Pitzer’s equation

Plummer, L. N., D.L. Parkhurst, G. W. Fleming and S. A. Dunkle, 1988, PHRQPITZ, a computer program incorporating Pitzer s equations for calculation of geochemical reactions in brines. US Geological Survey Water-Resources Investigations Report 88—4153. 

Generally, agreement has been found between our correlations and those of Pitzer, and others (1972, 1973, 1974, 1975, 1976) and Rard, and others (1976, 1977). Many of our correlations agree fairly well with Robinson and Stokes, (1965) and Harned and Owen, (1958) but in most cases a much larger data base and more recent measurements have been incorporated into the evaluations. It has been observed that agreement with Pitzer s equations is found below moderate concentrations (several molal), but often deviate at higher concentrations where the Pitzer equations do not contain enough parameters to account for the behavior of the activity (or osmotic) coefficient. 

The agreement with Pitzer s equations for sulfuric acid is reasonably good up to nearly 5 mol kg l. This agreement is depicted in Fig. 1, where the square symbols show values from Pitzer s equations, the crosses are experimental results, and the solid line is my evaluation (Staples, 1980). 

The activity of water is obtained by inserting Eq. (6.12) into Eq. (6.11). It should be mentioned that in mixed electrolytes with several components at high concentrations, it is necessary to use Pitzer s equation to calculate the activity of water. On the other hand, uhjO is near constant (and = 1) in most experimental studies of equilibria in dilute aqueous solutions, where an ionic medium is used in large excess with respect to the reactants. The ionic medium electrolyte thus determines the osmotic coefficient of the solvent. 

In more complex solutions of high ionic strengths with more than one electrolyte at significant concentrations, e.g., (Na, Mg, Ca " ) (Cl, SOl ), Pitzer s equation may be used to estimate the osmotic coefficient the necessary interaction coefficients are known for most systems of geochemical interest. 

Figure 18.3 Comparison of osmotic coefficients at T= 298.15 K for three different electrolytes as calculated from Pitzer s equations (solid lines) with the experimental results (symbols).

Thermal Properties Pitzer s equations for ln7 and 4> [equations (18.18) to (18.26)] can be used to obtain relative partial molar enthalpies L and L2, and relative partial molar heat capacities8 7j and J2, by taking derivatives. For... 

We have applied Pitzer s equations at T = 298.15 K, but they are not limited to that temperature and can be applied at any temperature where the coefficients are known.k Table I8.l (and Table A7.1 of Appendix 7) gives the Debye-Hiickel coefficients AA, Ah, and Aj as a function of temperature, but the coefficients specific to the electrolyte are tabulated in Appendix 7 only at T = 298.15 K. The usual solution to this problem is to express the coefficients as... 

Starting with Pitzer s equations, internally consistent generalized equations can be written to express the thermodynamic properties of aqueous electrolytes as a function of pressure, as well as temperature and molality. For example, Archer10 gives the equations for calculating 7 , , L, 4>CP, V, K and E for aqueous NaCl solutions1 as a function of p, T, and m over the temperature range from... 

Figure 18.5 Graph of (a) the osmotic coefficient and (b) the activity coefficient for NaCl(aqueous) at p = 0.1 MPa as a function of temperature. The curves were obtained by using temperature-dependent coefficients in Pitzer s equations. The dotted line is for r=273.15 K and the dashed line is for T = 298.15 K. The solid lines are for T= 323.15, 373.15, 423.15, 473.15, 523.15, and 573.15 K, with both and 7 decreasing with increasing temperature.

The log K values shown in Figure 18.10 are the values that best reproduce all of the heat of mixing curves.v The J1 values are obtained by estimating initial values using the activity coefficients for NaCl(aq).16 These initial values of Jy are then readjusted, as the value for Km is optimized, by adjusting the coefficients of Pitzer s equations, whose form is described in the previous section. Pitzer s equations are, of course, internally consistent so that adjustments to the activity or osmotic coefficient parameters result in adjustments to the thermal parameters (L, L2, 4>J, or J2), and hence, to the heat effects. 

Note the similarity between this term and the one obtained in Pitzer s equations for simpler electrolytes. 

In this appendix, we summarize the coefficients needed to calculate the thermodynamic properties for a number of solutes in an electrolyte solution from Pitzer s equations.3 Table A7.1 summarizes the Debye-Huckel parameters for water solutions as a function of temperature. They provide the leading terms for Pitzer s equations, and can also be used to calculate the Debye-Huckel limiting law values from the equations... 

Tables A7.2 to A7.6 summarize other coefficients needed to apply Pitzer s equations for the calculation of 7 and (f> for various 1 1, 2 1, 3 1, 4 1, 5 1, and 2 2 electrolytes. The equations for calculating the osmotic coefficient are...

Chapter 18 describes electrolyte solutions that are too concentrated for the Debye-Hiickel theory to apply. Gugenheim s equations are presented and the Pitzer and Brewer tabulations, as a method for obtaining the thermodynamic properties of electrolyte solutions, are described. Next, the complete set of Pitzer s equations from which all the thermodynamic properties can be calculated, are presented. This discussion ends with an example of the extension of Pitzer s equations to high temperatures and high pressures. Three-dimensional figures show the change in the thermo-... 

The three appendices in this volume give selected sets of thermodynamic data (Appendix 5), review the statistical calculations covered in Principles and Applications (Appendix 6), and summarize the equations and parameters required to calculate the properties of electrolyte solutions, principally from Pitzer s equations (Appendix 7). 

Plummer LN, Parkhurst DL, Fleming GW, Dunkle SA (1988) A computer program incorporating Pitzer s equation for calculation of geochemical reactions in brines. U S Geol. Surv. Water Resour.Inv.Rep. 88-415 3... 

Plummer L. N., Parkhurst D. L., Fleming G. W., and Dunkle S. A. (1988) A Computer Program Incorporating Pitzer s Equations for Calculation of Geochemical Reactions in Brines. US Geol. Surv. Water-Resour. Invest. Report 88-4153, 310pp. 

Na-Mg) interactions for mixtures (NaCl -h MgCl2) are related to and the ternary interactions (Na-Mg-Cl) are related to The Pitzer s equations thus incorporate Young s rule in all of its formulations. This general approach, although somewhat complicated, can account for all the possible interactions in a stepwise manner. Computer codes have been written that can be used to estimate the physical-chemical properties of natural waters over a wide range of temperatures (0-100 °C) and ionic strengths (0-6 m) (Millero, 2001). 

Felmy A. R. and Rai D. (1999) Application of Pitzer s equations for modeling aqueous thermodynamics of actinide species in natural waters a review. J. Solut. Chem. 28(5), 533-553. 

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