Estimating the thermodynamic properties of phosphate minerals at high and low temperature from the sum of constituent units

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 contribution of the distinct units, with more accuracy than the methods published up until now.


Introduction
The estimation of the thermodynamic properties of silicate minerals has been considered by many investigators.Wilcox & Bromley (1963), Slaughter (1966), Karpov & Kashik (1968), Tardy & Garrels (1974), Nriagu (1975), Chen (1975), Robinson & Haas (1983), Sposito (1986) and La Iglesia & Aznar (1986, 1990) have published methods based on the hypothesis that the free energy or enthalpy of formation is related to the sum of free energies of formation of their constituent oxide or hydroxide components.This summation hypothesis has a firm theoretical backing in that the macroscopic mineral thermodynamic properties are the cumulative result of the lattice vibrational energies of the various elements in the lattice, Grimwall (2001).
Following the polyhedral units model of Hazen (1985), Chermak & Rimstidt (1989, 1990), determined by multiple linear regression the Gibbs free energy of formation (g i ) and enthalpy of formation (h i ) of polyhedral units in silicate minerals: ∆G°f = Σn i g i (1) and ∆H°f = Σn i h i (2) 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 where n i is the number of moles of ith component per formula unit.With this method ∆G°f or ∆H°f of many silicates can be calculated with an uncertainty of less than 1%.Using this approach, La Iglesia & Felix (1994) calculated values of g i and hi for carbonates to create a model that predicts their ∆G°f and ∆H°f with an uncertainty of less than 0.5%.Tardy & Garrels (1976, 1977) published a method of estimation of the thermodynamic properties of minerals based on the fact that the free energy and enthalpy of formation within a family of compounds from their constituent oxides are linearly dependent on a parameter called ∆O M 2-, which is a function of the electronegativity of the constituent cations (M 2+ ).This correlation provides a means of estimating values for unmeasured minerals.So, Tardy & Gartner (1977) and Tardy & Vieillard (1977) have obtained empirical equations that estimate the Gibbs free energy of formation and enthalpy of formation of carbonates, nitrates, sulfates and phosphates with a deviation less than 3% with respect to the experimental values.For the phosphates, these authors obtain the following equations: and in which n 1 and n 2 are in a given compound, the number of oxygens required to balance the cation and phosphorous, respectively.Nriagu (1976), employing the hypothesis of sum of the free energy of constituents, proposed the following equation, which permit to estimate the ∆G°f of complex phosphates and hydroxyphosphates: where ∆G°f (ri) is the Gibbs free energy of formation of the ith reactant hydroxide or phosphate component within the hydroxyphosphate matrix and Q is a correction factor.According to Chen (1975), Q = RT Σ(n i lnn i ) where n i is the reaction coefficient for the ith hydroxide or phosphate.Using equation ( 5), values of ∆G°f (hydroxyphosphate) with deviations less than 3% with regard to experimental values, can be obtained.
In this paper, employing a method of leastsquares, the contribution of 19 constituent units to the Gibbs free energy and 15 units to the enthalpy of phosphates has been calculated.From these data we can estimate the thermodynamic properties of other phosphates with more accuracy than the methods published up until now.

Calculation of the Gibbs free energies. enthalpies and entropies of phosphate units at 298 °K
To evaluate the contribution of the units to the Gibbs free energy and enthalpy of phosphate minerals 41 crystalline phases (table 1) were selected.A system of 31 linear equations was established to calculate g i and 23 linear equations were used to calculate h i .To solve both systems we used the program MAPLE V release 2. The g i and h i coefficients were determined using a least-squares method.
The values of g i and h i obtained for each of the polyhedral units, together with their uncertainty ranges, are presented in table 2. Considering the easy substitution of O ligand by OH, F or Cl in these compounds, average values of g H 2 O (OH), g F, g Cl, h H 2 O (OH), h F and h Cl have been also calculated.These values represent the difference between the energy of polyhedral units M-O n and M-O n-1 (OH) or M-O n-1 F. Therefore, the procedure to calculate g Ca(OH) 2 for example, would be: g Ca(OH) 2 = g CaO + g H 2 O (OH).From the values of table 2, the ∆G°f or ∆H°f of any phosphate can be easily calculated using the equations (1) and (2).Thus, for example, to calculate the standard thermodynamic properties of ammonium taranakite, (NH 4 ) 3 Al 5 (P0 4 ) 8 H 6 18H 2 0: These results are in perfect agreement with the values of -16,129.15kJ mo1 -1 and -18,532.60 kJ mo1 -1 given by Vieillard & Tardy (1984).The deviations have been calculated using the equation: To evaluate the accuracy of method a new set of individual contributions for g i and h i using another 18 crystalline phases has been calculated and compared to results of table 2. The averages of differences are 0.62% for g i and 0.43% for h i (N = 10 values considered each).
The values of g i and h i of table 2 have also been checked for internal consistency versus values of Robie et al. (1979) of ∆G°f and ∆H°f free oxides obtaining the following relations: g i = 1.042 ∆G°f(oxides) -80.115 kJ mo1 -1 (r = 0.965) h i = 1.047 ∆H°f(oxides) -87.455 kJ mo1 -1 (r = 0.967) equivalents to the equations ( 3) and (4).In the same way in which the free energy or enthalpy of a compound has just been defined as the sum of the free energies or enthalpies of its constituent units, we can also define the entropy as: S = Σn i s i , where s i is the entropy of the constituent unit.Taking into account the thermodynamic equation: ∆G = ∆H -T∆S; it is easy to obtain the equation that relates the entropy of polyhedral units with free energy and enthalpy: s i = (h i -g i )/T, but it is recommendable to derivate these partial entropies following Holland (1989).This leads to lower propagated uncertainty on the resulting estimates.
Due to the scarce number of data found in the bibliography for ∆G°f and ∆H°f (phosphate), the contribution of the cation coordination number to the Gibbs free energy values of polyhedral units (which is important in calcium, aluminium and lead phosphates where different types of coordination  take place) could not be considered, as was done for the case of silicates and carbonates (Chermak & Rimstidt, 1989& 1990and La Iglesia & Felix, 1994).For this reason, the precision of the values calculated for g i and h i phosphate, is lower than that of silicates and carbonates.
Calculation of the free energies of polyhedral phosphate units at high temperature Chermak & Rimstidt (1990) published a method that permits the estimation of free energy of any silicate at high temperature, based on the approximation ∆C/δT = 0, which gives for the free energy at the polyhedral units versus temperature: This equation allows the calculation of ∆G°f.T of any silicate, in the temperature range 298-650 K, with an uncertainty of 0.25% (in relation to experimental value of Robie et al. (1979)).For temperatures higher than 650K the estimated values of ∆G°f.T can carry an uncertainty of 0.5%.This method has been applied for the estimation ∆G°f.T in carbonates, in the temperature range 400 to 1.000K with difference less 0.60% (La Iglesia & Felix, 1994).This is therefore an excellent method for estimating the ∆G°f.T of silicates and carbonates.
Using Equation 7, we have calculated the temperature function g i.T of the phosphate units, which appears in table 3.

Results
Tables 4 and 5 compare the standard free energy and enthalpy values predicted in this paper with the bibliographic data; the differences are shown as percent of relative error (% residual), R = 100(∆G cal -∆G bibl )/∆G bibl .When several experimental values of ∆G°f were found for one crystalline species we have calculated the residual error in relation to their average value (∆G bibl or ∆H bibl ).For ∆G°f values, the calculated average residual error is: R = 0.029, and standard deviation, ∆ n = 0.619 (N = 82 values considered).For ∆H°f, R = -0.003,and ∆ n = 0.525 (N = 58 values considered).From the 82 values of ∆G°f estimated, 74 values have an error less than 1%, 7 values are between 1 and 2% and only 1 value is more than 2%.Similar distribution is found for ∆H°f.All values are lower than 2%, and only 3 are over 1%.The preceding results confirm the goodness of the proposed method for the estimation of the thermodynamic properties of phosphates.Figures 1 and 2 present the frequency histogram of percent residual differences between bibliographic and calculated values for ∆G°and ∆H°.In both cases, a Gaussian distribution is obtained.
The accuracy of the proposed method was tested by predicting the thermodynamic properties of the phosphates not used to calculate g i and h i contribution (in tables 4 and 5 the values of ∆G°f and ∆H°f used to calculate g i and h i have been marked with *).The average residual error and standard deviation calculated from the above data for ∆G°f and ∆H°f are: R = 0.002 and ∆ n = 0.697 (N = 51 values considered) and R = 0.075 and ∆ n = 0.583 (N = 35 values considered), respectively.The uncertainty ranges of these estimates data are larger that those obtained by La Iglesia & Felix (1994) in the estimation of thermodynamic properties of carbonates.It was not possible to obtain better results for the following two reasons: 1) as indicated before, the contribution of the cation coordination to the Gibbs free energy values of the polyhedral units could not be considered as in the case of silicates and carbonates, and 2) the poorer agreement amoung the reported thermodynamic data for the phosphates.In this way, we have calculated the residual errors, in relation to the average values of ∆G°f and ∆H°f for some bibliographic data of fluorapatite and hydroxyapatite selected by Tacker & Stormer (1989) obtaining the following values of standard deviation: σ n = 0.666 (for ∆G°f, N = 17 values considered) and σ n = 0.645 (for ∆H°f, N = 19  1968), ( 4) Latimer (1952), (5) Parker et al. (1971), ( 6) Naumov et al. (1971), ( 7) Duff (1971a), (8) Rossini et al. (1952), ( 9) Duff (1972), (10) Robie et al. (1979), ( 11) Duff (1971b), ( 12) Tacker & Stormer (1989), ( 13) Al-Borno & Tomson (1994), ( 14) Woods & Garrels (1987).
values considered), an uncertainty level similar to the method here proposed!Table 6 compares the calculated values of free energy at high temperature, using equation 8, to the published experimental data of Robie et al. (1979) for berlinite, whitlockite, fluorapatite and hydroxiapatite in the temperature range of 400 to 700 K.In all of the studied cases, the relative error is less than 0.9%, with R = -0.044and σ n = 0.428 (N = 16 values considered).

Conclusions
The method proposed by Chermak & Rimstidt (1989) for calculation of free energy and enthalpy of silicates, by summing the contributions of polihedral units, is applicable to the estimation of ∆G°f and ∆H°f of phosphate minerals.This method allowed us to calculate the thermodynamic properties of phosphates with more accuracy than the methods published until now, with the following aditional advantages: a) use of a simple easy computed mathematical procedure, and b) the ability to obtain a wider field of application among the phosphates.
The calculation of free energy at high temperature proposed by Chermak & Rimstidt (1990) for silicates is also applicable to phosphates, giving values very close to the experimental data.

Table 2
.-g i and h i values of each basic unit calculated by the method of least-squares (kJ mo1 -1 )

Table 5 .
-Comparison of ∆H f

Table 5 .
-Comparison of ∆H f