FB4-09 La Iglesia 21/12/09 08:29 Página 109 Estudios Geológicos, 65(2) julio-diciembre 2009, 109-119 ISSN: 0367-0449 doi:10.3989/egeol.39849.060 Estimating the thermodynamic properties of phosphate minerals at high and low temperature from the sum of constituent units Estimación de las propiedades termodinámicas de fosfatos minerales a alta y baja temperatura por suma de sus unidades constituyentes A. La Iglesia1 RESUMEN Usando el modelo de unidades poliédricas de Hazen, y empleando un método de ajuste por mínimos cuadrados, se ha calculado la contribución de diecinueve unidades constituyentes a la energía libre y de otras quince unidades a la entalpía de los fosfatos minerales en condiciones termodinámicas normales (298 °K de temperatura y 1 bar de presión) y a altas temperaturas (400-1.000 °K). Con los valores de gi y hi calculados por este método se puede estimar las propiedades termodinámicas de cualquier fosfato mineral con mayor precisión que la conseguida hasta ahora por otros métodos. Palabras clave: Energía libre de Gibbs, Entalpía, Fosfatos minerales, Propiedades termodinámicas. ABSTRACT Using the polyhedral units model of Hazen and employing a method of least-squares, the contribution of nineteen constituent units to the free energy and fifteen units to the enthalpy, at 298 K and 1 bar of pressure, have been calculated for mineral phosphates. The contribution of these constituent units to the free energy at higher temperatures has also been calculated. From these data we can estimate the thermodynamic properties of phosphates by summing the con- tribution of the distinct units, with more accuracy than the methods published up until now. Key words: Gibbs Free Energy, Enthalpy, Phosphate minerals, Thermodynamic properties. Introduction theoretical backing in that the macroscopic mineral thermodynamic properties are the cumulative result The estimation of the thermodynamic properties of the lattice vibrational energies of the various ele- of silicate minerals has been considered by many ments in the lattice, Grimwall (2001). investigators. Wilcox & Bromley (1963), Slaughter Following the polyhedral units model of Hazen (1966), Karpov & Kashik (1968), Tardy & Garrels (1985), Chermak & Rimstidt (1989, 1990), deter- (1974), Nriagu (1975), Chen (1975), Robinson & mined by multiple linear regression the Gibbs free Haas (1983), Sposito (1986) and La Iglesia & energy of formation (gi) and enthalpy of forma- Aznar (1986, 1990) have published methods based tion (hi) of polyhedral units in silicate minerals: on the hypothesis that the free energy or enthalpy of ∆G° = Σn g (1) formation is related to the sum of free energies of f i i formation of their constituent oxide or hydroxide and ∆ Σ components. This summation hypothesis has a firm H°f = nihi (2) 1 Instituto de Geología Económica. CSIC. Facultad de Ciencias Geológicas. Universidad Complutense. 28040 Madrid. Email: [email protected] FB4-09 La Iglesia 21/12/09 08:29 Página 110 110 A. La Iglesia where ni is the number of moles of ith component In this paper, employing a method of least- ∆ ∆ per formula unit. With this method G°f or H°f of squares, the contribution of 19 constituent units to many silicates can be calculated with an uncertainty the Gibbs free energy and 15 units to the enthalpy of less than 1%. Using this approach, La Iglesia & of phosphates has been calculated. From these data Felix (1994) calculated values of gi and hi for car- we can estimate the thermodynamic properties of ∆ bonates to create a model that predicts their G°f other phosphates with more accuracy than the meth- ∆ and H°f with an uncertainty of less than 0.5%. ods published up until now. Tardy & Garrels (1976, 1977) published a method of estimation of the thermodynamic proper- ties of minerals based on the fact that the free ener- Calculation of the Gibbs free energies. gy and enthalpy of formation within a family of enthalpies and entropies of phosphate compounds from their constituent oxides are linear- units at 298 °K ∆ 2– ly dependent on a parameter called OM , which is a function of the electronegativity of the constituent To evaluate the contribution of the units to the cations (M2+). This correlation provides a means of Gibbs free energy and enthalpy of phosphate miner- estimating values for unmeasured minerals. So, als 41 crystalline phases (table 1) were selected. A Tardy & Gartner (1977) and Tardy & Vieillard system of 31 linear equations was established to (1977) have obtained empirical equations that esti- calculate gi and 23 linear equations were used to mate the Gibbs free energy of formation and calculate hi. To solve both systems we used the pro- enthalpy of formation of carbonates, nitrates, sul- gram MAPLE V release 2. The gi and hi coeffi- fates and phosphates with a deviation less than 3% cients were determined using a least-squares with respect to the experimental values. For the method. phosphates, these authors obtain the following The values of gi and hi obtained for each of the equations: polyhedral units, together with their uncertainty ranges, are presented in table 2. Considering the ∆ ∆ G°f (phosphate) = G°f(oxides) – easy substitution of O ligand by OH, F or Cl in ∆ 2– (3) 1.40n1n2( GOM + 76.9)/(n1 + n2) (kcal/mol) these compounds, average values of g H2O(OH), g F, g Cl, h H O h F and h Cl have been also calcu- and 2 (OH), lated. These values represent the difference ∆ ∆ between the energy of polyhedral units M-On and H°f (phosphate) = H°f(oxides) – ∆ 2– (4) M-On-1(OH) or M-On-1 F. Therefore, the procedure 1.41nln2( HOM + 87.3)/(nl + n2) (kcal/mol) to calculate g Ca(OH)2 for example, would be: g in which n and n are in a given compound, the Ca(OH)2 = g CaO + g H2O(OH). From the values of 1 2 ∆ ∆ number of oxygens required to balance the cation table 2, the G°f or H°f of any phosphate can be and phosphorous, respectively. easily calculated using the equations (1) and (2). Nriagu (1976), employing the hypothesis of sum Thus, for example, to calculate the standard ther- of the free energy of constituents, proposed the fol- modynamic properties of ammonium taranakite, ∆ (NH ) Al (P0 ) H 18H 0: lowing equation, which permit to estimate the G°f 4 3 5 4 8 6 2 of complex phosphates and hydroxyphosphates: ∆ G°f(NH4)3Al5(P04)8H618H2O = 3/2 g (NH4)2O + ∆ 5/2 g A12O3 + 4g P2O5 + 3 g H2O(H) + l8 g H2O(cryst.) = G°f(hydroxyphosphate) = Σ∆ ∆ –16,106.33 ± 41.60 kJ mo1-1. G°f(ri) + n G°fH20 – Q (5) ∆ ∆ H°f(NH4)3Al5(P04)8H618H2O = 3/2 h (NH4)2O + where G°f(ri) is the Gibbs free energy of formation 5/2 h Al2O3 + 4h P2O5 + 3h H2O(H) + l8h H2O(cryst.) = of the ith reactant hydroxide or phosphate compo- -1 nent within the hydroxyphosphate matrix and Q is a –18,484.51 ± 38.45 kJ mo1 . correction factor. According to Chen (1975), These results are in perfect agreement with the Σ Q=RT (nilnni) where ni is the reaction coefficient values of –16,129.15 kJ mo1-1 and –18,532.60 kJ for the ith hydroxide or phosphate. Using equa- mo1-1 given by Vieillard & Tardy (1984). The devi- ∆ tion (5), values of G°f(hydroxyphosphate) with devia- ations have been calculated using the equation: tions less than 3% with regard to experimental val- σ Σ σ 2 1/2 ues, can be obtained. = [ (ni i) ] (6) Estudios Geol., 65(2), 109-119, julio-diciembre 2009. ISSN: 0367-0449. doi:10.3989/egeol.39849.060 FB4-09 La Iglesia 21/12/09 08:29 Página 111 Estimating the thermodynamic properties of phosphate minerals 111 Table 1.—Crystalline phases selected to calculate gi and hi values Li3PO4 (Lithiophosphatite) hi Ca10(P04)6F2 (Fluorapatite) gi + hi Li4P2O7 hi Ca10(P04)6C12 (Chorapatite) gi + hi Na4P2O7·10H2Ohi CaIIPO4·2H2O (Brushite) gi + hi Na5P3O10 gi + hi Ca2C1PO4 gi Na2HPO4·12H2Ogi + hi Fe3(P04)2·8H2O (Vivianite) gi + hi Na2H2P2O7 gi Fe3(PO4)2 gi Na3HP2O7 hi Co3(PO4)2 gi KMgPO4·6H2Ohi Co(UO2)2(PO4)2·7H2Ogi K2HPO4 hi Ni2P2O7 gi KUO2PO4·3H2Ogi Ni(UO2)2(PO4)2·7H2Ogi KAI2(P04)2OH·2H2O (Minyulite) gi Zn3(PO4)2·4H2O (Hopeite) gi + hi NH4MgPO4·6H2O (Struvite) gi Zn3(PO4)2 gi NH4A12(PO4)2OH·2H2O (Am. Minyulite) gi Zn2P2O7 hi NH4UO2PO4·3H2Ogi Cu3(PO4)2 gi + hi NH4H2PO4 hi Cu2P2O7 gi (NH4)2HPO4 hi Cu(H2PO4)2 hi Mg3(PO4)2 (Farringtonite) gi + hi Pb3(PO)2 gi + hi Mg4(PO4)2 gi Pb4(PO4)3 gi Mg5(PO4)3Fgi A14(PO4)3(OH)3 (Trolleite) gi + hi Mg3(PO4)2·8H2O (Bobierrite) gi A1PO4 (Berlinite) gi + hi Ca10(PO4)6(OH)2 (Hydroxyapatite) gi + hi To evaluate the accuracy of method a new set of Table 2.—gi and hi values of each basic unit -1 individual contributions for gi and hi using another calculated by the method of least-squares (kJ mo1 ) 18 crystalline phases has been calculated and com- pared to results of table 2. The averages of differ- basic unit gi error ± hi error ± ences are 0.62% for gi and 0.43% for hi(N = 10 va- P2O5 –1,636.94 8.44 –1,726.84 7.22 lues considered each). Li2O — — –817.30 12.97 Na2O –665.22 4.54 –719.50 3.40 The values of gi and hi of table 2 have also been K2O –751.28 1.24 –774.50 3.26 checked for internal consistency versus values of (NH4) O –359.62 6.12 –541.28 9.72 ∆ ∆ 2 Robie et al.