**3.3. Thermodynamic properties of apatite**

**Fig. 23.** EBDS analysis of investigated fluorapatite specimen.

142 Apatites and their Synthetic Analogues - Synthesis, Structure, Properties and Applications

**Fig. 24.** The Kikuchi sphere showing the orientation of apatite crystal break plane.

The Kikuchi sphere in **Fig. 24** shows the orientation of the sample that is in agreement with indistinct cleavage of fluorapatite to the direction given by Miller-Bravais indices of {0001}.

DROUET [170] gives the comprehensive review on experimental and predicted thermodynam‐ ic properties of phosphate apatites and oxyapatites, where O2− ion replaces 2X<sup>−</sup> in general M10(PO4)6X2 formula of apatite phase and publishes the summary of available thermodynam‐ ic data including standard formation Gibbs energy (∆*G*<sup>f</sup> °), ∆*H*<sup>f</sup> ° and *S*° at the temperature of 298 K (25°C) and the pressure of 1 bar (105 Pa), which are listed in the periodic table of phosphate apatites in **Fig. 25**.

**Fig. 25.** Comprehensive periodic table of phosphate apatites provided by C. DROUET [170].

The discrepancies between published thermodynamic data probably arise from variable crystallinity states, polymorphs (either hexagonal or monoclinic, those not being systemati‐ cally identified in literature reports), nonstoichiometry, hydration state and/or the presence of undetected impurities. When experimental-based data are not available (or are questionable), the so-called prediction of thermodynamic properties of solids becomes relevant. For exam‐ ple, it may allow an understanding of some unsuccessful experiment aiming at obtaining a desired hypothetical composition, or it may fill the gap between reported and needed thermodynamic values for the evaluation of equilibria constants or for the establishment of phase diagrams [170]. There are many methods developed for this purpose the summary of which can be found in works [171],[172].

For double oxides, A*x*B*y*O*z* in the system AO-BO was established the dependence [171]:

$$
\Delta H^{\circ}\_{f/\text{ac}} = \left( A\_{\text{x}} B\_{\text{y}} O\_{\text{z}} \right) = f \left( \Delta \overline{H}^{\circ}\_{f} \right) \tag{15}
$$

where ∆*H*fcc°(A*x*B*y*O*z*) is the standard enthalpy of the formation of double oxide AxByOz from the component oxides AO and BO and Δ*<sup>H</sup>*¯ *<sup>f</sup>* represents the sum of molar fraction enthalpies of component oxides AO and BO according to the following relationship:

$$
\Delta \overline{H}\_f^\circ = \mathbf{x}\_{\text{AO}} \Delta H\_f^\circ \left(\text{AO}\right) + \mathbf{x}\_{\text{BO}} \Delta H\_f^\circ \left(\text{BO}\right) \tag{16}
$$

∆*H*°(AO) and ∆*H*°(BO) are the standard enthalpies of the formation of component oxides from the elements, and XAO and XBO are the molarfractions of component oxides in the double oxide A*x*B*y*O*z* with a given composition [171].

The entropy of a solid compound is a function of masses of constituent atoms and the forces acting between these atoms: the greaterthe mass and the lowerthe force, the largerthe entropy. The entropy of ionic solid will also depend upon the magnitude of the ionic charges. For compounds, the specific heat of which has reached the DULONG and PETIT [173] value of 6 cal. per gram-atom [174],[175],27 the mass is the principal factor, and in 1921, the authors gave an equation for the contribution of each element to the entropy of the compound [176].

$$\text{LS}^{\circ} \text{(298 K)} = \bigvee\_{2}^{} R \text{ 1nat.wt.} - 0.94 \tag{17}$$

For simple salts, such as alkali halides, the entropy may be estimated with fair accuracy as the sum of the entropies of constituent elements as given by this equation. However, the forces in solid salts are largely the ionic attractions, and the effect of the ionic radii upon the force constants and the vibrational frequencies is appreciable; in general, the entropy of a large ion is increased and the entropy of a small ion is decreased compared to the values given by **Eq. 17**. [176].

#### **3.3.1. Volume-based thermodynamic predictive method**

The volume-based thermodynamic approach (VTB), the so-called first-order method, has especially received much attention because the method is rather easy to use and has been shown in some cases to lead to output data well related to experimental results [170].

<sup>27</sup> One calorie is 4.184 J (joules). Gram-atom [gm] (and gram-molecule) was used to specify the amount of chemical elements or compound. These units had a direct relation with "atomic weights" and "molecular weights," which are in fact relative masses. "Atomic weights" were originally referred to the atomic weight of oxygen, by general agreement taken as 16. Although physicists separated the isotopes in a mass spectrometer and attributed the value of 16 to one of the isotopes of oxygen, chemists attributed the same value to the (slightly variable) mixture of isotopes 16, 17, and 18, which was for them naturally occurring element oxygen. Finally, an agreement between the International Union of Pure and Applied Physics (IUPAP) and the International Union of Pure and Applied Chemistry (IUPAC) brought this duality to an end in 1959/1960. Physicists and chemists have ever since agreed to assign the value 12, exactly, to the so-called atomic weight of the isotope of carbon with the mass number 12 (carbon 12, 12C), correctly called the relative atomic mass Ar(12C). The unified scale thus obtained gives the relative atomic and molecular masses, also known as the atomic and molecular weights, respectively [174]. The law is also known as Dulong and Petit principle, which can be expressed in modern unit as: atomic weigh × specific heat ≈ ∂(3*kTN*A)/∂*T* 3*kN*<sup>A</sup> ≈ *c*<sup>V</sup> ≈ 25 J·K−1mol−1, i.e. the atomic weight of solid element multiplied by its molar specific heat is a constant [175].
