The Melting Point - Relation to Pressure
Attached herewith is paper number ten in my liquid series which shows that the melting points under pressure can be accurately computed by means of a very simple equation. This paper makes a highly significant contribution to the work as a whole, as the melting point equation is not only a purely theoretical expression derived directly from the basic postulates of the work, but it is also the same equation that was used to calculate solid compressibilities in previous papers. It is difficult to visualize the possibility that such an equation could successfully reproduce the full range of experimental values in either field unless it were an essentially correct representation of the physical facts; the possibility that an incorrect expression could accomplish these results in both fields is wholly inconceivable.
It appears to me that this occasion, on which I am submitting some striking new evidence of the validity of the fundamental postulates on which my work is based, is an appropriate time to make some comments regarding those aspects of currently accepted physical theory with which my findings are in conflict. It is true; of course, that these conflicts are quite numerous, and there is a very understandable reluctance to believe that accepted theory can be wrong in so many important respects. But it is not difficult to prove this point. It is quite a task to prove that any of the current theories are wrong, because long years of effort on the part of the theorists have enabled them to adjust their theories to fit the facts in most cases and to devise means of evading most of the many contradictions that still remain, but it is quite simple to show that the particular theories in question are lacking in factual support and therefore can be wrong. As long as this is true, it is clearly in order to consider conflicting theories.
The case in favor of my new theoretical structure rests primarily on the contention that not a single one of the thousands of necessary and unavoidable consequences of my fundamental postulates is inconsistent with any positively established fact. It becomes very important, therefore, to distinguish clearly between items that are definitely known to be factual and those, which do not qualify. The enclosed memorandum is a discussion of this situation as it applies to the points of conflict. The memorandum does not attempt to discuss all of the points at issue, nor to make an exhaustive analysis of any case; nothing short of a book-length presentation could cover that much ground. However, the various categories of pseudo-facts commonly encountered are described and illustrated by examples, and it should not be difficult to see that every one of the accepted ideas with which my findings are in conflict is one of these pseudo facts: a hypothesis, assumption, or extrapolation masquerading in the guise of an established fact.
Perhaps it will be regarded as presumptuous for me to suggest that the strict and critical tests which will be applied to my new theory be applied to the conflicting parts of currently accepted theory as well, but I believe that by this time the new developments which I have covered in the liquid papers add up to an impressive enough total to justify a full and comprehensive examination of the underlying theories, including a careful consideration of this question as to the true status of conflicting ideas.
D. B. Larson
Strictly speaking, the melting point is a phenomenon of the solid state rather than of the liquid state and in general its behavior follows the solid state pattern, but since this temperature constitutes the point of transition between the two physical states it has some close relations with various liquid properties which justify giving it some consideration in this survey of the liquid state,
If the melting point were actually a property of the liquid, we could expect that it would be linearly related to the pressure, just as the true liquid component of the volume and other liquid properties are so related. Solid state properties, however, are directly or inversely related to the square root of the pressure, rather than to the first power, for reasons that are explained in the author's book "The Structure of the Physical Universe", which develops the general physical theories underlying the liquid principles that form the basis for this series of papers. It has already been brought out in a paper which preceded this liquid series5 that the solid state equivalent of Bole's Law, PV = k, is PV2 = k, from which we obtain the relation V/V0 = P01/2(P +P0)1/2. In terms of density this becomes
The previous solid compressibility paper showed that this equation is able to reproduce the experimental values of the compression, within the existing margin of uncertainty, over the entire range of temperatures, pressures and substances for which data are available.
According to the basic theory, this is a characteristic relationship applying to solid state properties in general, rather than merely a volume relation, and we may therefore rewrite the equation, substituting Tm, the normal melting point, for d0, and Tp, the melting point under an externally applied pressure P, for d. We then have
It will be noted that this expression is identical with the well-known Simon melting point equation, except for the substitution of a constant for one of the two variables of the Simon equation, which the originator expresses as
Simon's factor a corresponds to the initial pressure, as is generally recognized, His factor c (unexplained theoretically) has usually been assigned a value somewhere between 1.5 and 2 in application to organic compounds and other low melting point substances, and replacing this variable factor by the constant value 2 is not a major modification of the equation, so far as the effect on the calculated values is concerned, as these values are not very sensitive to changes in c if accompanied by corresponding changes in a. The situation with respect to the high melting point elements, which are often assigned considerably higher values of the factor c. will be discussed later.
No satisfactory theoretical explanation has thus far been discovered for the Simon equation, but this expression agrees with the experimental results over a wide range of pressures and substances, and it is generally conceded that such a theoretical explanation must exist. Strong and Bundy state the case in these words, "Simon's fusion equation has now endured a considerable amount of experimental and theoretical examination. Because it applies in so many cases... it must contain fundamentally correct concepts concerning some of the properties of matter".50
In the areas previously covered by this series of papers, the new equations, which have been developed, are to a large degree filling a vacuum, as no generally applicable mathematical representations of these properties have hitherto been available. It is therefore quite significant that when we reach an area where an equation of recognized standing does already exist, the new development does not produce something totally new; the general liquid principles on which the work is based lead to a melting point expression which is essentially a modified form of the previously existing equation. Here, as in so many places outside the liquid field, genuine knowledge already in existence coincides with the products of the development of the postulates of this work, and can simply be incorporated into the new theoretical structure with nothing more than minor modifications. What the new development actually does, in essence, is to establish the exact nature of those "fundamentally correct concepts" to which Strong and Bundy refer.
But even though the required modification of the Simon equation is minor, it does not necessarily follow that it is unimportant. As brought out in paper IX of this series, the more restrictive the mathematical expression of a physical property can be made, the more likely it is to be a correct representation of the true physical facts, providing, of course, that it produces results which agree with the experimental values within the margin of uncertainty of the latter. Replacement of one of the two adjustable factors in the Simon equation by the constant value 2 as required by the new theory is an important move in this direction.
Now that this value has been fixed, the only additional requirement for a complete and unequivocal definition of the pressure-melting point relation for each substance is a means of calculating the initial pressure applicable in each case. In the solid compressibility paper previously mentioned, the following equation for the initial pressure applicable to compression of solids was developed:
The initial pressure applicable to liquid compressibility is considerably lower and paper IV in this series expressed this relation as
Since the melting point is at the boundary between the liquid and solid states, it is to be expected that the initial pressure applicable to this property will lie somewhere between the true liquid and true solid values, and a study of this situation leads to an equation of an intermediate type:
The symbols in this equation have the same significance as in the expressions for the true liquid and true solid initial pressures. The factors a, b, and c are the effective displacements in the three dimensions of space, a concept that is explained in the author's book previously mentioned. V0 is the initial specific volume of the liquid as defined in paper II of this series, and n is the number of independent units in the molecule at the melting temperature.
The values of n applicable to the solid-liquid transition are usually less than those applicable to liquid compressibility, as would be expected since the number of effective units per molecule is normally less in the solid, particularly at low temperatures, than it is in the liquid, and an intermediate value is appropriate for the boundary state. There is also a marked tendency toward a constant value in each of the various homologous series of compounds, at least in those portions of these series for which experimental data are available. Thus the value for most of the aliphatic acids is 4, and for the normal alcohols it is 3. Most elements have n = 1, the principal exceptions being such elements as sulfur and phosphorus which have quite complex liquid structures.
In the majority of substances on which experimental results are available for comparison including most of the common organic compounds, the a and c factors take the theoretical maximum values 4 and 8 respectively. The factor b is usually 1 at low pressures, except for the elements in the middle of each periodic group, which have the same tendency toward higher values that was noted in the case of solid compressibility.
Some of the low melting point elements have a-c-b values at or near the theoretical minimum, a point which is of particular interest; first, because it provides a definite reference point for these factors which helps to demonstrate that they have a real physical significance, and second, because the wide spread between the possible values of the factors at the lower end of the scale makes identification of the applicable factors a very simple matter. Helium, for example, takes the minimum values, 1-1-1. The next higher combination that is theoretically possible 1½-1-1 would result in a reduction of more than 20 percent in the melting temperature at the upper end of the experimental temperature range. This is, of course, far outside the margin of experimental uncertainty, which is normally in the neighborhood of one or two percent and the 1-1-1 factors are therefore definitely the ones that are applicable.
The situation with respect to the other elements of very low melting point is similar, and the theoretical melting point pattern for these substances is therefore positively established. It does not necessarily follow, however, that the divergence between the experimental melting points and the values thus calculated is always chargeable to experimental error. The theoretical values are those which would result from the application of pressure only, without any "second order" effects such as those due to the presence of impurities, to consolidation of molecules under pressure, to polymorphic transitions, etc, and they will not necessarily coincide exactly with the results of accurate measurements made on a substance which is subject to extraneous influences of this kind.
It should also be recognized that correlation of the theoretical and experimental values is not as simple a matter in the melting point field as it is for a property such as surface tension, on which we have a large volume of reasonably accurate experimental data. Only a comparatively small amount of work has been done on the melting curves, and most of that has been confined to the range below 1000 atm. Outside of the recent work with the metallic elements and the elements of very low melting point, Bridgman's investigations are practically the only source of information at the higher pressures. This, of course, introduces some serious uncertainties into any correlations that we may attempt. If the calculated and experimental values agree, each serves to some extent as a corroboration for the other, but where there is a divergence it is not immediately apparent which of the two is in error.
The correlations of theory against experiment in the areas covered by previous papers in this series have been of the, massive type, Calculations have been carried out for hundreds of substances of many different classes and, although it has not been possible to show all of these data in the tabulations accompanying the papers, a reasonably good sample has been included in each case. Where the experimental data are scarce and to a large degree unconfirmed 2 as in the present instance) it will be necessary to use a more selective technique, and to examine the evidence of the validity of each phase of the theoretical relationship separately, rather than verifying the entire development in one operation by a massive demonstration of agreement with the results of observation.
The first point, which we will want to consider, is the validity of the square root relationship. For this purpose the most significant experimental results are those in which the percentage increase in the melting point is the greatest. Where the ratio of the melting point at the upper end of the experimental pressure range to the normal melting point is less than 2, the difference between a square root curve and some other possible exponential curve, or a linear curve, is small, and unless the experimental values are extremely accurate it is difficult to determine which relation these values actually follow. The divergence of the curves increases rapidly, however, as the ratio rises, and if this ratio is 4 or more the nature of the curve is readily ascertained.
For this particular purpose, therefore, the work at the highest pressures is of no particular value. Even where pressures in the neighborhood of 150,000 atm. have been reached in the study of the metallic elements, the corresponding melting point ratio is only about 1.2. In the range from 1.0 to 1.2 the difference due to even a fairly large change in the exponent of the melting point expression is negligible. It is not surprising, therefore, that there is much difference of opinion as to just what this exponent should be. In the case of iron, for example, Gilvarry arrives at an exponent of 1.9 for the Simon formula, Simon himself selects 4, and Strong gives us the value 8 (which corresponds to 1/8 on the basis of equation 16).51 On first consideration this seems to be an extreme case of disagreement, but if the value of Simon's constant a is adjusted empirically (as is always done, of course), the differences between these various exponential curves in this range are so much less than the experimental uncertainty that the curves are for all practical purposes coincident. For example, the square root of 1.1 is 1.0488 and the square root of 1.2 is 1.0954. If the curve from 1.0 to 1.2 were linear, the value at 1.1 would be 1.0477, which differs from 1.0448, the value on the square root basis, by only one-tenth of one percent. A similar calculation using an exponent of 1/8 (equivalent to Strong's value 8) shows that the deviation from the linear curve is still less, only about ½0 of one percent. Where the normal melting point is in the vicinity of 20000, a change from Gilvarry's exponent 1.9 to Strong's exponent 8 changes the position of the midpoint of the curve only about one degree if the value of a is fitted to the maximum experimental value of the melting temperature. It is therefore clear that within the experimental pressure range all of the exponents selected by previous investigators are in agreement with each other and with the exponent of equation 16, But where the experimental data fit everything they prove nothing.
The definite verification of the square root relationship is furnished by the elements of very law melting point, the most conclusive demonstration coming from helium and hydrogen. Helium does not melt at all except under pressure, and its melting curve cannot be referred to the normal melting point in the usual manner, but a study of the situation indicates that the melting point of this element can be calculated from equation 16 by the use of a pseudo-melting point which has been evaluated empirically as 11.1° K. We first calculate the melting point under pressure just as if 11.1° were the normal melting point, and then we subtract 11.1° from the result. At 5000 atm. for example, we find that the quantity (P +P0)1/2/P01/2 amounts to 4.516. Multiplying by 11.1° we obtain 50.13°, and subtracting 11.1° we arrive at a theoretical melting point of 39.03° K. A measurement at this pressure is reported as 39° K. If the value of the expression
is less than 1.0 (that is, if the true melting point is below the pseudo-melting point), the true melting point is proportional to the 2/3 power of the foregoing expression instead of the first power. The reasons for this behavior are not clear, although it is not surprising to find that the values below the reference temperature, which correspond in some degree to negative temperatures, are abnormal, Table X-1 compares the calculated and experimental melting points of helium. Here we see that although the maximum pressure of observation is only 7270 atm, the melting point ratio (designated as R in this and the following tables) at this pressure is 4.4, which is well above the minimum requirement for positive identification of the nature of the melting curve.
Also included in this table are the values for He3 which are computed in the same manner, except that the pseudo-melting point is slightly higher, 11.5º K, and the 5/6 power is substituted for the 2/3 power below the pseudo-melting point. It will be noted that for both isotopes the differences between the theoretical and experimental values are abnormally high in the vicinity of the pseudo-melting point. This is a mathematical effect of the distribution of molecular velocities in the neighborhood of a transition point, similar to the effect on the fluidity values discussed in paper VIII, page 3, and rough calculations indicate that when the transition is studied in detail so that the proper corrections for this effect can be determined, the agreement at these temperatures will be found just as close in this range as it is where the transition effect is absent.
Aside from hydrogen, which will be discussed later, there is no other substance on which the melting curve has been followed farther than a ratio of about 2.5. In the range from around 2.0 to 2.5 we find such substances as nitrogen, which shows an agreement within 1º over the full experimental pressure range (if we use Bridgman's values up to his pressure limit); carbon tetrachloride, which agrees within 1º to 6000 kg/cm², with somewhat larger deviations above this pressure ethyl bromide, which agrees within 2º to 25,000 kg/cm² and shows a deviation of 5º at 30,000 kg/cm², beyond what appears to be a transition of some kind; chloroform, which agrees with the results of one set of measurements by Bridgman to within 2º but differs substantially from another set of results; and two of the normal alcohols, ethyl and butyl, for which the agreement is within 4º and 3º respectively up to 25,000 kg/cm², beyond which the values diverge. It is not clear whether this divergence is due to experimental error or to a transition to a new value of P0 similar to the transitions, which were found in the study of compressibility.
The calculated and experimental melting points for these substances are listed in Table X-2,. Also included in this table are values for a few other substances which have been observed up to 11,000 kg/cm² or higher) but only to melting point ratios between 1.25 and 2.0, Even though the information available within the range of significant melting point ratios is quite limited, the comparisons in this table should be sufficient to add considerable weight to the conclusions reached on the basis of the helium values
The melting curves of a large number of substances have been determined with precision to pressures in the neighborhood of 1000 atm. For reasons previously discussed, these determinations are of no value from the standpoint of verifying the square root relation, but now that the validity of this relation has been confirmed by other means, the values in the lower pressure range can be utilized as a test of equation 17, and a number of comparisons of melting points in this range are given in Table X - 3. Since all other factors that enter into the determination of the melting points of the common organic compounds are definitely fixed, identity of the values of n applicable to related compounds) or obvious regularities in the values for such compounds, are strong evidence of the validity of equations 16 and 17 and of the theoretical principles from which these equations were derived. For example, melting points for the first seven of the aliphatic acids computed on the basis of n = 4 agree with the experimental results within 1° in four cases, and in only one of these compounds (propionic acid) is there any deviation as large as 3°
Benzene and some of its simple derivatives contribute additional evidence of the same kind. Benzene itself has n = 5, and a large percentage of the closely related compounds for which melting curves are available take the same values of this factor. Among these are toluene, two of the xylenes, naphthalene, benzophenone, nitrobenzene, and two of the nitrobenzene. Representative examples of both the benzenes and the aliphatic acids are included in Table X-3.
The data for water, Table X - 4, are particularly interesting. The a-c-b values are 4-8-1 as usual, except that there is a transition to 4-8-2 between 8000 and 9000 kg/cm² affecting ice VI only. Disregarding the abnormal forms of ice that exist below 2000 kg/cm², we find that the entire melting point pattern of water, complex as it is, can be reproduced simply on the assumption that the factor n, the number of effective units in the molecule, increases step by step as we pass from one form of ice to the next: ice III - 1, ice V - 2. ice VI - 3, ice VII - 4. (The status of ice IV is questionable; it may not even exist). Except in the range from 20,000 to 28,000 kg/cm², where the effects of a polymorphic transition that takes place at 22,400 kg/cm² are in evidence, the agreement between the calculated and experimental values is within 2º in all cases.
The increase in the factor b from 1 to 2 which was found at a pressure of approximately 9000 kg/cm² in ice VI is one of the very few instances where the existence of a transition of this kind, involving an increase in the initial pressure, appears to be definitely confirmed. The pattern of increase in this factor found in the study of solid compressibility suggests, however, that such transitions may be normal, and that their infrequent appearance in melting phenomena is merely due to the relatively narrow pressure range that has thus far been covered experimentally.
A similar transition in the opposite direction occurs in hydrogen and the inert gases above helium. Here we find that the initial a-c-b values are 1½-1-1, but subsequently these factors drop to the minimum level 1-1-1, What we may regard as the normal pattern for this transition is illustrated by krypton and xenon, Table X-5. In these elements the 1½-1-1 factors prevail up to 170º K (one-third of the liquid temperature unit, 510º). From 170º to 340º (two-thirds of the temperature unit) there is a linear transition to 1-1-1, and above 340º the effective factors remain at this minimum level. The experimental results on argon are erratic and inconclusive, but not inconsistent with values calculated on the same basis. Neon and hydrogen follow the same general pattern, but the transition temperatures are fractional values of those normally applicable. In the case of hydrogen, the transition begins at 28.3º K, one-sixth of the normal 170º, and is completed in 113.3º, two-thirds of the normal interval. Table X-6 compares the hydrogen melting points calculated on this basis with a set of values compiled from experimental data. As the table shows, the two sets of values agree within 1/4 degree up to the two highest pressures of observation, and even in these cases the difference is less than ½ degree.
Table X-7 lists the values of the various factors entering into the calculation of initial pressures and melting points for all of the substances included in the preceding tabulations.
5. Larson, D. B., Compressibility of Solids, privately circulated paper available from the author on request.
50. Strong, H. M., and Bundy, F. P., Phys. Rev., 115-278.
51. See discussion by Strong in Nature, 183-1381.
52. MiIls, R. L., and Frilly, E. R., Phys. Rev., 99-480. Numerical values supplied by Dr, Mills in private communication.
53. Holland, Huggill and Jones, Proc. Roy, Soc. (London), A 207-268.
54. Bridgman, P. W., various, For a bibliography of Bridgman's reports see his book "The Physics of High Pressure". 0. Bell & Sons., London, 1958. All values in Table X-2 are from Bridgman unless otherwise specified.
55. Robinson., D. W., Proc. Roy. Soc. (London), A 225-393. Butuzov, V. P., Doklady Akad. Nauk, S.S.S.R., 91-1083
57. Deffet, Ll, Bull, Soc. Chim. Belg., 44-41.
58. Lahr, P. H. and Eversolet W. 0., J. Chem. Engr. Data, 7-42.
59. Compiled from original sources by Wooley, Scott and Brickwedde, J. Res. N.B.S., 41-3790