Adapting molar data (without density) for molal models

ARTICLE IN PRESS

Computers & Geosciences 33 (2007) 829–834 www.elsevier.com/locate/cageo

Short Note

Adapting molar data (without density) for molal models$ Giles M. Marion Desert Research Institute, 2215 Raggio Parkway, Reno, NV 89512, USA Received 6 July 2006; received in revised form 10 October 2006; accepted 25 October 2006

Abstract Theoretical geochemical models for electrolyte solutions based on classical thermodynamic principles rely largely upon molal concentrations as input because molality (wt/wt) is independent of temperature and pressure. On the other hand, there are countless studies in the literature where concentrations are expressed as molarity (wt/vol) because these units are more easily measured. To convert from molarity to molality requires an estimate of solution density. Unfortunately, in many, if not most, cases where molarity is the concentration of choice, solution densities are not measured. For concentrated brines such as seawater or even more dense brines, the difference between molarity and molality is significant. Without knowledge of density, these brinish, molar-based studies are closed to theoretical electrolyte solution models. The objective of this paper is to present an algorithm that can accurately calculate the density of molar-based solutions, and, as a consequence, molality. The algorithm consist of molar inputs into a molal-based model that can calculate density (FREZCHEM). The algorithm uses an iterative process for calculating absolute salinity ðS A Þ, density ðrÞ, and the conversion factor ðCF Þ for molarity to molality. Three cases were examined ranging in density from 1.023 to 1.203 kg(soln.)/l. In all three cases, the SA , r, and CF values converged to within 1ppm by nine iterations. In all three cases, the calculated densities agreed with experimental measurements to within 0:1%. This algorithm opens a large literature based on molar concentrations to exploration with theoretical models based on molal concentrations and classical thermodynamic principles. r 2007 Elsevier Ltd. All rights reserved. Keywords: Density; Molarity; Molality; Salinity; Geochemical models; Electrolyte solutions; Algorithm

1. Introduction Theoretical geochemical models for electrolyte solutions based on classical thermodynamic principles rely largely upon molal concentrations as input because molality (wt/wt) is independent of temperature and pressure, which greatly facilitates model $

Code available from server at http://www.iamg.org/ CGEditor/index.htm. Tel.: +1 775 673 7349; fax: +1 775 673 7485. E-mail address: [email protected]. 0098-3004/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cageo.2006.10.009

development (e.g., Marion et al., 2005; Millero, 2001; Nordstrom and Munoz, 1994, Pitzer, 1991, 1995). These theoretical models can further our understanding of natural processes by exploring facets such as degree of mineral saturation and evolution of solutions with changing temperature, pressure, and gas concentrations. On the other hand, there are countless studies in the literature where concentrations are expressed as molarity (wt/vol) because these units are more easily measured (e.g., Bockheim, 1997; Kohut and Dudas, 1994; Lyons et al., 2005; Ouellet et al., 1989; Skarie

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et al., 1987). To convert from molarity to molality requires an estimate of density (r). Unfortunately, in many, if not most, cases where molarity is the concentration of choice, aqueous solution densities are not measured. In dilute aqueous solutions [ionic strength ðIÞp0:1 m], molarity and molality are nearly equivalent ½r  1:00 kgðsoln:Þ=l, and lack of density is a nonsignificant issue. However, for concentrated brines such as seawater or even more dense brines, the difference between molarity and molality is significant. Without knowledge of density, these brinish, molar-based studies are closed to theoretical aqueous solution models. For example, in Lyons et al. (2005), molal data from nonsaline Lakes Fryxell and Hoare (Antarctica) were used with the FREZCHEM model to explore saline Lake Bonney processes rather than Lake Bonney data because only molar data were available

Table 1 A FORTRAN code for calculating density and molal concentrations from molar data (see text for definition of terms) I ¼ 1 RHO(I) ¼ 1.0 SA ¼ SL/RHO(I) CF ¼ (1/RHO(I))*(1/(1-SA/1000)) DO J ¼ 1,N MOLAL(J) ¼ CF*MOLAR(J) END DO I ¼ I+1 CALL DENSITY (MOLAL(N),N,RHO(I)) IF((RHO(I)-RHO(I-1)).GT.1E-6) GO TO 10 CONTINUE

10

for Lake Bonney (see Table 1 in Lyons et al., 2005). The objective of this paper is to open these closed cases to theoretical models by demonstrating how density can be estimated based on molar inputs used in conjunction with a theoretical molal model that can estimate density. 2. Theory The conversion of molarity ðM j Þ to molality ðmj Þ of the jth species is given by    1 kgðsoln:Þ Mj (1) ¼ mj , r kgðwaterÞ where kgðsoln:Þ ¼ kgðwaterÞ



1:00 1:00  S A =1000

 (2)

and S A is absolute salinity [g salt/kg(soln.)]. In what follows, the expression    1 kgðsoln:Þ CF ¼ (3) r kgðwaterÞ will be referred to as the conversion factor (see Eq. (1)). The approach to estimate the density of aqueous molar solutions requires a geochemical model that can estimate solution density based on molal concentrations. The model that will be used in this study is called FREZCHEM and is based on the Pitzer approach (Pitzer, 1991, 1995) for estimating activity coefficients, the activity of water, density, and solubility products as functions of temperature

Table 2 A comparison of model inputs (molar concentrations and S L ) and resulting model outputs (SA , r, and conversion factors) Lake Bonney, West Lobea

Lake Bonney, East Lobea

Seawaterb

Na (mol/l) K (mol/l) Mg (mol/l) Ca (mol/l) Cl (mol/l) SO4 (mol/l) Alkalinity (equil./l) SL (g salt/l)

1.734 0.041 0.378 0.041 2.433 0.051 0.078 148.47

3.2844 0.071 1.074 0.037 5.499 0.037 0.0044 305.45

0.47995 0.01045 0.05406 0.01060 0.55955 0.02890 0.00237 35.99

SA [g salt/kg(soln.)] rcalc: [kg(soln.)/l] rexpt: [kg(soln.)/l] Conversion factor [l/kg(H2 O)] Convergence (1 ppm) iterations

134.84 1.1011 1.102 1.04972 6

253.99 1.2026 1.203 1.11465 9

35.17 1.023302 1.023344 1.01285 5

a

Data from Torii and Yamagata (1981) and Lyons et al. (2005). Data from Feistel (2003).

b

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Input ρinit, SL, Mj

i=1 Calculate Salinity SAi = SL / ρi No IF

Calculate Conversion Factor CFi =

1 ρi

( )((1 - S

1

Ai

)

/1000)

ρi - ρ(i-1) < convergence criterion

Yes

Print Results, Exit

Calculate Molality mij = CFi * Mj

i=i+1

Calculate density (ρi) with model

Fig. 1. Algorithm flowchart for estimating absolute salinity, density, and conversion factor for molarity to molality.

(70 to 25  C), pressure (1–1000 bars), and composition (81 solid phases, with maximum Ip20 m) (Marion, 2001, 2002; Marion and Farren, 1999; Marion et al., 2003, 2005, 2006). Fig. 1 is a flowchart for this mathematical algorithm. As input, we need to specify an initial estimate of density ðrinit Þ, molar salinity [SL (g salt/l)], and molar concentrations [M j (mol/l)]. For the three examples discussed below, we set rinit ¼ 1:00 kgðsoln:Þ=l. Initially, there are four unknowns in the algorithm: r, S A , CF, and mj ; but only two of these unknowns are mathematically independent. We use rinit and S L to estimate S A and CF, which allows a ‘‘first approximation’’ of molal concentrations ðmj Þ, which then are used as input into the FREZCHEM model to calculate a new density (Fig. 1). This procedure iterates until ri , SAi ,

CF i , and mij converge. This is a simple algorithm that only takes a few lines of code at the beginning of a computer program. For example, Table 1 contains a FORTRAN code for implementing this algorithm. 3. Results and discussion We evaluated three electrolyte solutions based on estimates of molarity (Table 2). The first eight rows are algorithm inputs; the last rows contain algorithm outputs of SA , r, and CF. To estimate molality, you need to simply multiply the molar concentrations by their respective conversion factor. The molar concentrations (mol/l) of Lake Bonney (Antarctic lakes) in Table 2 were calculated from g/ kg(soln.) data (Torii and Yamagata, 1981) multiplied

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DENSITY [kg(soln-)/L] or CONVERSION FACTOR [L/kg(H2O)]

832

1.45 RHO-L.B.W.L.

1.40

CF-L.B.W.L. RHO-L.B.E.L.

1.35

CF-L.B.E.L. 1.30 1.25 1.20 1.15 1.10 1.05 1.00 0

1

2

3

4 5 ITERATION

6

7

8

9

Fig. 2. Iterative convergence of density ðrÞ and conversion factor ðCF Þ for the two Lake Bonney cases (L.B.W.L. ¼ Lake Bonney West Lobe, L.B.E.L. ¼ Lake Bonney East Lobe).

by density [kg(soln.)/l] at 24  C (Torii and Yamagata, 1981; Torii et al., 1977) and molar masses (mol/g). All of the Lake Bonney data were from Torii and Yamagata (1981), except for alkalinity (equil./l) that came from Lyons et al. (2005). The original Lake Bonney data (Torii and Yamagata, 1981; Lyons et al., 2005) were out of charge balance by 5.7–7.8%. Because these two solutions are dominated by Na and Cl, we created a perfect charge balance, which is necessary for model calculations dealing with alkalinity, by adjusting the Na concentration. We chose this option for charge balance on the assumption that Cl is more likely to be accurate than Na (Millero and Sohn, 1992). Below we will also examine an alternative assumption that adjusts both Na and Cl equally to bring the solution into charge balance. Fig. 2 demonstrates how r and CF change with iterations for these two Lake Bonney examples. Initially r was set equal to 1.00. Calculating CF based on r ¼ 1 leads to apparent CF values of 1.17 and 1.44 in Iteration 1. Using these CF values to calculate molality (Fig. 1) and running the FREZCHEM model with these molal concentrations leads to new density estimates of 1.11 and 1.25 kg(soln.)/l at the end of Iteration 1. These new densities are

inputs to Iteration 2, which causes precipitous drops in CF and small drops in r which, in turn, are inputs to Iteration 3. What is clear from Fig. 2 is that r and CF (and S A , not shown) for a particular case converged rapidly to within 1 ppm by six to nine iterations. The number of iterations needed to converge within 1 ppm is a function of density (Fig. 2, Table 2). Our model estimates for the densities of the West and East Lobes of Lake Bonney are 1.1011 and 1.2026 kg(soln./l), which are in excellent agreement with experimental measurements of 1.102 and 1.203 kg(soln.)/l (Table 2). We also estimated the density of the West and East Lobes of Lake Bonney by splitting the charge imbalance difference ðDÞ between Na and Cl (Na is increased by D=2 and Cl is decreased by D=2). In this case, the model calculated densities are 1.0975 kg(soln.)/l for the West Lobe and 1.1972 kg(soln.)/l for the East Lobe. For the West Lobe in the first case (Na solely adjusted), our model estimate of r was low by 0.08% (Table 2), and in the second case (Na and Cl equally adjusted), our model estimate of r was low by 0.41%. Similarly for the East Lobe in the first case, our model estimate was low by 0.03% (Table 2), and in the second case, our model estimate was low by 0.48%. These comparisons would argue in favor of the initial assumption that

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Cl is more accurate than Na. In either case, however, given the experimental uncertainties in charge balance (5.7–7.8%), these errors are within the limits of the assumptions used in these calculations. While these Lake Bonney comparisons are typical of field studies (charge imbalances of 5–10% are not uncommon), a sounder way to assess the validity of this algorithm is to compare it to a more accurately-known electrolyte solution such as seawater. The last column in Table 2 has seawater molar concentrations estimated from the Feistel (2003) model for S (seawater salinity, ppt)¼ 35:00, T ¼ 25  C, and P (gage pressure) ¼ 0:00 bars. The standard error in estimating solution densities for the Feistel model is about 10 ppm compared to experimental measurements. Using the molar concentrations ðM j Þ, S L ¼ 35:99 g salt=l, and rinit ¼ 1:00 kgðsoln:Þ=l as initial inputs leads within five iterations to convergence onto r ¼ 1:023302 kgðsoln:Þ=l. This value differs from the Feistel (2003) model value by 42 ppm (Table 2) and differs by 41 ppm from another estimate of this seawater density by Millero (2001). Given that the FREZCHEM model is not, per se, a seawater model, this agreement is within the error limits of the model. Note that the original SL value of 35.99 g salt/l eventually leads to S A ¼ 35:17 g salt=kgðsoln:Þ (Table 2), which is in perfect agreement with experimental measurements for S ¼ 35:00 (Millero, 2001). Had we used molal concentrations as input directly into the FREZCHEM model, as is normally done, our model estimate of the density of this solution is 1.023301 kg(soln.)/l, which is virtually identical to the indirect estimate [1.023302 kg(soln.)/l] based on molar inputs, the new algorithm (Fig. 1), and calculated molalities. What limits the accuracy of this algorithm for calculating the density of molar solutions and the conversion of molar to molal concentrations are the accuracies of the geochemical data, the geochemical model, and temperature. Temperatures were known for the densities of the three examples in Table 2. But it may be necessary, at times when the temperature of the molar solution is not given, to assume a temperature like ‘‘room’’ temperature ð 21  CÞ. For the Lake Bonney, East Lobe, our estimate of density at 24  C was 1.2026 kg(soln.)/l (Table 2); assuming a room temperature of 21  C leads to a density of 1.2039 kg(soln.)/l, which is 0.11% different than at 24  C. The magnitude of error can also be estimated via the thermal

expansion coefficient ðaÞ, which is defined as   1 dr . a¼ r dT

833

(4)

For seawater at T ¼ 25  C, S ¼ 35:0, and P ¼ 0:0 bars, a ¼ 0:0002976ð1=KÞ and r ¼ 1:023344 kg ðsoln:Þ=l (Feistel, 2003), which results in dr=dT ¼  0:00030455 kg=l=K. An error of 3 K would cause an error in density of seawater of 0.09%. For most practical applications, errors in calculated densities and molalities of  0:1% are insignificant. Appending this simple algorithm at the beginning of a molal-based, density model would enable program input in either molar or molal units. Alternatively, one could use the algorithm to calculate density and molal concentrations and export this data to one’s favorite geochemical model. This algorithm opens a large literature based on molar concentrations to exploration with theoretical models based on molal concentrations and classical thermodynamic principles. After this paper was accepted, I found an earlier paper by Monnin (1994) that uses a similar, but not identical, algorithm for estimating the density of molar solutions. The Monnin paper contains a complete FORTRAN program for estimating solution densities from molar data at 25  C and 1 bar pressure. Alternatively, one can append the simple FORTRAN code of Table 1 to a broader T-Pcomposition model as was done in this paper. Acknowledgements Funding was provided by a NASA Planetary Geology and Geophysics Project, ‘‘An Aqueous Geochemical Model for Cold Planets,’’ and a NASA EPSCoR Project, ‘‘Building Expertise and Collaborative Infrastructure for Successful Astrobiology Research, Technology, and Education in Nevada.’’ I thank Lisa Wable for assistance in preparing the manuscript, and Kirk Nordstrom, Ron Spencer, and Frank Millero for responding to inquiries concerning similar algorithms. I also thank two anonymous reviewers for constructive comments. References Bockheim, J.G., 1997. Properties and classification of cold desert soils from Antarctica. Soil Science Society of America Journal 61, 224–231. Feistel, R., 2003. A new extended Gibbs thermodynamic potential of seawater. Progress in Oceanography 58, 43–114.

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