**3.4. Dissolution of apatites**

At the fundamental level, the reactions between solids and liquids involve a coupled se‐ quence of mass transport, adsorption/desorption phenomena, heterogeneous reactions, chemical transformations of intermediates, etc., the identification, separation and kinetic quantification of which are all necessary if the mechanism of the process is to be fully understood and described [111],[187]. It was generally accepted that the process during the dissolution of lattice ions includes the following [187],[188]:

**a.** Detachment of species (ion) from a kink site


**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

,298.15 K ,298.15 K ,298.15 K e.g.,

**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*)

unit. For example, for the series of alkaline aluminate, it can be written as

AlO2 2 AlO 2 3 AlO Al O Al O 2 24 3 6 + +

Since each higher anion is formed by the addition of primary ion (AlO2)

® ®¼ -

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‐

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‐

At the fundamental level, the reactions between solids and liquids involve a coupled se‐ quence of mass transport, adsorption/desorption phenomena, heterogeneous reactions, chemical transformations of intermediates, etc., the identification, separation and kinetic quantification of which are all necessary if the mechanism of the process is to be fully understood and described [111],[187]. It was generally accepted that the process during the

ered. It can be generalized to any end-member in the form 9CaO + 3P2O5 + XF2 [170].

(25)

, Li+ and Na+ obtained from their standard entropies in an infinitely

SrCuO ,298.15 K SrO,298.15 K CuO,298.15 K

= +

pm pm pm pm 2 pm pm

*C ABx y xC A yC B C CC*

*x A s yB s A B s* ( ) + = ( ) *x y* ( ), e.g., SrO CuO SrCuO (*ss s* ) + = ( ) <sup>2</sup> ( ) (23)

<sup>o</sup> (24)

*<sup>z</sup>*<sup>−</sup> in the formula

<sup>−</sup> unit, higher anion

constituent oxides (Neumann–Kopp rule, NKR):

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

or from the structural contribution [179].

o

tion for cation K+

diluted solution [179].

**3.4. Dissolution of apatites**

dissolution of lattice ions includes the following [187],[188]:

**a.** Detachment of species (ion) from a kink site

**d.** Diffusion into the bulk solution

The dissolution of apatites under steady-state conditions, in pure water or in aqueous acidic media, includes the following simultaneous steps [187],[189]:


These steps are likely to be more complicated, e.g. the processes I and V include chemical transformation ofionic species during diffusion because the pH of solution is known to depend on the distance from the solid/liquid interface. In other words, the value of pH is higher near to the surface of apatite and decreases with increasing distance from the surface.

When apatite gets in contact with undersaturated solution, the dissolution states from 1 to 5 mentioned above take place. In order to provide detailed description of the process, the following assumption must be introduced [187]:


Since the dissolution models have limitations and drawbacks, none of them was able to describe the dissolution of apatite in general. Furthermore, the most of models were elaborat‐ ed for the apatite dissolution in slightly acidic or nearly neutral solution (4 < pH < 8), for relatively small values of solution undersaturation and for the temperatures in the range from 25°C to 37°C, nothing is known about their validity forthe dissolution of apatite in strong inorganic acid such as HCl, HNO3 and H2SO4 and at temperatures above 70°C [187].

The classification of congruent (stoichiometric)/incongruent (nonstoichiometric) dissolution is based on direct measurements of either ionic concentrations in the solution or the surface composition of apatite during the dissolution [187]:

**1. Congruent dissolution**: ions in solids are dissolved simultaneously with the dissolution rates proportional to their molar concentrations, e.g. for Ca5(PO4)3Z it should be written [190],[191]:

$$\frac{\left[\text{Ca}\right]\_{\iota}}{\left[\text{P}\right]\_{\iota}-\left[\text{P}\right]\_{0}}-R=0\tag{26}$$

where [Ca]t , [P]t and [P]0 denotes actual (at time *t*) concentration of calcium and phos‐ phorus in the solution and initial concentration of phosphorus (at time *t* = 0), respective‐ ly. The value of R is given by ideal stoichiometric ration of Ca:P = 5/3 in the formula of apatite (**Table 7** in **Chapter 1**).

**2. Incongruent dissolution**: the dissolution rate is different for each ion. That leads to the formation of surface layer with chemical composition different from the bulk of solid apatite phase.

The behavior of surface of apatite during the dissolution according to DOROZHKIN [112] is shown in **Fig. 26**. Fluorine from fluorapatite or hydroxylfrom hydroxyapatite dissolves most probably as the first. This can be explained by their position in the channels of crystal lattice. The dissolution starts with replacements of fluorine for water. Proton(s), chemisorbed on the nearest phosphate group(s), most probably catalyze this process. Local positive charge on apatite is formed as the result (*b*). Obtained local positive charge is removed by the detach‐ ment of one of the nearest calcium cations: Ca(2) is more likely to be detached first (*c*) since Ca(1) is located ratherfarfrom the channel. Acidic anions presentin the solution most probably participate in this process. Later, proton(s) from the bulk solution replace other calcium cation(s) around the nearest phosphate group. Very thin surface layer of acidic calcium phosphates is formed as the result [112].

Since the dissolution models have limitations and drawbacks, none of them was able to describe the dissolution of apatite in general. Furthermore, the most of models were elaborat‐ ed for the apatite dissolution in slightly acidic or nearly neutral solution (4 < pH < 8), for relatively small values of solution undersaturation and for the temperatures in the range from 25°C to 37°C, nothing is known about their validity forthe dissolution of apatite in strong

The classification of congruent (stoichiometric)/incongruent (nonstoichiometric) dissolution is based on direct measurements of either ionic concentrations in the solution or the surface

**1. Congruent dissolution**: ions in solids are dissolved simultaneously with the dissolution rates proportional to their molar concentrations, e.g. for Ca5(PO4)3Z it should be written

0

phorus in the solution and initial concentration of phosphorus (at time *t* = 0), respective‐ ly. The value of R is given by ideal stoichiometric ration of Ca:P = 5/3 in the formula of

**2. Incongruent dissolution**: the dissolution rate is different for each ion. That leads to the formation of surface layer with chemical composition different from the bulk of solid

The behavior of surface of apatite during the dissolution according to DOROZHKIN [112] is shown in **Fig. 26**. Fluorine from fluorapatite or hydroxylfrom hydroxyapatite dissolves most probably as the first. This can be explained by their position in the channels of crystal lattice. The dissolution starts with replacements of fluorine for water. Proton(s), chemisorbed on the nearest phosphate group(s), most probably catalyze this process. Local positive charge on apatite is formed as the result (*b*). Obtained local positive charge is removed by the detach‐ ment of one of the nearest calcium cations: Ca(2) is more likely to be detached first (*c*) since Ca(1) is located ratherfarfrom the channel. Acidic anions presentin the solution most probably participate in this process. Later, proton(s) from the bulk solution replace other calcium cation(s) around the nearest phosphate group. Very thin surface layer of acidic calcium

, [P]t and [P]0 denotes actual (at time *t*) concentration of calcium and phos‐


inorganic acid such as HCl, HNO3 and H2SO4 and at temperatures above 70°C [187].

[ ] [ ] [ ]<sup>0</sup> Ca

P P *t*

*t*

composition of apatite during the dissolution [187]:

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

[190],[191]:

where [Ca]t

apatite phase.

apatite (**Table 7** in **Chapter 1**).

phosphates is formed as the result [112].

**Fig. 26.** Schematic illustration of the surface dissolution mechanism of apatite at the nanolevel: (a) part of the initial surface of apatite; (b) replacement of one fluorine (or hydroxyl) anion with water molecule resulting in local positive charge formation; (c) removal of one of the nearest calcium cations; (d) sorption of next proton; (e) removal of another calcium cation with simultaneous formation of acidic calcium phosphate; (f) detachment of one phosphate anion to‐ gether (or simultaneously) with third calcium cation. A jump-wise shift of the dissolution step occurs simultaneously at stage f. (●) Fluorine for fluorapatite or hydroxyl for hydroxylapatite; (○) Ca(II) on the first plane, (○) Ca(II) on the back plane and Ca(I) on the back plane; (\*(+)) molecule of water and local positive charge; (∆) PO4 3− tetrahedra; H+ ∆H<sup>+</sup> and ∆H<sup>+</sup> represent the surface tetrahedral anions of H2PO4 − and HPO4 2−, respectively. Chemisorbed protons, water mol‐ ecules and acidic anions are omitted for simplicity. Note that crystal structure of apatite is shown very schematically: it should be hexagonal, while here it looks more or less like cubic [112].

When all (or almost all) the nearest calcium cations have been replaced with protons accord‐ ing to the reactions [112]:

$$\rm CaHPO\_4 + H^+ \rightarrow Ca^{2+} + H\_2PO\_4^- \tag{27}$$

$$\text{CaHPO}\_4 + 2\text{H}^+ \rightarrow \text{Ca}^{2+} + \text{H}\_3\text{PO}\_4 \tag{28}$$

phosphate anions (H2PO4 − , CaH2PO4 + , or H3PO4) also detach (*f*). As the result, the dissolution step moves forward jump-wise over a distance equal to the dimension of phosphate anion, of approximately 3 Å. The detachment of phosphate anions and calcium cations results in the formation of hole. The dimension of this hole should be close to the lattice parameters of apatite. Most probably, it is a dissolution nucleus on which the polynuclear dissolution mechanism is based [112].

#### **3.4.1. Dissolution of fluorapatite**

The adsorption of H+ onto the surface of apatite (**Fig. 26**(**b**)–(**d**)) resulted in the aqueous pH increasing from 5.60 to 8.45 within the first hour of dissolution. Ions of H+ were adsorbed onto oxygen ions of phosphate groups as well as onto ions of fluoride [187],[191].

The reaction of stoichiometric (congruent) dissolution of pure stoichiometric apatite can be expressed by the reaction30 [112],[187],[192]:

$$\text{Ca}\_{\text{s}}\text{(PO}\_{4}\text{)}\_{\text{3}}\text{Z} \rightarrow \text{5 Ca}^{2+} + \text{3 PO}\_{4}^{3-} + \text{Z}^{-}\tag{29}$$

where *Z* = OH<sup>−</sup> and F<sup>−</sup> . Assuming the unit activity of the solid phase (*a*Ca5(PO4)3*<sup>Z</sup>* = 1), the equilibrium constant of dissolution (*K*) can be expressed via the solubility product or ion activity product (IAP) of apatite31 [187],[193],[194],[195],[196]:

$$K = \mathbf{I} \mathbf{A} \mathbf{P} = a\_{\mathbf{Ca}^{2+}}^{\circ} a\_{\mathbf{PO\_4^\cdot}}^{\circ} a\_{\mathbf{Z}} \qquad \text{ [eq.]} \tag{30}$$

where *a*<sup>i</sup> denotes the thermodynamic activity of aqueous species. The standard Gibbs (free) energy of the reaction related to the standard temperature (298.15 K) and pressure (0.101 MPa) is given by the formula32 :

*K*sp = *a*(Ca2+) <sup>10</sup> *a*(PO4 3−) 6 10−2*p*OH = *a*(Ca2+) <sup>10</sup> *a*(PO4 3−) 6 10−2(14−*p*H) = *a*(Ca2+) <sup>10</sup> *a*(PO4 3−) <sup>6</sup> *K*<sup>w</sup> 2 102*p*H.

The activity of ionic species is the product of ion molar concentration ([196]) and ion activity coefficient, e.g. (*a*(Ca2+) = ([Ca2+] / [Ca2+]°) *γ*(Ca2+) , where the standard state [Ca2+]° = 1 mol·dm−3 can be chosen), which can be calculated, e.g. the example via **Debye–Hückel**, **extended Debye–Hückel**, or **modified Davies equation** (in dependence on ionic strength):

log *γ*<sup>i</sup> = -*AZ*<sup>i</sup> 2 √*I* (*I* < 10−3),

log *γ*<sup>i</sup> = [-*AZ*<sup>i</sup> 2 √*I* / (1+*B*·*α*<sup>i</sup> √*I*)] (10−3 < *I* < 0.1),

or log *γ*<sup>i</sup> = [-*AZ*<sup>i</sup> 2 √*I* / (1+*B*·*α*<sup>i</sup> √*I*)] + 0.3*I* (*I* > 0.1), respectively.

*A* and *B* are the temperature-dependent constants, *Z*<sup>i</sup> is the charge number of *i*th ion, *α*<sup>i</sup> is the radius of hydrated *i*th ion, and *I* is the number of ions: *I* = ½ *∑* ([*C*<sup>i</sup> ]*Z*<sup>i</sup> 2 ).

The solubility product defined as the product of concentration of compound constituents ions, which are released during the dissociation is often used: *K*sp = [Ca2+] 10 [PO4 3−] 6 [Z<sup>−</sup> ] 2 .

<sup>30</sup> Reaction **Eq. 7** and **Eq. 12** are widely used for the description of dissolution process of apatite [112] using stoichiometry pertinent single or double apatite formula and Z = F or OH.

<sup>31</sup> Out of equilibrium state, *IAP* is not equal to *K* (see the discussion to **Eq. 20**). Double formula of apatite is assumed then with respect to the apatite stoichiometry; the law for ionic activity product has the following form:

*K*sp = *a*(Ca2+) <sup>10</sup> *a*(PO4 3 −) <sup>6</sup> *a*(Z−)<sup>2</sup> .

For example, in hydroxylapatite, where *Z*<sup>−</sup> = OH<sup>−</sup> , the activity of OH<sup>−</sup> anion can be expressed by using ionic product of water (25°C): *K*w = *a*(H+ ) *a*(OH<sup>−</sup> ) = 1·10−14 (mol·dm−3) 2 and then *a*(OH<sup>−</sup> ) = *K*w/ *a*(H+ ) . Since *p*H = -*log* [H+ ] and *p*OH = -*log* [OH<sup>−</sup> ] the [H+ ] = 10−pH and [OH<sup>−</sup> ] = 10−pOH and *pK*w = *p*H + *p*OH = 14:

<sup>32</sup> ∆*rG*° = -RT ln *K* (in the equilibrium state) and ln *K* = ln 10 log *K*. From that, it should be derived that ∆*rG*° = -8.314·298.15 ln 10 log *K* = -5708 log *K* [J·mol−1], where ln 10 ≈ 2.303. Since log10*K* = log *K* = lg *K*, it can also be written as ∆*rG*° = -5708 lg *K*.

Identification, Characterization and Properties of Apatites http://dx.doi.org/10.5772/62211 153

$$
\Delta\_r G^\circ = -5.707 \text{ \log K} \qquad \left[ \text{kJ} \cdot \text{mol}^{-1} \right] \tag{31}
$$

For reaction 29, the following equation33 can be derived [187]:

$$\begin{aligned} \Delta\_r G^\circ &= \ $\,\Delta G\_f^\circ \Big(\text{Ca}^{2+}\Big) + \$ \,\Delta G\_f^\circ \Big(\text{PO}\_4^{3-}\Big) + \Delta G\_f^\circ \Big(Z^-\Big) \\ &- \Delta G\_f^\circ \Big(\text{Ca}\_5\Big(\text{PO}\_4\Big)\_3 Z\Big) \end{aligned} \tag{32}$$

**Eq. 32** can be further treated as follows:

**3.4.1. Dissolution of fluorapatite**

onto the surface of apatite (**Fig. 26**(**b**)–(**d**)) resulted in the aqueous pH

. Assuming the unit activity of the solid phase (*a*Ca5(PO4)3*<sup>Z</sup>* = 1), the

Ca PO Z *K aaa* IAP eq. = = <sup>+</sup> (30)

, the activity of OH<sup>−</sup> anion can be expressed by using ionic product of

] and *p*OH = -*log* [OH<sup>−</sup>

] the

=

. Since *p*H = -*log* [H+

were adsorbed onto

(29)

increasing from 5.60 to 8.45 within the first hour of dissolution. Ions of H+

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

[112],[187],[192]:

oxygen ions of phosphate groups as well as onto ions of fluoride [187],[191].

The reaction of stoichiometric (congruent) dissolution of pure stoichiometric apatite can be

equilibrium constant of dissolution (*K*) can be expressed via the solubility product or ion

[187],[193],[194],[195],[196]:

energy of the reaction related to the standard temperature (298.15 K) and pressure

30 Reaction **Eq. 7** and **Eq. 12** are widely used for the description of dissolution process of apatite [112] using stoichiometry

31 Out of equilibrium state, *IAP* is not equal to *K* (see the discussion to **Eq. 20**). Double formula of apatite is assumed then

) = *K*w/ *a*(H+ )

, where the standard state [Ca2+]° = 1 mol·dm−3 can be chosen), which can be calculated, e.g. the

and then *a*(OH<sup>−</sup>

<sup>10</sup> *a*(PO4 3−) <sup>6</sup> *K*<sup>w</sup> 2 102*p*H.

The activity of ionic species is the product of ion molar concentration ([196]) and ion activity coefficient, e.g. (*a*(Ca2+)

example via **Debye–Hückel**, **extended Debye–Hückel**, or **modified Davies equation** (in dependence on ionic strength):

*A* and *B* are the temperature-dependent constants, *Z*<sup>i</sup> is the charge number of *i*th ion, *α*<sup>i</sup> is the radius of hydrated *i*th ion,

The solubility product defined as the product of concentration of compound constituents ions, which are released during

2 3 [ ] <sup>4</sup>

denotes the thermodynamic activity of aqueous species. The standard Gibbs (free)

( ) 2 3 Ca PO Z 5 Ca 3 PO Z 5 4 <sup>3</sup> <sup>4</sup> ®++ + --

5 3

:

with respect to the apatite stoichiometry; the law for ionic activity product has the following form:

= OH<sup>−</sup>

2

10−2(14−*p*H) = *a*(Ca2+)

√*I*)] + 0.3*I* (*I* > 0.1), respectively.

10 [PO4 3−] 6 [Z<sup>−</sup> ] 2 . <sup>32</sup> ∆*rG*° = -RT ln *K* (in the equilibrium state) and ln *K* = ln 10 log *K*. From that, it should be derived that ∆*rG*° = -8.314·298.15 ln 10 log *K* = -5708 log *K* [J·mol−1], where ln 10 ≈ 2.303. Since log10*K* = log *K* = lg *K*, it can also be written as

]*Z*<sup>i</sup> 2 ).

= 1·10−14 (mol·dm−3)

<sup>10</sup> *a*(PO4 3−) 6

√*I*)] (10−3 < *I* < 0.1),

] = 10−pOH and *pK*w = *p*H + *p*OH = 14:

The adsorption of H+

expressed by the reaction30

and F<sup>−</sup>

activity product (IAP) of apatite31

(0.101 MPa) is given by the formula32

) *a*(OH<sup>−</sup> )

10−2*p*OH = *a*(Ca2+)

pertinent single or double apatite formula and Z = F or OH.

where *Z* = OH<sup>−</sup>

where *a*<sup>i</sup>

*K*sp = *a*(Ca2+)

*K*sp = *a*(Ca2+)

[H+

log *γ*<sup>i</sup> = -*AZ*<sup>i</sup> 2

log *γ*<sup>i</sup>

or log *γ*<sup>i</sup>

<sup>10</sup> *a*(PO4 3 −) <sup>6</sup> *a*(Z−)<sup>2</sup> . For example, in hydroxylapatite, where *Z*<sup>−</sup>

water (25°C): *K*w = *a*(H+

([Ca2+] / [Ca2+]°) *γ*(Ca2+)

 = [-*AZ*<sup>i</sup> 2 √*I* / (1+*B*·*α*<sup>i</sup>

∆*rG*° = -5708 lg *K*.

 = [-*AZ*<sup>i</sup> 2 √*I* / (1+*B*·*α*<sup>i</sup>

] = 10−pH and [OH<sup>−</sup>

<sup>10</sup> *a*(PO4 3−) 6

√*I* (*I* < 10−3),

and *I* is the number of ions: *I* = ½ *∑* ([*C*<sup>i</sup>

the dissociation is often used: *K*sp = [Ca2+]

$$\begin{aligned} \Delta G\_f^\circ \left( \text{Ca}\_\circ \text{(PO}\_4 \text{)}\_3 \text{Z} \right) &= 5 \text{ } \Delta G\_f^\circ \text{(Ca}^{2+} \text{)} + 3 \text{ } \Delta G\_f^\circ \text{(PO}\_4^{3-} \text{)}\\ + + \Delta G\_f^\circ \left( \text{Z}^- \right) &- \Delta\_r G^\circ \end{aligned} \tag{33}$$

HAROUIYA et al [197] assumes that the dissolution of apatite in the temperature range from 5°C to 50°C, and the pH from 1 to 6 can be expressed by the following formula:

$$\mathrm{Ca\_{s}\left(PO\_{4}\right)\_{3}} \\ \mathrm{F} + \mathrm{3\ H^{+}} \rightarrow \mathrm{5\ Ca^{2+}} + \mathrm{3\ HPO\_{4}^{2-}} + \mathrm{F^{-}} \tag{34}$$

With regard to assumed standard state, the equilibrium constant of reaction (**Eq. 34**) can be written as34

$$K'=\boldsymbol{a}\_{\text{Cu}^{2+}}^{\text{s}}\,\boldsymbol{a}\_{\text{HPO}\_{4}^{2-}}^{\text{3}}\,\boldsymbol{a}\_{\text{F}}\,\boldsymbol{a}\_{\text{H}^{\*}}^{-\text{3}}\tag{35}$$

The chemical affinity35 (*A*) of **Eq. 34** is given by the law:

$$A = -RT \ln\left(\frac{K'a\_{\rm H^{+}}^{\rm \cdot}}{a\_{\rm Cu^{2+}}^{\rm \cdot}, a\_{\rm HPO\_{4}^{2-}}^{\rm \cdot}a\_{\rm F}}\right) \tag{36}$$

In the closed-system experiment, the dissolution rates are generally obtained from the slope of concentration of reactive solution versus the time:

<sup>33</sup> ∆*rG*° = ∑*ν*<sup>i</sup> ∆*G*°f,i, where *ν*<sup>i</sup> denotes the stoichiometric coefficient of given species and ∆*G*°f,i its standard enthalpy of formation.

<sup>34</sup> Since the saturation of solution with respect to Ca5(PO4)3Z means that *K′* = 0 (equilibrium state), it can be derived that *a*(Ca2+) <sup>5</sup> *a*(PO4 3−) <sup>3</sup> *a*(Z−) – *K´a*(H+)3 = 0 and then ∆rG° = -RT ln (*a*(Ca2+) <sup>5</sup> *a*(PO4 3−) <sup>3</sup> *a*(Z−) / *K´a*(H+ ) 3 ). Since *A* = -∆r*G*° (please see note 35), A = -RT ln (*K´a*(H+ ) 3 / *a*(Ca2+) <sup>5</sup> *a*(PO4 3−) <sup>3</sup> *a*(Z−)).

<sup>35</sup> The relationship between the reaction Gibbs energy and chemical affinity: *A* = -∆r*G*° was introduced by T. DE DONDER.

$$r = \frac{\partial \mathcal{C}\_i}{\partial t} \frac{M\_r}{\nu\_i s} \tag{37}$$

where *r* refers to the dissolution rate of apatite, *c*<sup>i</sup> denotes the concentration of *i*th element, *t* is the time, *M*r designates the mass of fluid in the reactor, *ν*I is the stoichiometric coefficient and *s* designates the total surface area of sample in the reactor. The slope of the plot may not be constant and may increase or decrease with time from the following reasons [195],[197]:


$$r = r\_+ \left( 1 - \exp\left[\frac{-A}{\sigma RT}\right] \right) \tag{38}$$

The symbol *r*+ symbolizes the far from equilibrium dissolution rate, which may depend on the composition of solution, *A* is the affinity of reaction of dissolution, *σ* stands for the Temkin's average stoichiometric number equal to the ration of rate of destruction of the activated or precursor complex relative to overall rate, *R* designates universal gas constant and *T* is the temperature on the absolute scale. Overall rate (*r*) is equal to forward rate (*r*+) when *A* >> *σ RT*. As one of the approaches of equilibrium, overall rates gradually decrease and reach zero at equilibrium where *A* = 0. The value of *r* is within 10% of *r*+ when *A*/*σ RT* > 2.3 which is equivalent to *A* > 1.36 *σ* kcal·mol−1. It indicates that the parameter *σ* plays a crucial role in the variation of dissolution rates at near to equilibrium conditions [195],[197].

The value of *r*+ depends on the pH according to the following equation [197],[198]:

$$r\_\* = k \text{ } \text{pH}^{-\*}\tag{39}$$

where *k* refers to the tare constant and *n* stands for the reaction order determined as the slope of linear dependence of *ln r*+ on pH. The dependence of *k* on the temperature is given by the Arrhenius law:

$$k = A\_{\Lambda} \exp\left(\frac{-E\_{\Lambda}}{RT}\right) \tag{40}$$

where *A*A is the preexponential (frequency) factor and *E*A is the activation energy of the process. The combination of **Eqs. 38**, **39** and **40** leads to the equation for the dissolution rate as follows [197]:

*i r i c M <sup>r</sup> t s* n

the time, *M*r designates the mass of fluid in the reactor, *ν*I is the stoichiometric coefficient and *s* designates the total surface area of sample in the reactor. The slope of the plot may not be constant and may increase or decrease with time from the following reasons [195],[197]:

**1.** Changes in the reactive fluid volume, which may occur due to the evaporation of solvent

**3.** Approach to equilibrium, where the dissolution rate decreases and reaches zero at equilibrium. This approach is described by the transition state theory as follows:

> s*RT* <sup>+</sup>

The symbol *r*+ symbolizes the far from equilibrium dissolution rate, which may depend on the composition of solution, *A* is the affinity of reaction of dissolution, *σ* stands for the Temkin's average stoichiometric number equal to the ration of rate of destruction of the activated or precursor complex relative to overall rate, *R* designates universal gas constant and *T* is the temperature on the absolute scale. Overall rate (*r*) is equal to forward rate (*r*+) when *A* >> *σ RT*. As one of the approaches of equilibrium, overall rates gradually decrease and reach zero at equilibrium where *A* = 0. The value of *r* is within 10% of *r*+ when *A*/*σ RT* > 2.3 which is equivalent to *A* > 1.36 *σ* kcal·mol−1. It indicates that the parameter *σ* plays a crucial role in the variation of

1 exp *<sup>A</sup> r r*

The value of *r*+ depends on the pH according to the following equation [197],[198]:

pH *<sup>n</sup> r k* -

where *k* refers to the tare constant and *n* stands for the reaction order determined as the slope of linear dependence of *ln r*+ on pH. The dependence of *k* on the temperature is given by the

> <sup>A</sup> exp RT *<sup>E</sup> k A* æ ö - <sup>=</sup> ç ÷

A

dissolution rates at near to equilibrium conditions [195],[197].

where *r* refers to the dissolution rate of apatite, *c*<sup>i</sup>

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

or regular sampling of reactive fluid

**2.** Nonzero order reaction kinetics

Arrhenius law:

¶ <sup>=</sup> ¶ (37)

æ ö é ù - = - ç ÷ ê ú è ø ë û (38)

<sup>+</sup> = (39)

è ø (40)

denotes the concentration of *i*th element, *t* is

$$r = A\_{\Lambda} a\_{\text{H}^{+}}^{\*} \exp\left(E\_{\Lambda} / \text{RT}\right) \left(\text{l} - \exp\left[\frac{-A}{\sigma \text{RT}}\right]\right) \tag{41}$$

For acidic dissolution of calcium fluorapatite, ions of *F* were found to dissolve faster (or prior to) when compared to calcium and phosphate. A similar phenomenon of prior (or faster) dissolution of calcium when compared to that of phosphate was also found [187],[199]. The release of calcium and phosphate ions from the surface of apatite seems to be affected by the presence of salts, such as Na2SO4, CH3COONa, or NaCl, in the solution. The concentration of phosphate in the solution increases in the following order [187]:

$$\text{Na}\_2\text{SO}\_4 > \text{CH}\_3\text{COON}\_a > \text{NaCl}.$$

On the contrary, the concentration of calcium ions decreases in the same order.

The undersaturation (US) and relative undersaturation (USr) of apatite solvent dissolved upon is defined as follows36 [193]:

$$\text{USS}\_r = \text{l} - \text{US} = \text{l} - \left(\frac{\text{IAP}}{K}\right)^{1/18} \tag{42}$$

where *K* is the equilibrium constant of reaction 29 and IAP is the ion activity product. The law is written with regard to the stoichiometry of double formula of apatite, where *∑ν*<sup>i</sup> in **Eq. 30** is 2 × (5 + 3 + 1) = 18. The value of *IAP*/*K* ratio is as follows:


This ration is also used to calculate the saturation index (SI) for the reaction of dissolution [200]:

$$\text{SI} = \log\left(\frac{\text{IAP}}{K}\right) \tag{43}$$

Depending on the saturation index, the following states of solutions are recognized:

<sup>36</sup> Since the system is not in the equilibrium state IAP ≠ K.

