Many organic and inorganic compounds with globular molecules form so-called plastic crystals near their melting temperatures [1] . The definition of “globular” implies that either the molecules (or atoms) of these compounds have high symmetry (2,2-dimethylpropane, CH₄, SF₆, Kr, Xe) or when molecular rotation around one of the molecular axes of symmetry leads to some figure of rotation whose shape is close spherical (for instance, cyclopentane, cyclohexane, 2,2,3-trimethylbutane, PH₃, HCl). In addition, near the melting temperature, the plastic-crystalline phase of plastic crystals is characterized by orientational disorder in the direction of the axes of rotation of the molecules that form it, while long-range order is still preserved in the time-averaged location of the centers of mass of its molecules [2] . Thus, the nature of the movement of molecules in plastic crystals is similar to the movement of molecules in liquids: their molecules oscillate around equilibrium positions due to interaction with neighboring molecules, and also rotate, however, maintaining the long-range order of their mutual arrangement, as in ordinary crystals.

It should be noted here that the characteristic of rotation of molecules in the plastic crystalline phase is determined primarily by the absolute temperature T of this phase, as well as the magnitude of the energy barriers caused by intermolecular interactions between neighboring particles [3] . If the temperature T is sufficiently high and the above energy barriers are small enough, then practically free rotation of the molecules of the plastic phase (around any of the three possible rotation axes for each molecule) takes place. The presence of the such free rotations leads to the lower values of the fusion enthalpies for the solid chemical compounds, existing in their plastic crystalline phase, than for compounds which do not have the given phase (see the work [3] ). As the examples of the plastic-crystalline chemical compounds whose molecules have practically free rotation near their melting points, we can cite adamantane [4] , octafluorocyclobutane [5] , camphor [6] . However, a more common case of molecular motion in the plastic phase is the situation when the magnitude of the energy barriers that must be overcome to realize free rotation slightly exceeds the time-averaged value of thermal kinetic energy for each rotational degree of freedom of any molecules of the plastic phase under consideration. In this case, the molecules of the plastic phase under consideration undergo librational movements, which, due to fluctuations in kinetic energy caused by intermolecular interactions, can lead to their sudden spatial reorientation (molecular tumbling) and, accordingly, to their orientational disorder. So, for example, a study of the plastic crystalline phase of sulfur hexafluoride (see [7] ) has shown that the average one-particle isotropic librational angle is equal to 14˚. Obviously, the closer the absolute temperature T of the plastic phase is to the melting temperature of the compound under study, the more often such jumps (tumblings) occur. Upon reaching the appropriate melting temperature, these molecular jumps transform into free one-particle rotations in the bulk of the corresponding liquid phase. So, for instance, for tetramethylsilane, cyclohexane, cyclopentane, and 2,3-dimethylpropane, the values of the energy barriers (required for molecular tumbling in their plastic crystalline phases) are equal to 30.36, 9.63, 4.19 and 3.68 kJ/mol respectively [8] - [10] .

Here it is to be noted that most often plastic crystals have cubic crystalline structures [11] . Also the plastic crystalline phase can appear in the case when non-globular or elongated molecules possess degrees of rotational freedom about a preferred axis. Then the free rotation around this axis may be excited in crystals while still retaining three-dimensional positional order. Such structures are called ‘‘hexagonal plastic crystals”. For example, the such situation takes place for near melting solid phases of nitrogen and hydrogen (see [12] [13] ). One can say that plastic crystals possess weak intermolecular interactions and are liquid-like. So, the surface layer of the solid phase of hydrogen possesses the clear-cut quasi-liquid nature, when the molecules of this layer can take part in their translational motion, which realizes in the form of their molecular jumps [14] .

Thus, taking into account the circumstances mentioned above, such as the proximity of the plastic crystalline phase of chemical compounds to their liquid one (considered at their corresponding melting points) and the small values of their fusion enthalpies ΔH, the aim of the present work is to find an equation relating ΔH to the physicochemical quantities of the liquid phase of these compounds, which arises as a result of the melting of their plastic-crystalline phase.

In order to find the sought equation for the quantity ΔH let us consider a system consisting of a mole of some chemical compound having a plastic crystalline phase. We shall also assume that, firstly, this phase is at its melting point; and, secondly, that the process of its melting is incomplete. In this case the system will contain both: its solid phase and its liquid one along with the interface between these two phases. Further, it was shown in the work [15] that for different liquids considered at their normal freezing points, their liquid phase contains the so-called “hot” (exited) molecules. Unlike the other molecules of the considered near-freezing liquid phase these “hot” molecules can move throughout the entire volume of the given phase. At the same time (see [15] ) any of these “hot” molecules is situated and moves within some its spherical cavity (cage), whose radius is proportional to the quantity Vc (where Vc is the critical volume of the chemical compound under study. It was also shown in [15] that for one mole of any non-associated near-freezing liquid the total number N of such “hot” molecules can be defined by means of the following equations:

$N=\xi N_A$(1)

where N_A is Avogadro’s number, and the coefficient of proportionality ξ is defined as:

$\xi =\exp \left(-g\sigma V^{2/3}/T\right)$(2)

In Equation (2) σ is surface tension of a liquid, V and T are, respectively, its molar volume and its freezing point, and ξ is the constant, whose numerical value is equal to 9.35 × 10⁶ (J⁻¹∙K∙mol^2/3). The constant ξ does not depend on T or on the nature of the liquid phase [15] .

It is to be noted here that the presence of these “hot” moving molecules in the above-mentioned cavities is the first (and the main) difference between the melt of the plastic crystalline phase and the corresponding solid plastic crystalline phase. The second difference is connected with the fact, that the considered “hot” molecules (unlike the molecules of the solid plastic crystalline phase) take part in their translational and free rotational motions within the limits of their one-particle spherical cavities. In their turn, all the molecules of the solid plastic crystalline phase take part only in their vibrational and librational motions. Nevertheless, their librations can be accompanied with the corresponding (jump-like) one-particle reorientations, creating the orientational disorder, which is characteristic for the plastic crystalline phase.

Thus, bearing in mind all the above-mentioned, we shall consider that the process of the melting of the investigated plastic crystalline phase is being realized on the interface between the solid and liquid phases and is reduced to the appearance on this surface the above considered “hot” molecules. Then the newly formed “hot” molecules along with those molecules of this interface (which have kept their vibrational and librational motions) pass into the liquid phase. It is clear that the formation of any of these “hot” molecules requires both the conditions: the overcoming of the potential energy barriers, describing its three-dimensional vibrations (and librations) and the formation in the liquid phase (near the interface) the mentioned spherical cavities which will contain the “hot” molecules under study. Further, we shall consider only the liquid phase of the studied two-phases melting system. Obviously, during the process of the melting, the amount of the liquid phase will increase. Let, further, some number $d{N}_i$ of the molecules have passed from the above-considered interface into the liquid phase. We can write the number dN_i as:

$d{N}_i=d{N}_{1i}+d{N}_{2i}$(3)

Here $d{N}_{1i}$ are $d{N}_{2i}$ are, respectively, the number of the passed “hot” molecules and that of the vibrating ones (which have kept their vibrational motion by passing from the solid phase to the liquid one). Now we can writhe the following expression for the enthalpy changing $d{N}_i$ which accompanies the above molecular passing of the number $d{N}_i$ molecules:

$d{H}_i=Td{S}_i+VdP+{\mu }_{1}d{N}_{1i}+{\mu }_{2}d{N}_{2i}.$(4)

In Equation (4) $d{S}_i$ is the change (the differential) of the entropy of the considered liquid phase which is caused by the passing of the above $d{N}_i$ molecules from the solid phase to the liquid one. The quantity V is the volume of the liquid phase under study; and P is the external (normal) atmospheric pressure (P = const = 0.1 MPa). As P is constant, so the second term in Equation (4) is equal to zero. The chemical potentials μ₁ and μ₂ are, respectively, the one-particle chemical potentials of the “hot” molecules and the vibrating (see above) ones.

Since the one-particle chemical potential is equal to a certain amount of energy that must be spent in order to add one new molecule of a chemical compound to the system under study, we shall further consider the values of μ separately for “hot” and vibrating molecules which abandon the interface between the two phases and enter the liquid phase. Besides, the character of molecular motion of the vibrating molecules is not changed by the above passing, so, hereafter, we shall suppose that their transition into the considered liquid phase does not require any energy consumption; and therefore, the quantity μ₂ is equal to zero. Thus, we can rewrite Equation (4) in the form:

$d{H}_i=Td{S}_i+{\mu }_{1}d{N}_{1i}.$(5)

Considering that all the solid phase is transformed into the liquid one (the melting process is over) and bearing in mind that enthalpy and entropy are additive quantities, we can do the summation over the index i for all the quantities $d{H}_i$. Since the volume concentration of the “hot” molecules in the studied liquid phase remains constant, therefore we consider that the quantity μ₁ remains also constant. Then, for the total sum of all $d{H}_i$ the following equation holds:

$\Delta H=T\Delta S+{\mu }_{1}N.$(6)

In Equation (6) ΔH and ΔS are, respectively, the molar fusion enthalpy and the molar fusion entropy which accompany the melting of the compound under study. Here we have also taken into account that the sum of $d{N}_{1i}$ equals to N, where the number N is being defined by means of Equations (1) and (2).

Now, we have to find the quantity μ₁ for the “hot” molecules which appear in the liquid phase during the investigated melting process. For this purpose, let us firstly define the average one-particle energy which is required for the formation in the liquid phase (near the interface) of the above mentioned one-particle spherical cavity. It was shown in the work [16] that the following equation holds for the radius R of this cavity:

$R={\left(3V_c/4\text{π}{N}_A\right)}^{1/3}.$(7)

In Equation (7) V_c is the molar critical volume of the chemical compound under study.

Then the internal surface area Q of the cavity will be 4πR². As it was also shown in the work [16] the energy E required for the formation of such cavity is proportional to the given area Q and the quantity σ of the surface tension of the studied liquid phase. Thus, we can write the following equation for E:

$E=\chi \sigma Q.$(8)

Further, basing on the results received in the works [15] - [17] , in the present work we shall consider that the coefficient of proportionality χ between E and the product σ × Q does not depend on T and is constant for different near-freezing liquids. Now, using Equation (7), we can write the expression for E as:

$E=\chi {\left(36\pi \right)}^{1/3}{\left(N_A\right)}^{-2/3}\sigma V_c^{2/3}=\phi \sigma V_c^{2/3}.$(9)

In Equation (9) the coefficient of proportionality $\phi \left(\phi =\chi {\left(36\pi \right)}^{1/3}{\left(N_A\right)}^{-2/3}\right)$does depend on T and is constant for various near freezing liquids. However, the value of E defined by means of Equation (9) presents only some part of the total energy which is required for entering of one separate molecule from the solid phase into the liquid one. Really, for any considered molecule some additional one-particle energy is required for the overcoming of the potential energy barriers which are connected with its three oscillatory motions and one librational movement.

In the present work we shall suppose that all these motions are harmonic, i.e. the total energy of any of them can be described by means of the two corresponding quadratic terms. In this case (see [18] ) the average one-particle kinetic energy for any of the above-mentioned motions will be numerically equal to the corresponding potential well. Further, basing on the energy equipartition theorem, one can suppose (see [18] [19] ) that for any of the given motions its kinetic energy equals to kT (where k is Boltzmann’s constant, and T is the melting temperature). Therefore, for any of the molecules of the solid phase, the one-particle energy value required for the overcoming of all its potential energy barriers (in order to make it free and put it into its cavity) will be 4 kT. However, there are a number of cases, when the value of the given one-particle energy is less than 4 kT. First of all, it is about the melting of the solid plastic phase of the inert gases. Since the molecules of these gases are monatomic, so we can consider only their vibrational motions because their librational motion is absent. So, for these chemical compounds the above-mentioned one-particle energy value will be equal to 3 kT. The similar situation takes place for the high-temperature melting of the chemical compounds, whose molecules have the practically free rotation in their solid plastic phase. For instance, it occurs for the melting of such compounds as adamantane or camphor (see the Introduction). Therefore, for the such compounds the above-mentioned one-particle energy value is also 3 kT. Thus, hereafter, we shall consider that for any molecule passing from the solid phase into the liquid one, the thermal energy spent for this process is associated with overcoming four or three of its energy barriers and is equal to 4 kT or 3 kT, respectively.

Further, using Equation (9), we can write the expression for μ₁ in the following final form:

${\mu }_1=nkT+\phi \sigma V_c^{2/3}.$(10)

In Equation (10) the number n is the coefficient at kT which may be equal to 3 or 4; it is the total number of overcome one-particle potential energy barriers related to one-particle vibrational and librational motions.

Here we have to clarify some details connected with involving of two boundary solid molecular layers into the corresponding liquid phase. Let the first of the above two layers coincide with the interface between two phases, and the second one be the corresponding solid neighboring layer. Obviously, there is some density gradient whose direction is normal to the interface and whose magnitude changes in time. Now we shall consider motion of the molecules in the second solid layer. If the density change of the first surface layer is small enough to affect one-particle vibrational movements in the second layer and at the same time it is sufficient to transform one of the one-particle librational modes into the corresponding free rotation of molecules belonging to this second layer, then these molecules will have also free rotation. That’s why the number n for the molecules that initially did not have free rotation is equal to four.

Now we have to define the quantity ΔS entering Equation (6). It can be done basing on the following reasons. First of all, it should be noted that for the studied liquid system, i.e. one mole of some chemical compound at its freezing point, not all N_A molecules belong to the above near-freezing liquid phase. Really, it was shown in the works [20] [21] that the molecular layers situated near the surface of the given liquid form the so-called quasi-crystalline phase. This phase contains only vibrating molecules [20] - [22] . The thickness of this phase equals to zero at the corresponding critical temperature of the studied compound. At the same time, the thickness of the considered phase reaches its maximum when T goes to the normal freezing point of the compound.

Further we can note that the real liquid phase includes only some number N_L molecules (but not all the N_A ones). Hereafter, basing on the principle of the corresponding states, we shall suppose that the number N_L equals to ϰN_A, where the coefficient ϰ does not depend on the nature of a liquid. Thus, among the above N_L molecules there are the number ξϰN_A of the “hot” molecules (see Equations (1) and (2)) and the number (ϰN_A - ξϰN_A) of the vibrating ones.

Nevertheless, the spatial positions for all the NL molecules remain practically the same as they were in the corresponding solid phase. Even for the above considered “hot” molecules their time-average positions remain in the centers of their molecular cavities (cages). Here it is to be noted, that the results of the work [23] indicate that the maximum possible displacements, describing one-particle jumps of the “hot” molecules within their cavities are essentially less than average intermolecular distances. Further, as it was found in [24] , the molecular structure of any liquid phase (taken near its normal freezing point) can be considered as some lattice system, which is constructed from the two sub-lattices: the first is formed by the vibrating molecules, and the second includes the above “hot” ones. Moreover, the vibrating molecules are being transformed into the new “hot” ones after passing of some impulse from the former “hot” molecules, which, obviously, become the newly formed vibrating ones (see [24] ). It means that we can suppose that there is some movement of one-particle molecular cavities (whose number equals ξN_L or ξϰN_A and any of these cavities is occupied only by one molecule) involving all the NL molecules. In the other words, the given cavities may be considered as some quasi-particles, which can move through the studied bulk liquid phase. Then the new microstates of the investigated liquid system will arise due to the movement of these cavities among the N_L knots of the two above mentioned sub-lattices.

Now we can write the following expression for the thermodynamic probability W, describing the macro-state of the system:

$W=\prod \left[N_L-i+1\right]=\prod \left[\varkappa N_A-i+1\right]$.(11)

Here the symbol Π denotes the product of the corresponding multipliers, whose index i changes from 1 up to ξϰN_A. Using Equation (1), Equation (11) can be also rewritten in the following form:

$W=N_L!/\left(N_L-\xi \varkappa N_A\right)!=\left(\varkappa N_A\right)!/\left(\varkappa N_A-\xi \varkappa N_A\right)!$(12)

Here the symbol “!” means the corresponding factorial. Now, considering that in the solid state there are none of the above-considered one-particle cavities, including the “hot” molecules, and using the well-known Boltzmann’s entropy formula (S = klnW) we can write the following equation for the quantity ΔS:

$\Delta S=S_L-S_S=S_L=k\ln \left[\left(\varkappa N_A\right)!/\left(\varkappa N_A-\xi \varkappa N_A\right)!\right]$(13)

In Equation (13) S_L and S_S are, respectively, the entropy values connected with distribution of the above one-particle cavities in the liquid and solid phases of the system under study. Further, using Stirling’s formula, one can easily show that:

$\Delta S=k\xi \varkappa N_A\ln \left(\varkappa N_A/e\right)=\xi \varkappa R\ln \left(\varkappa N_A/e\right).$(14)

Here the product of the constants k and N_A is denoted as R (R is the universal gas constant), and e is the base of the natural logarithm. Denoting $\varkappa R\ln \left(\varkappa N_A/e\right)$ as δ, we receive the simple final expression for ΔS:

$\Delta S=\delta \xi$.(15)

In Equation (15) the coefficient of proportionality δ does not depend on temperature and is constant for various near-freezing liquids.

Now, uniting Equations (6), (10), (15) and bearing in mind that $N=\xi \cdot N_A$ (see Equation (1)), we can write for the quantity ΔH the following final equation:

$\Delta H=\xi \left(nRT+\psi \sigma V_c^{2/3}+\delta T\right).$(16)

Here the quantity ξ is defined by means of Equation (2) and the constant ψ (which does not depend on T and on the chemical nature of the melting plastic crystalline phase under study) equals to $\phi \cdot N_A$.

The practical usage of Equation (16) requires the numerical values of the constants ψ and δ. In order to determine their values, the six chemical compounds presented in Table 1 were considered at their normal freezing points T. The values of V_c, V, and σ for these compounds were taken from [25] - [32] .

It should be noted that the compounds given in Table 1 were chosen for the following reasons: firstly, they are well studied experimentally and, secondly, they have various chemical structures, which differ significantly. A part of them belongs to organic compounds, another one is presented by inorganic substances. For nitrogen the number n equals to 3. It follows from the experimental fact that for the high-temperature β-phase of solid nitrogen (near its melting point) the reorientation motion of its molecules becomes practically free (see [33] ).

Further, the numerical values of the constants ψ and δ were defined by means of the multiple regression analysis applied to the data set of Table 1 . The found values of ψ and δ are equal to 7.7035 × 10⁷ (J/mol) and 8.6087 (J/[mol∙K]), respectively. The results of the usage of Eq. (16) with the above found values of ψ and δ are presents in Table 2 for the large number of various organic and inorganic compounds. The experimental values of V_c, V, σ needed for these calculations were taken from the works [13] [15] [27] - [32] [34] - [53] . The comparison of the values ΔH calculated by means of Equation (16) and the corresponding experimental ones [13] [15] [27] - [32] [34] - [53] shows that this equation describes the fusion enthalpies of all the studied compounds adequately. Here it should be noted that the numbers n equal to 3 for the following cases. Firstly, for the compounds, whose molecules rotate practically free within their high-temperature solid plastic phases considered near their normal melting points (see the Introduction) and, secondly, for the mono-atomic inert gases. The rest compounds of Table 2 have n = 4. The only exception is hydrogen, which has n = 2. It is due to the mentioned in the Introduction fact that the solid phase of hydrogen (which is near the interface between its solid and liquid phases) can be considered as some quasi-liquid phase [14] . The molecules, belonging to this phase, have both the types of their molecular motion, namely, the free rotation and the one-dimensional molecular jumps. It means, that for any of these molecules one of its three vibrational potential barriers has been overcome. Thus, for hydrogen the number n is equal to two.

The values of ΔH calculated by means of Equation (16) are given in Table 2 along with the values of ΔH obtained by the so-called “Leave-one-out” Cross-Validation Scheme (LOOCV). The given statistical method consists in consequent removal of one of the studied compounds from the regression analysis and the following calculation of the predicted quantity for the removed compound. It can be seen that the values of the fusion enthalpies in the last two columns of Table 2 are very close to each other and to the corresponding experimental ones.

It should be noted that the mechanism described above for the occurrence of one-particle free rotation at the interface in the presence of a density gradient normal to it can occur not only for molecular, but also for ionic crystals. So, for example, as the temperature increases, free rotation of ions can appear both in single-crystalline samples at the boundaries between ionic layers [54] and in polycrystalline samples [55] on the outer boundary surface of polycrystals (see [56] , [57] ). That is, in principle, an equation similar to Equation (16) can also hold for ionic plastic crystals.

The following conclusions can be drawn. Equation (16) allows one to perform the adequate calculation of the fusion enthalpy for various organic and inorganic compounds possessing the high-temperature plastic crystalline phase. The obtained equation may also be used to determine the possibility of free one-particle molecular rotations in the plastic crystalline phase for other substances with the same crystalline phase. In addition, our preliminary results indicate that the functional dependence similar to Equation (16) can hold also for organic compounds whose high-temperature crystalline phases differ from the plastic one.