The description of heat capacity by the Debye – Mayer – Kelly hybrid model

The universal Debye – Mayer – Kelly hybrid model was proposed for the description of the heat capacity from 0 K to the melting points of substance within the experimental uncertainty for the first time. To describe the heat capacity, the in-house software on the base of commercial one DELPHI-7 was used with a 95% confidence level. To demonstrate the perfect suitability of this model, a thermodynamic analysis of the heat capacities of the fourth group elements, and some compounds of the AB and AB phases was carried out. It produced good agreement within the experimental uncertainty. There is no a similar model description in literature. The Similarity Method is a convenient and effective tool for critical analysis of the heat capacities of isostructural phases, which was used as an example for diamond-like compounds. Phases with the same sum of the atomic numbers of elements (Z), such as diamond and B0.5 N0.5 (cub) (Z = 6); pure silicon (Si) and Al0.5 P0.5 (Z=14); pure germanium (Ge) and Ga0.5 As0.5 (Z = 32)); pure grey tin (alpha-Sn) and In0.5 Sb0.5, and Cd0.5 Te0.5 (Z = 50) have the same heat capacity experimental values in the solid state. The proposed Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 26 August 2020 doi:10.20944/preprints202008.0576.v1 © 2020 by the author(s). Distributed under a Creative Commons CC BY license. models can be used to both different binary and multicomponent phases. It helps to standardize the physicochemical constants.


Introduction
Despite the fact that significant experimental data on various physicochemical constants have been accumulated to date, the problem of standardizing these constants has arisen because mathematical modeling only exacerbates this problem. The concept of the relation of physicochemical constants of isostructural compounds with the Periodic Law is taken as a basis for this discussion.
Our previous review articles [1] evidenced the relation of thermodynamic data with Periodic Law and established a strict relationship between the enthalpy of formation, melting point and the atomic numbers of components in the semiconductor A III B V phases, with the diamond-like structure of sphalerite and wurtzite types.
The proposed model was used for the critical assessment of the thermodynamic properties of isostructural compounds. The relationship between the reduced enthalpy, standard entropy, reduced Gibbs energy and the sum of the atomic numbers (Zi = ZA + ZB) has been used for the critical assessment of the thermodynamic properties of A III B V phases. In this work, the Similarity Method was applied to the critical analysis of heat capacities in the solid state of the diamond-like phases. The relationship of the heat capacities of A III B V phases vs. the logarithm of the sum of atomic numbers of elements of the A III B V phases (sphalerite and wurtzite types) were used to estimate the continuum above 298 K [2]. In the present work, the Debye -Mayer -Kelly hybrid model was proposed for the first time. It allows one to describe the specific heat of solid phases from 0 K to melting points within experimental errors.

Relation of physicochemical constants of isostructural compounds using the Periodic Law.
The basis of this Law is the atomic number of the element (Z), which is equal to the number of protons in the atomic nucleus. This atomic number defines the chemical properties and most of the physical properties of the atom. This law can be extended and applied to different isostructural chemical compounds, both binary and multicomponent systems.
Let us consider, as an example, the heat capacity of diamond-like elements of the fourth group (C, Si, Ge, Sn) and their isostructural analogues of the A III B V and A II B VI phases ( 3 ) in the solid state over a wide temperature range from 0 K to their melting points.

Kelly Equation
The isochoric heat capacity Cv can be presented by the linear combinations of the Debye's functions [Debye,[3] and it is proposed to approximate the experimental data on the heat capacities of diamond-like phases in the form: In this case, the function To is an adjustable parameter that provides the smooth transition of heat capacity Cv to Cp in the equation (3) within a range from 1 to a coefficient b of the Mayer-Kelly equation.
The coefficients "a" and "b" were calculated with using the Mayer-Kelly equation: Cp' = a + bT -3 + cT -2 (4) So Cp extends to Cv at low temperature, and it extends to the Mayer-Kelly equation at high temperatures. The least-squares m ethod is one of the common methods for the calculation of the minimum errors of deviation of approximate functions from experimental data.
Here n is the number of experimental temperatures -heat capacity pairs of the selected phase (or phases).
To describe the heat capacity, the in-house software on the base of commercial one DELPHI-7 was used with a 95% confidence level.
The selected values of heat capacities of the fourth group (C, Si, Ge, -Sn and the analogue compounds AlP, GaAs, InSb, and CdTe, the range of temperatures and n number of the using points from the different references were collected in Tables 1-4.

Results
The experimental data for the specific heat of Cp' compounds of diamond-like phases (at temperatures above 240 K) were processed by the least square method to obtain the coefficients a, b and c in the Mayer-Kelly formula. This formula is not applicable below 1200K for the diamond and cubic boron nitride. The calculated values of a, b and c are given in Table 5. The calculated parameters a, b, To, A1, Q1, A2, Q2, A3, Q3, the number of experimental pairs (TK-Cp) and standard errors are given in Table 6.

Table 5. Parameters of Mayer-Kelly equations C'p = a + b10 -3 T -c10 5 T -2 (J/(mol-at) -1 K -1 ) at the high temperature region
The coordinate of the minimum () for the coefficients of the Debye's functions was taken from the regions of 0.01 < Aj <1, and 30 K < Qj<2000 K. The calculation was stopped when the heat capacity became equal to the mean-root-square error sigma ( The remaining equations are auxiliary.    Table 1-6); 790 points

Table 6. Parameters of equation (3) in the range 0 K -Tm K
The heat capacities of the fourth group phases (in J/mole-at K) obtained by using the Debye-Mayer-Kelly equation (3) are presented in Table 8 within the range of temperature 5 -300 K. (See the columns II, III-VI, and equations 1, 2, 4, 6, 8 from Table 6.) We also calculated the heat capacities of diamond and cubic boron nitride using the polynomial equation 1 (Table 7), which are presented in column III (Table 8).
The values of columns II and III are consistent with each other within the range of experimental error. The heat capacities above 300 K can be determined with equation (3) or Mayer-Kelly equations (4) ( Table 5).
5. Discussion. The heat capacity measurements and sources of errors.

Diamond and c-BN; graphite and h-BN
The wurtzite, hexagonal, rhombohedral, turbostratic) [34] contain various impurities due to the specifics of their synthesis [36].
So, it was found in [21], when studying the influence of the alloying additives on the heat capacity of germanium at temperatures of 20 -200 K, that the heat capacity of germanium with an aluminum content of up to 0.006 at.% gives an increase of 0.17 J/(mol-at) -1 K -1 at temperature of 100 K compared to pure germanium (See Fig.3).
. The heat capacity of diamond was studied on industrial samples with a content of 0.2 wt.
It was noted in [35] that cubic boron nitride used in the measurements contained up to 4 wt. % hexagonal boron nitride and up to 1% metallic impurities. According to [34], cubic BN included up to 0.15 wt. % impurities. Nevertheless, the low-temperature heat capacities of cubic BN [35] are closer to the heat curve of diamond than the data of [34].

Fig. 3.
Comparison of the heat capacity:  -pure Ge [21],  -Ge with 0.006 at %Al [21], line is a fitted Ge (Table 6). The recent calculated results of the specific heat of cubic BN by the Monte Carlo method [37] showed good agreement with the specific heat of diamond [4,5,6,10] in the range of 100-600K (see Fig.2). with a layer of graphite [38]. The presence of impurities can contribute to the catalytic graphitization of diamond. In [39] the kinetics of graphitization of thin diamond-like carbon (DLC) films was studied at 773K coated with Ni metallic nanoparticles [39].
Among the three low index planes, the {111} plane can be graphitized easily [39].
In [5] while measuring the heat capacity of diamond a silver container was used. Silver, its vapors and impurities up to 0.2% can contribute to the graphitization of diamond above 800 K. In Fig. 4, the heat capacity of diamond (1) with points + [5] intersects the heat capacity c-BN (2) [34] and goes into the heat capacity region of graphite above 800 K (compare with Fig. 1), which contradicts commonly accepted beliefs. The compiled heat capacity curve of the diamond was shown in Fig. 5. This curve of the diamond was described by equation (3), the parameters of which are presented in Table 6.
According to our concept, there should be no overlap of the heat capacity curves, since all isostructural phases form independent groups in accordance with the sum of the atomic numbers of the elements or the structure of the phases. Let us compare the heat capacities of graphite and hexagonal boron nitride [34], although the data [34] are unreliable due to the presence of impurities. According to our concept, hexagonal boron nitride should have a specific heat capacity similar to graphite. It is necessary to note that the data [34] of the heat capacities of cubic boron nitride begin to correspond to diamond and that of hexagonal boron nitride to graphite at temperatures above 1000K. In our opinion, the impurity of the samples ceases to affect the measured heat capacity due to their removal from the container at high temperatures and the transformation of a substance in a normal state.  Table 1 and 6)
Aluminum phosphide is a highly toxic substance; contact with water or moist air releases toxic phosphine [40]. That is why the study of its physicochemical properties is limited.
The heat capacity of aluminum phosphide was studded in [19], [20] at high temperature (See Table 2, 5, 6). We did not use the heat capacity data of the A III B V compounds [20] due to their high error. Nevertheless, in some cases, in the absence of other data, they can be used. So, the heat capacity data [20] showed satisfactory agreement when optimizing the Al-P system in [41]. per mole-atom.
The experimental points of the heat capacity of silicon Cp (T) in the solid state was described by equation (4) with precision = 0.11 J/(mol-at) -1 K -1 in the range of temperature 5 -2000 K. Parameters of this equation were presented in Table 6. Below 50 K the heat capacity of germanium were described by the polynomial equation Cp = x1T 3 + x2T 5 + x3T 6 with error = 0.07 J/(mol-at) -1 K -1 (See the Table 7).
The set of experimental points of the heat capacity of Ge and GaAs Cp (T) was described by equation (4) with precision = 0.16 J/(mol-at) -1 K -1 and by polynomial equation with precision = 0.08 J/(mol-at) -1 K -1 in the same range of temperature. The AlSb, InP, CdS, and ZnSe compounds with a sum of atomic numbers 32 per moleatom have a Cp (T) dependence close to that of germanium. The discrepancy error of heat capacity may be associated with a deviation from the stoichiometry of the forming phases [43,44] and the influence of impurities [21] (see Fig.3) as well as the difference in the isotopic composition of elements [45,46,47]. There is also a great influence of the high partial pressure of the vapor components of the substance on the heat capacity at high temperature. The heat capacity of indium phosphide at high temperatures was studied in three papers (See Table 9 and Fig.10) , and germanium (solid red line) (see Table 6).
If the low-temperature heat capacities of germanium and indium phosphide are in good agreement with each other, their high-temperature values are contradictory. Variants of the DSC method [48,49] are not suitable for studying the heat capacity of indium phosphide due to the high partial pressure of phosphorus at high temperatures (see Fig. 11). Drop calorimetry data of the AIIIBV compounds in [20] are usually overestimated. Therefore, the high-temperature heat capacities of indium phosphide were not used in our calculations.
The experimental points of the heat capacity Cp (T) of -Sn in solid state was described by equation (3) with precision = 0.12 J/(mol-at) -1 K -1 in the range of temperature 5 -2000 K. Parameters of this equation are presented in Table 6. Below 14 K the heat capacity of gray tin were described by the polynomial equation Cp = x1T 3 + x2T 5 + x3T 6 with error = 0.03 J/(mol-at) -1 K -1 (See the Table 7). The calculated values of gray tin above 287.15 K correspond to its metastable state.
The heat capacities of InSb and CdTe are the same as the gray tin below 287.15 K within the limitation of the experimental data. The

Conclusion
The carried out thermodynamic analysis showed that the series of considered isostructural substances consist of five groups: it is preferable to use the heat capacity of pure elements, which should be considered as standard substances.
Our proposed model eliminates all the shortcomings of the description when extrapolating data. The reliable heat capacity data are required in industry for growing single crystals and films.
The model can be used in both binary and multicomponent systems of different substances. This model also helps the standardization of physicochemical constants.
This work is indispensable for both the prognostic calculations in the chemical thermodynamics and applied research projects.