**3.3.2. Additive estimation methods**

where ∆*H*fcc°(A*x*B*y*O*z*) is the standard enthalpy of the formation of double oxide AxByOz from

∆*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

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.

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**.

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

27 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

shown in some cases to lead to output data well related to experimental results [170].

equation for the contribution of each element to the entropy of the compound [176].

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

multiplied by its molar specific heat is a constant [175].

the mass is the principal factor, and in 1921, the authors gave an

( ) <sup>3</sup> 298 K lnat.wt. 0.94 <sup>2</sup> *S R* <sup>=</sup> - <sup>o</sup> (17)

D=D +D *H xH xH ff f* AO (AO) BO (BO) oo o

of component oxides AO and BO according to the following relationship:

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

*<sup>f</sup>* represents the sum of molar fraction enthalpies

(16)

the component oxides AO and BO and Δ*<sup>H</sup>*¯

A*x*B*y*O*z* with a given composition [171].

per gram-atom [174],[175],27

[176].

Additive estimation or contributive methods are probably the simplest approach based on the following [170],[177]:

**1. Atomic and ionic contribution**: the technique based on the method proposed by KELLOGG [178]:

$$\begin{aligned} \text{C}\_{\text{pm}}^{\circ} \left( A\_{\text{x}} B\_{\text{y}}, \text{298.15 K} \right) &= \text{x} \begin{aligned} ^{\circ} \text{C}\_{\text{pm}}^{\circ} \left( A \right) + \text{y} \begin{aligned} ^{\circ} \text{C}\_{\text{pm}}^{\circ} \left( B \right) & \text{e.g.}: \\ \text{C}\_{\text{pm}}^{\circ} \left( \text{BaCl}\_{2}, \text{298.15 K} \right) &= \text{C}\_{\text{pm}}^{\circ} \left( \text{Ba} \right) + 2 \text{ } ^{\circ} \text{C}\_{\text{pm}}^{\circ} \left( \text{Cl} \right) \end{aligned} \tag{18}$$

The approach was later revised by KUBASCHEWSKI [179],[180]. These authors also pro‐ posed the method for the estimation of parameters A, B and C in the temperature dependence of *C*pm°(T)28 :

$$C\_{\rm pu}^{\circ} \left( T \right) = A + BT + \frac{C}{T^2} \tag{19}$$

$$A = \frac{10^{-3} T\_{\pi} \left[ C\_{\text{pm}}^{\circ} \left( 298.15 \text{ K} \right) + 4.7 n \right] - 1.25 n \cdot 10^{5} \left( T\_{\text{w}} \right)^{-2} - 9.05 n}{10^{-3} T\_{\text{w}} - 0.298} \tag{20}$$

$$B = \frac{25.6n + 4.2n \cdot 10^{8} \left(T\_{\text{m}}\right)^{-2} - C\_{\text{pm}}^{\circ} \left(298.15 \text{ K}\right)}{10^{-3}T\_{\text{m}} - 0.298} \tag{21}$$

$$C = -4.2n\tag{22}$$

where *n* is the number of ions (contributions) in the formula unit. The described ap‐ proach is worthy for the substances with melting point temperatures (*T*m) bellow 2300 K. The data on cationic and anionic contributions to heat capacity at 298 K are published in works [177], [179],[181],[182],[183].

For ionic compounds, the entropy can be calculated29 from additive data given in **Table** 4, empirically found for cation and anion constituents of the compound (increments method of LATIMER [184]) [172].


<sup>28</sup> The full equation for the temperature dependence is *C*pm° = A + BT + C/T2 + DT2 + F/T1/2 [J·K−1·mol−1].


<sup>29</sup>**Table 4** [172] refers to the values of entropy contribution from work [176] where some data for PO<sup>4</sup> 3− ion are given in brackets. In order to verify this value (for the charge of cation, i.e. Ca2+, the contribution of PO4 3− anion is 17 [calories] × 4.184 = 71.13 J·K−1·mol−1, it is possible to calculate it from recommended thermodynamic data from **Fig. 25**. If (for example) three apatite end members were used, it is possible to calculate the contribution to PO4 3− anion in hydroxylapatite, fluorapatite, chlorapatite, and bromapatite as follows:

( ) ( ) ( ) 1 1 HAP : 10 39.1 6 2 18.83 780 58.56 J K mol *x x* - - × + +× = Þ= × ×

( ) ( ) ( ) 1 1 FAP : 10 39.1 6 2 17.00 728 50.50 J K mol *x x* - - × + +× = Þ= × ×

( ) ( ) ( ) 1 1 ClAP : 10 39.1 6 2 31.80 835 63.40 J K mol *x x* - - × + +× = Þ= × ×

$$\text{BrAP}: \quad \text{(10.-39.1)} + \text{(6x)} + \text{(2.-45.70)} = 870 \quad \Rightarrow \text{ x} = 64.60 \quad \text{J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$$

It is also possible to calculate it from the contribution data for Ca3(PO4)2 or Mg3(PO4)2, where S°(298.15K) = 235.998 and 189.2 J·K−1mol−1 (HSC software v.7.1), respectively:

( ) ( ) 1 1 3 39.1 2 235.998 59.35 J K mol *x x* - - × + = Þ= × ×

( ) ( ) 1 1 3 23.40 2 189.2 59.5 J K mol *x x* - - × + = Þ= × ×

Therefore, it is suggested to use average value from these calculations, i.e. PO4 3− (for M2+ cation) ≈ 59 J·K−1mol−1. It is then possible that the application of contribution techniques to apatite leads to positive error in estimated thermodynamic data.


**Table 4.** The contribution of cations and anions to *C*pm° and *S*m° (298.15 K) [172],[180].

**Element** *C***pm°** *S***m° Element** *C***pm°** *S***m° Element** *C***pm°** *S***m°**

Al 19.66 23.4 Hg 25.10 59.4 Rb 26.36 59.2 As 25.10 45.2 Ho 23.01 56.0 Sb 23.85 58.9 Au – 58.5 In 24.27 55.0 Se 21.34 60.5 B – 23.5 Ir (23.85) 50.0 Si – 35.2 Ba 26.36 62.7 K 25.94 46.4 Sm 25.10 60.2 Be (9.62) 12.6 La (25.52) 62.3 Sn 23.43 58.2 Bi 26.78 65.0 Li 19.66 14.6 Sr 25.52 48.7 Ca 24.69 39.1 Lu – 51.5 Ta 23.01 53.8 Cd 23.01 50.7 Mg 19.66 23.4 Te – 69.0 Ce 23.43 61.9 Mn 23.43 43.8 Th 25.52 59.9 Co 28.03 34.1 Mo – 35.9 Ti 21.76 39.3 Cr 23.01 32.9 Na 25.94 37.2 Tl 27.61 72.1

3− ion are given in

3− anion is 17 [calories] ×

3− anion in hydroxylapatite,

3− (for M2+ cation) ≈ 59 J·K−1mol−1. It is then

<sup>29</sup>**Table 4** [172] refers to the values of entropy contribution from work [176] where some data for PO<sup>4</sup>

4.184 = 71.13 J·K−1·mol−1, it is possible to calculate it from recommended thermodynamic data from **Fig. 25**. If (for example)

( ) ( ) ( ) 1 1 HAP : 10 39.1 6 2 18.83 780 58.56 J K mol *x x* - - × + +× = Þ= × ×

( ) ( ) ( ) 1 1 FAP : 10 39.1 6 2 17.00 728 50.50 J K mol *x x* - - × + +× = Þ= × ×

( ) ( ) ( ) 1 1 ClAP : 10 39.1 6 2 31.80 835 63.40 J K mol *x x* - - × + +× = Þ= × ×

( ) ( ) ( ) 1 1 BrAP : 10 39.1 6 2 45.70 870 64.60 J K mol *x x* - - × + +× = Þ= × ×

It is also possible to calculate it from the contribution data for Ca3(PO4)2 or Mg3(PO4)2, where S°(298.15K) = 235.998 and

( ) ( ) 1 1 3 39.1 2 235.998 59.35 J K mol *x x* - - × + = Þ= × ×

( ) ( ) 1 1 3 23.40 2 189.2 59.5 J K mol *x x* - - × + = Þ= × ×

possible that the application of contribution techniques to apatite leads to positive error in estimated thermodynamic

brackets. In order to verify this value (for the charge of cation, i.e. Ca2+, the contribution of PO4

three apatite end members were used, it is possible to calculate the contribution to PO4

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

fluorapatite, chlorapatite, and bromapatite as follows:

189.2 J·K−1mol−1 (HSC software v.7.1), respectively:

data.

Therefore, it is suggested to use average value from these calculations, i.e. PO4

[J·K−1·mol−1]

**2. Structural and simple oxide contribution**: the value of *C*pm°(298.15 K) or the parame‐ ters of the temperature dependence in **Eq. 19** can be calculated from the contribution of constituent oxides (Neumann–Kopp rule, NKR):

$$\text{Ax } A(\text{s}) + \text{y} \\ B(\text{s}) = A\_x \text{B}\_\text{y}(\text{s}), \quad \text{e.g.}, \quad \text{SrO}(\text{s}) + \text{CuO}(\text{s}) = \text{SrCuO}\_2(\text{s}) \tag{23}$$

$$\begin{aligned} \left(C\_{\text{pn}}^{\circ}\left(A\_{\text{s}}B\_{\text{p}}, 298.15\,\text{K}\right) = \text{x}\,\,\,^{\circ}\_{\text{pm}}\left(A, 298.15\,\text{K}\right) + \text{y}\,\,\,^{\circ}\_{\text{pm}}\left(B, 298.15\,\text{K}\right) & \text{e.g.},\\ \left(C\_{\text{pn}}^{\circ}\left(\text{SrCuO}\_{2}, 298.15\,\text{K}\right) = C\_{\text{pn}}^{\circ}\left(\text{SrO}, 298.15\,\text{K}\right) + C\_{\text{pn}}^{\circ}\left(\text{CuO}, 298.15\,\text{K}\right) & \end{aligned} \tag{24}$$

or from the structural contribution [179].

**3. Prediction method for homological series and groups of chemically related substan‐ ces (oxides)**: based on the approach of ALDABERGENOV et al [185] and GOSPODINOV and MIHOV [186]. The molar heat capacity in homological series as Am(B*x*O*y*)n, is a linear function of *n*, i.e. the coefficient which specifies the number of complex anions (B*x*O*y*) *<sup>z</sup>*<sup>−</sup> in the formula unit. For example, for the series of alkaline aluminate, it can be written as

$$\text{AlO}\_2 \stackrel{\text{\*}\text{AlO}\_2}{\rightarrow} \text{Al}\_2\text{O}\_4^{2-} \stackrel{\text{\*}\text{AlO}\_2}{\rightarrow} \text{Al}\_3\text{O}\_6^{3-} \dots \tag{25}$$

Since each higher anion is formed by the addition of primary ion (AlO2) <sup>−</sup> unit, higher anion is considered to form *n*-multiples of primary ion, the value of which is determined from the available experimental data for KAlO2, LiAlO2 and NaAlO2 and from ions contribu‐ tion for cation K+ , Li+ and Na+ obtained from their standard entropies in an infinitely diluted solution [179].

Apatite phase may be treated in the first approximation as the sum of contributions arising from the constitution of binary oxides/compounds. For example, in the case of fluorapatite (Ca10(PO4)6F2), a decomposition into contribution of 9CaO + 3P2O5 + CaF2 could be consid‐ ered. It can be generalized to any end-member in the form 9CaO + 3P2O5 + XF2 [170].
