atoms in the probability distribution that have specific heats above this level cannot conform to the oneunit line but must follow a line that rises at a much slower rate. The lower specific heat of these atoms reduces the average specific heat of the aggregate, and causes the aggregate curve to diverge more and more from the straight line relation as the proportion of atoms reaching the transition point increases. The divergence reaches a maximum at the transition temperature, after which the specific heat of the aggregate gradually approaches the upper atomic curve. Because of this divergence of the measured (aggregate) specific heats from the values applicable to the individual atoms the specific heat of silver at 75º K is 4.10 instead of 4.69. Figure 5: Specific Heat SilverA similar effect in the opposite direction can be seen at the lower end of the silver curve. Here the specific heat of the aggregate (the average of the individual values) could stay on the oneunit theoretical curve only if it were possible for the individual specific heats to fall below zero. But there is no negative thermal energy, and the atoms which are individually at temperatures below the point where the curve intersects the zero specific heat level all have zero thermal energy and zero specific heat. Thus there is no negative deviation from the average, and the positive deviation due to the presence of atoms with individual temperatures above zero constitutes the specific heat of the aggregate. The specific heat of a silver atom at 15º K is zero, but the measured specific heat of a silver aggregate at an average temperature of 15º K is 0.163. Evaluation of the deviations from the linear relationship in these transitional regions involves the application of probability mathematics, the validity of which was assumed as a part of the Second Fundamental Postulate of the Reciprocal System. For reasons previously explained, a full treatment of the probability aspects of the phenomena now under discussion is beyond the scope of this work, but a general consideration of the situation will enable us to arrive at some qualitative conclusions which will be adequate for present purposes. At the present stage of development of probability theory there are a number of probability functions in general use, each of which seems to have advantages for certain applications. For the purpose of this work the appropriate function is one that expresses the results of pure chance without modification by any other factor. Such a function is strictly applicable only where the units involved are all exactly alike, the distribution is perfectly random, the units are infinitely small, the variability is continuous, and the size of the group is infinitely large. The ordinary classes of events around which most presentday probability theory has been constructed, such as coin and dice experiments, obviously fail to meet these requirements by a wide margin. Coins, for instance, are not continuously variable with an infinite number of possible states. They have only two states, heads and tails. This means that a major item of uncertainty has become almost a certainty, and the shape of the probability distribution curve has been altered accordingly. Strictly speaking, it is no longer a true probability curve, but a combination curve of probability and knowledge. The basic physical phenomena do conform closely to the requirements of a system in which the laws of pure chance are valid. The units are nearly uniform, the distribution is random, the variability is continuous, or nearly continuous, and the size of the group, although not infinite, is extremely large. If any of the probability functions in general use can qualify as representing pure chance the most likely prospect is the socalled “normal” probability function, which can be expressed as
Tables of this function and its integral to fifteen decimal places are available.6 It has been found in the course of this work that sufficient accuracy for present purposes can be attained by calculating probabilities on the basis of this expression, and it has therefore been utilized in all of the probability applications discussed herein, without necessarily assuming the absolute accuracy of this function in these applications, or denying the existence of more accurate alternatives. For example, Maxwell’s asymmetric probability distribution is presumably accurate in the applications for which it was devised (a point that has not yet been examined in the context of the Reciprocal System), and it may also apply to some of the phenomena discussed in this work. However, the results thus far obtained, particularly in application to the liquid properties, favor the normal function. In any event it is clear that if any error is introduced by utilizing the normal function it is not large enough to be significant in this first general treatment of the subject matter. On the foregoing basis, the distribution of molecules with different individual temperatures takes the form of a probability function ø_{t}, where t is the deviation from the average temperature. The contribution of the ø_{t} molecules at any specified temperature to the deviation of the specific heat from the theoretical value corresponding to the average temperature depends not only on the number of these molecules but also on the magnitude of the specific heat deviation attributable to each molecule; that is, the difference between the specific heat of the molecule and that of a molecule at the average temperature of the aggregate. Since the specific heat segment from which the deviation takes place is linear, this deviation is proportional to the temperature difference t, and may be represented as kt. The total deviation due to the ø_{t} molecules at temperature t is then ktø_{t}, and the sum of all deviations in one direction (positive or negative) may be obtained by integration. It is quite evident that the deviations of the experimental specific heat curves from the theoretical straight lines, both at the zero level and at the transition point have the general characteristics of the probability curves. However, the experimental values are not accurate enough, particularly in the temperature range of the lower transition, to make it worth while to attempt any quantitative correlations between the theoretical and experimental results. Furthermore, there is still some theoretical uncertainty with respect to the proper application of the probability function that prevents specifying the exact location of the probability curve. The uncertain element in the situation is the magnitude of the probability unit. Equation 61 is complete mathematically, but in order to apply it, or any of its derivatives, to any physical situation it is necessary to ascertain the physical unit corresponding to the mathematical unit. One pertinent question still lacking a definite answer is whether this probability unit is the same for all substances. If so, the lower portion of the curve, when reduced to a common temperature base, should be the same for all substances with the 1.32 initial level. On this basis, the specific heat of the aggregate at the temperature T_{0}, where the theoretical curve intersects the zero axis, should be a constant. Actually, most of the elements with the 1.32 initial level do have a measured specific heat in the neighborhood of 0.20 at this point, but a few others show substantial deviations from this value. It is not yet clear whether this is a result of variability in the probability unit, or reflects inaccuracies in the experimental values. Whether all of the curves with the same maximum deviation (0.20) are coincident below T_{0} is likewise still somewhat uncertain. There is a greater spread in the observed specific heats below 0.20 than can be ascribed to errors in measurement, but most of the scatter can probably be explained as the result of lack of thermal equilibrium. At these low temperatures it no doubt takes a long time to establish equilibrium, and even an accurate measurement will not produce the correct result unless the aggregate is in thermal equilibrium. It is significant that the specific heats of the common elements which have been studied most extensively deviate only slightly from a smooth curve in this low temperature region. Fig.6, which shows the measured values of the specific heats of six of these elements on a temperature scale relative to T_{0}, demonstrates this coincidence. If the probability unit is the same for all, or most, of the elements, as these data suggest, the deviation of the experimental curve from the theoretical curve for the single atom at the first transition point, T_{1}, should also have a constant value. Preliminary examination of the curves of the elements that follow the regular pattern indicates that the values of this deviation actually do lie within a range extending from about 0.55 to about 0.70. Considerable additional work will be required before these curves can be defined accurately enough to determine whether these is complete coincidence, but present indications are that the deviation at T_{1} is, in fact, a constant for all of the regular elements, and is in the neighborhood of three times the deviation at T_{0.} Figure 6: Specific Heat Low TemperaturesWith the benefit of the foregoing information as to the general nature of the deviations from the theoretical curves of Chapter 5 due to the manner in which the measurements are made, we are now prepared to examine the correlation between the theoretical curves and the measured specific heats. In order to arrive at a complete definition of the specific heat of a substance it is not only necessary to establish the shapes of the specific heat curves, the objective at which most of the foregoing discussion is aimed, but also to define the temperature scale of each curve. Although the theoretical conclusions with respect to these two theoretical aspects of the specific heat situation, like all other conclusions in this work, are derived by developing the consequences of the fundamental postulates of the Reciprocal System of theory, they are necessarily reached by two different lines of theoretical development. For this reason a more meaningful comparison with the experimental data can be presented if we deal with these two aspects independently. In this chapter, therefore, the experimental values will be compared graphically with theoretical curves in which the temperature scales are empirical. Chapter 7 will complete the definition of the curves be deriving the relevant temperature magnitudes. The curves of Fig.7 are typical of those of most of the elements.7 As indicated in Fig.4, the final straight line segment of each curve occupies the greater part of the temperature range of the solid state in the case of the high melting point elements. The significant features of the curves are therefore confined to the lower temperatures, and in order to bring them out more clearly only the lower temperature range (up to 300º K) is shown in the illustrations that follow. the remaining sections of the curves of Fig.7 are extensions of the lines shown in the diagram, except in the case of tungsten, which undergoes a transition to the fourunit status at about 325º K. Figure 7: Specific HeatFig.8 is a similar group of specific heat curves for four of the electronegative elements with the 0.66 initial level. Aside from this higher initial level these curves are identical with those of Fig.7 when all are reduced to a common temperature scale. The transition to the twounit vibration takes place at 4.63 (2_{¹/3} R) regardless of the higher initial level. This point will be given further consideration in Chapter 7. The upper portions of the lead and antimony curves, which are not shown on the graph, are extensions of the lines in the diagram. Arsenic and silicon have transitions at temperatures above 300º K.

Internal
Thermal Units 
1st 
Initial Levels 
2nd 
Specific Heat
at T_{1} 

Propane 
0

2.64

2.64

9.27


Butane 
0

2.64

2.64

9.27


Pentane 
2

2.64

6.62

13.90


Hexane 
3

2.64

7.95

16.22


Heptane 
4

3.96

9.27

18.54


Octane 
5

3.96

10.59

20.86


Nonane 
6

5.30

11.92

23.18


Decane 
7

5.30

13.24

25.49


Hendecane 
8

5.30

14.57

27.81


Dodecane 
9

6.62

15.89

30.12


Tridecane 
10

6.62

17.22

32.44


Tetradecane 
11

6.62

18.54

34.76


Pentadecane 
12

6.62

19.86

37.08


Hexadecane 
13

7.95

21.19

39.39

Propane and butane have only the two molecular thermal groups corresponding to the positive and negative ends of the molecules, and their specific heat at T_{1} is the normal twogroup value: 9.27. Beginning with two internal groups in pentane, each added CH_{2} structural group becomes and internal thermal unit, and adds 2.317 to the total specific heat of the molecule at the transition point. The initial level of the first segment of the specific heat curve is 2.64 (the twogroup value) in the lower compounds, and changes slowly, adding units of 1.32, as the length of the chain increases. The initial level of the second segment is 2.64 in butane and propane. In the higher compounds, each of which consists of n structural groups (CH_{2} and CH_{3}), this second initial level is 1.324 n.
The values thus derived theoretically are all consistent with the experimental curves. In a few cases the intersection of the two curve segments may not coincide with the calculated specific heat of the transition point, but these deviations, if they are real, are small enough to be explainable on the basis of changes in the temperature factors, the nature of which will be one of the subjects of discussion in Chapter 7.
Branching of a hydrocarbon chain tightens the structure and tends to reduce the number of internal thermal units. For example, octane has five internal thermal units, and a specific heat of 20.86 at the transition point. But 2,2,4trimethyl pentane, a branched compound with the same composition, has no internal motion at all, and the T_{1} specific heat of this compound is 9.27, identical with that of the C_{3} paraffin, propane. Ring formation has a similar effect. Ethylbenzene and the xylenes, which are also C_{8} compounds, have some internal motion, but their T_{1} specific heats are 11.59 (one internal unit) and 13.90 (two internal units) respectively, well below the octane level. In Fig.13 the specific heat curves of hexane (straight chain) and benzene (ring), both C_{6} hydrocarbons, are contrasted.
The subject matter of this and the preceding five chapters consists of various aspects of the volumetric and thermal relations of material substances. The study of these relations was the principal avenue of approach to the clarification of basic physical processes that ultimately led to the identification of the physical universe as a universe of motion, and the determination of the nature of the fundamental features of that universe. There relations were examined in great detail over a period of many years, during which thousands of experimental results were analyzed and studied. Incorporation of the accumulated mass of information into the theoretical structure was the first task undertaken after the formulation of the postulates of the Reciprocal System of theory, and it has therefore been possible to present a reasonably complete description of each of the phenomena thus far dicussed, including what we may call the smallscale effects.
Beginning with the next chapter, we will be dealing with subjects not covered in the inductive phase of the theoretical development. In this second phase, the deductive development, we are extending the application of the theory to all of the other major fields of physical science, in order to demonstrate that it is, in fact, a general physical theory. Obviously, where the area to be covered is so large, no individual investigator can expect to carry the development into great detail. Consequently, some of the conclusions expressed in the subsequent pages with respect to the smallscale features of the areas covered are subject to a degree of uncertainty. In other cases it will be necessary to leave the entire smallscale patter for some future investigation.