In the works [12, 15] we could have derived handy mathematical expressions for the entropy, internal energy—and therefore for the Helmholtz free energy—as functions of the absolute temperature and demonstrated that the actual statistical properties of the systems under consideration ought to obey the Beta probability distribution. This way it is throughout possible to treat mathematically formally and rigorously the ubiquitous fuzzy aspects of the problem under study.

Meanwhile, the next important theoretical problem is to how might we treat further important thermodynamic functions, namely, the enthalpy—and therefore the Gibbs free energy?

It is well known [16, 17] that enthalpy is connected with the heat capacity at constant pressures, C_p, which might well be approximated by C_V, but solely at low and intermediate temperatures, whereas at higher temperatures there are always discrepancies between the both. The actual reason for such a behavior has been thoroughly studied both empirically and computationally, and the point is that C_V gets anyway saturated at approximately Dulong-Petit level, whereas C_p is still noticeably temperature-dependent (see e.g., [1–3, 6, 18, 19] and the references therein).

This urges us to look in for the physical sense of the enthalpy notion in much more detail.

The interconnections between the C_V and C_p are well known as well (cf. e.g., [20]). Specifically, to work with an intensive property, we first define the specific heat capacity

Here C stands for the heat capacity of some physical-chemical system of interest, being built-up by a definite material; m is the mass of this system, whereas ρ and V are the corresponding density and volume, respectively. With this in mind, we might speak of the following basic relationship, first of all:

Here α stands for the thermal expansion/compression coefficient, whereas β_T is isothermal compressibility, p–pressure, T–absolute temperature. Moreover, there is ratio c_p/c_V as well

Here β_S stands for the adiabatic/isentropic compressibility, S is the entropy.

Further, in general terms, the enthalpy change ought to be expressed as follows:

Taking into account (Equations 1–3), we get:

Of considerable interest would be evaluating how enthalpy is dependent on the absolute temperature. With this in mind, first we would like to neglect the pressure dependence of enthalpy, in considering the case of the ideal gas. Otherwise, to avoid working with basically ideal systems, we might wish to consider some realistic isobaric situation. Then, dp = 0, implying that in our considerations we might control from outside the pressure imposed on our system, and wish now to ensure that the pressure imposed on our system is constant. This way, we might recast Equation (5) as follows:

As we do know from Linhart's work what the temperature dependence of c_V should be, of our primary interest is to correctly assign the temperature dependences of both the isothermal and the adiabatic compressibility.

In the literature we might fetch an extended row of papers dealing with this topic thoroughly and in detail (cf. e.g., [21–42] and the references therein). After taking into account this entire information, we dare to conclude that whereas the adiabatic compressibility might largely be considered a linear function of temperature, the isothermal compressibility is dependent on temperature in a much more non-trivial way, namely:

In these relationships, the temperature is in Kelvin, whereas isothermal compressibility is in (J · cm⁻³)⁻¹.

The work [26] shows that the both relationships given by Equation (7) properly guarantee satisfactory fitting the relevant experimental results, while using the exponential expression leads to noticeably better results than employing the algebraic equation. On the other hand, to our mind, the exponential functional dependence is much more difficult to treat theoretically, while the latter one enables us to evaluate and analyze the anti-derivative of Equation (6) in a straightforward way.

This is why we employ the algebraic formulation in our present work, implying that it delivers a handy functional approximation to the exponential one. That the latter one is possessed of the better numeric properties renders our conclusions a rather reasonable approximation.

With this in mind we shall present here our result, with sending the readership to the mathematical Appendix here for all the necessary technical details.

To sum up, we adopt here the relative absolute temperature like in our work [12], x ≡ T/T_ref, where T_ref is some reference temperature, the physical sense of which we shall clarify. Bearing this in mind; we arrive at the following result for the isobaric temperature dependence of enthalpy, H_p (“₂F₁” denotes below the conventional Gaussian hypergeometric function ₂F₁ (a, b; c, z)):

Analyzing Equation (8) enables to recast it in the following approximate form (see the Appendix in Supplementary Material for the technical details):

One striking feature might be of immediate interest, namely the similarity of the functional expressions of the enthalpy and entropy's temperature dependencies. Indeed, Linhart's result for the entropy's temperature dependence dictates [1–3, 6, 12] that

Here C_∞ stands for the limiting heat capacity (C_∞ ≅ 3N_Ak_B ≡ 3R, where N_A is the Avogadro number, k_B−the Boltzmann's constant and R−the universal gas constant) [1–12]. Figure 1 depicts typical behavior of entropy and enthalpy vs. temperature, and one can immediately recognize the convergent behavior of the both up to the value of x = 1, after which the graphs diverge.

Hence, it is throughout possible to interpret this as an enthalpy-entropy compensation (EEC), whereas x = 1, that is, T = T_ref defines the so-called “enthalpy-entropy compensation temperature” T_c, that is, T_ref ≡ T_c (see e.g., [43–52] and the references therein).

Noteworthy, the approximate considerations here (cf. Figure 1) support our conclusion that the generic functional form of the EEC ought to be rather non-linear, although linear function still appears to be a rather good approximation for the EEC anyway (for details, see [44, 45] and the references therein).

Howbeit, in the work [12] we discussed a possibility of interpreting the temperature values T_ref as points of some “hidden zero-order phase transition.” Such a conclusion is firmly supported by finding a discontinuity at x = 1 in the temperature dependencies of the internal energy U(T) and, accordingly, in the Helmholtz free energy F(T), without revealing it in the temperature dependencies of both C_V(T) and S(T).

Interestingly in this respect, according to our isobaric analysis above, the temperature dependence of the Gibbs' free energy G(T) does not exhibit any discontinuities for the whole range of x values from zero to unity:

Noteworthy, this is a very interesting result, for our earlier elementary thermodynamic consideration of the EEC phenomenon reveals a definite interrelationship between the Helmholtz' and Gibbs' free energies [51].

Meanwhile, the both free energy types are possessed of quite different mathematical properties, and therefore ought to reveal two quite different sides of one and the same phenomenon.

Indeed, the first idea coming to mind would be that although the enthalpy notion does contain the internal energy notion, adding the pV product to the latter ought to remove the discontinuity in the U as a function of T. This means that the absolute temperature dependence of the pV product does somehow compensate the discontinuity in the U(T). In more mathematical and physical detail we would like to analyze and discuss this point in the mathematical Appendix in Supplementary Material.

This train of thoughts brings us immediately to the old and good poser of writing the equation of state, as here we have just arrived at the fundamental problem of properly describing correlations among volume/density, pressure and temperature.

After reading the above headline our readership might immediately exclaim: Stop, there is but still no universal equation of state, see any serious handbook of thermodynamics (e.g., [53]).

Indeed, we open this book and read in its Chapter 6 on the Page 280 as follows:

Without going in for answering the question as to “what makes it so difficult to develop such an equation?”—This exciting topic will be discussed in a separate publication—we just give here the pointer to the ingenious works by Swedish colleagues Dr. Nils Engelbrektsson (1875–1963) and Karl Alexius Franzén (1882–1967), who had answered this poser quite positively and very constructively. Specifically, Dr. Engelbrektsson could have inferred such an equation by analyzing in detail the fundamental principles of thermodynamics, whereas Mr. Franzén could have successfully checked Dr. Engelbrektsson's inferences experimentally and proven their complete validity. Back in 1920 they both have published their seminal results as a separate book in Swedish, but until now nobody had anywhere come to pay attention to this publication. Dr. Engelbrektsson had devoted his whole life in working on the formulation of the basic thermodynamics, being also an interesting publicist as a hobby aside of this. His works might now be fetched in the Swedish National Library, and we would greatly appreciate presenting here his full publication list [54–68].

May the above be considered an answer to our poser in such a case? In our opinion, not to 100%, for Dr. Engelbrektsson was not studying interrelationships between thermodynamics and statistical mechanics. Instead, Dr. Linhart could have solely managed to show the way of developing the statistical mechanics in connection with thermodynamics, and thus duly pursuing Prof. Dr. J. W. Gibbs' train of thoughts. This is why; to answer the above poser we would actually need bridging the gap between Linhart and Engelbrektsson's approaches.

Still, our picture would be largely incomplete, if we would not mention here the works by another outstanding theoretical physicist, who was working in Germany: Prof. Dr. Max Bernhard Weinstein (1852–1918). Prof. Dr. Weinstein is presently known solely as a harsh censor of A. Einstein's approach to the relativity theory, whereas in fact, he was one of the serious and productive multitalented workers in the fields of theoretical metrology, physics and natural philosophy. Here we would like to present the full list of his works (the publications in the periodic media as well as monographs, including both his original works and German translations of the classical physical treatises) [69–101]. As it is apparent from his publication list, among other important problems, Prof. Dr. Weinstein was productively working on the formulation of the equation of state. These works by him have been noticed and taken into detailed consideration by Dr. Engelbrektsson.

Apart from the above, of immense interest for us here ought to be Prof. Dr. Weinstein's immense and productive efforts in the fields of thermodynamics and statistical mechanics' foundations. Specifically, to be mentioned are Prof. Weinstein's criticism of the then vogue in connection with the so-called “Third Basic Law of Thermodynamics” and the reaction to it from the scientific community [102, 103], as well as Prof. Weinstein's grounded, constructive criticism of Ludwig Boltzmann's approach to the statistical thermodynamics [104] (noteworthy, Prof. Weinstein has published this work under the pseudonym of “Mr. F. Cohen”), as well as the immediate reaction to this publication by Max Planck (revealing but not much more than his outrage and voluntarism in this particular case) [105].

Meanwhile, as we know now, Dr. Linhart could mathematically formally infer the famous, but mostly guessed Boltzmann-Planck formula S = k_B· ln W. He had demonstrated that revealing the important details of the function S ≡ S(T), where T is the absolute temperature, leads to the conclusion that S(0) = 0. As it is throughout possible to infer the latter fact in the mathematically rigorous way, it immediately loses its truly fabulous context of some Basic and Fundamental Natural Law…This interesting theme will be discussed elsewhere in detail.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.