Ind. Eng. Chem. Res. 2002, 41, 5109-5113
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Debye-Hu 1 ckel Model for Calculating the Viscosity of Binary Strong Electrolyte Solutions at Different Temperatures Manoel J. C. Esteves, Ma´ rcio J. E. de M. Cardoso,* and Oswaldo E. Barcia Laborato´ rio de Fı´sico-Quı´mica de Lı´quidos e Eletroquı´mica, Departamento de Fı´sico-Quı´mica, Instituto de Quı´mica, Universidade Federal do Rio de Janeiro, Centro de Tecnologia, Bloco A, sala 408, Cidade Universita´ ria, 21949-900, Rio de Janeiro, RJ, Brasil
In the present article, a recently published model (Esteves, M. J. C.; Cardoso, M. J. E. de M.; Barcia, O. E. Ind. Eng. Chem. Res. 2001, 40, 5021) for calculating the viscosity of binary strong electrolyte solutions, at 25 °C and 0.1 MPa, has been extended for calculating the viscosity of binary strong electrolyte solutions at different temperatures. A temperature dependence has been introduced into the two adjustable parameters of the original model. The empirical expression originally proposed by Silvester and Pitzer (J. Phys. Chem. 1977, 81, 1822) to take into account the temperature dependence of thermodynamic properties of aqueous electrolyte solutions has been adopted. The proposed model contains a total of five adjustable parameters that have been fitted by means of experimental viscosity data in the literature. The total number of 20 binary electrolyte systems (at 0.1 MPa and in the temperature range of -35 to 55 °C) with two different solvents (water and methanol) have been studied. The overall average mean relative standard deviation is 0.98% 1. Introduction It is possible to find in the literature several empirical and semiempirical models for calculating the viscosity of binary strong electrolyte solutions at different temperatures, mostly for water as the solvent.1 The most extensively used model has been the one proposed by Jones and Dole.1 Jenkins and Marcus2 present, in their review article on the Jones and Dole model, correlations for calculating the so-called JonesDole B parameter as a function of temperature for several ions in aqueous solutions. For both aqueous and nonaqueous solvents, they also show tables for the ionic B parameter at different temperatures. Carto´n et al.3 have made experimental determinations and have proposed an empirical correlation for calculating the viscosity of aqueous lithium sulfate solutions in the temperature range of 283.15 to 338.15 K. They have fitted 12 adjustable parameters of their correlation and have obtained a mean relative standard deviation of 0.37% between their experimental and correlated values. Mahiuddin and Ismail4 have measured the temperature and concentration dependence of the viscosity of aqueous sodium nitrate and sodium thiosulfate systems. The temperature dependence of their viscosity data has been represented by the so-called Vogel-TammanFulcher equation1, which is commonly used for highly concentrated electrolyte solutions. Recently, Pereira et al.5 have proposed a modification of the Kumar equation6 for calculating the kinematic viscosity of electrolyte solutions at different temperatures. The model parameters are calculated as a power series of temperature, and 8 adjustable parameters have been fitted by means of literature data for 20 binary strong, aqueous electrolyte solutions in the temperature range of 293.15 to 323.15 K. The fitted parameters were * Corresponding author. E-mail:
[email protected]. Phone: (55) (21) 25627172. Fax: (55) (21) 25627265.
utilized for calculating 20 two-solute and 11 three-solute aqueous eletrolyte solutions with an overall error of 2.5%. In a recent article, Esteves et al.7 have proposed a new model for calculating the viscosity of binary strong, aqueous and nonaqueous, electrolyte solutions at 25 °C and 0.1 MPa. The proposed model is based on Eyring’s absolute rate theory and a Debye-Hu¨ckel-type excess free energy model for calculating the excess free energy of activation of the viscous flow. The purpose of this article is to propose a temperature-dependence expression of the two adjustable parameters of the original Esteves et al.7 model to allow the calculation of the viscosity of binary, aqueous and nonaqueous, strong electrolyte solutions at different temperatures. The organization of the rest of the article is as follows. In section 2, we present a brief overview of the main equations of the Esteves et al.7 model. In section 3, we present the expressions chosen for taking into account the temperature dependence of the two parameters of the original model, and we discuss the calculation results. Finally, in section 4, we summarize our conclusions. 2. Debye-Hu 1 ckel Model for Viscosity In this section, we present a brief overview of our recently published model for calculating the viscosity of a strong electrolyte solution at 25 °C and 0.1 MPa7 in order to discuss the introduction of an explicit temperature dependence into the model adjustable parameters. As can be found elsewhere,7 the proposed model is based on the absolute rate theory of Eyring and co-workes and a Debye-Hu¨ckel-type expression for calculating the excess (electrostatic) free energy of activation of the viscous flow. The ratio between the viscosity of an electrolyte solution (at a given temperature T, pressure P, solvent chemical potencial µ1, and with ni moles of the solute
10.1021/ie020260k CCC: $22.00 © 2002 American Chemical Society Published on Web 08/31/2002
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species (ions) i) and the viscosity of an ideal dilute solution at the same T, P, ni, and µ1 as the real solution can be written as follows:7
( ) () NSOLU
η - η1 ηid - η1
)
∑ i)1
ci
exp
NSOLU
∑ i)1
ψ* E
2), and the parameter of the Guggenheim-type equation, B (eq 6). The final expressions used for calculating the temperature dependence of the A and B parameters are discussed in next section. 3. Results and Discussion
(1)
RT
ci,id
where η1 is the pure solvent dynamic viscosity, NSOLU is the total number of solute species, ci and ci,id are the molar concentrations of the solute species i of the solution and of the ideal dilute solution, respectively, R is the gas constant, and ψ* E is the excess free energy of activation of the electrolyte solution per mole of solute species. The following expression has been adopted for calculating the ideal dilute solution viscosity:7
The proposed modification of the model has been used for correlating viscosity data for 20 binary strong electrolyte systems at different temperatures in two different solvents: water and methanol. The viscosities of pure solvents at different temperatures were taken from ref 17 for water and from ref 15 for methanol. For the systems and temperature ranges studied, the following temperature-dependence expressions have been used for calculating the model parameters A and B, respectively:
A(T) ) PA1 + PA4 (T - TR) and
NSOLU
ηid ) η1(1 + A
∑ i)1
ci,id)
(2)
where A (L/mol) was taken as an empirical adjustable parameter. The excess free energy of activation (per mole of solute species), ψ* E , takes into account the electrostatic (longrange) interations of the ionic solute species in a continuous solvent and can be written as follows: *,DH + ψ*,G ψ* E ) ψE E
(3)
where ψ*,DH is the contribution calculated by means of E reprethe Debye-Hu¨ckel primitive model and ψ*,G E sents a contribution related to a Guggenheim-type expression, which allows the extension of the applicability of the model to higher electrolyte concentrations. Therefore, for a binary (one electrolyte plus one solvent) solution, one can write7
)ψ*,DH E
4.6581 × 107I3/2 τ(κa) D3/2T1/2νc2
(4)
with
[
τ(κa) ) 3(κa)-3 ln(1 + κa) - κa +
]
(κa)2 2
) ψ*,G E
RTBI2 νc2
B(T) ) PB1 + PB2
where I is the ionic strength of the solution (mol/L), κ is the inverse of the Debye length (m-1), a is the distance of closest approach of two ionic species (m), taken as the arithmetic mean of their crystal radii, D is the dielectric constant of the solvent, ν is the sum of the stoichiometric coefficients of the cation and anion of a given electrolyte (ν ) ν+ + ν-), c2 is the electrolyte concentration (mol/L), and B (L/mol) is taken as an empirical adjustable parameter. A temperature dependence has been introduced in the two adjustable parameters of the original model: the parameter of the ideal dilute solution expression, A (eq
)
( )
1 T 1 + P3B ln T TR TR
(8)
ND
F)
exp 2 (ηcal ∑ i - ηi ) i)1
(9)
where ND is the number of experimental data points and ηexp are, respectively, the calculated and and ηcal i i experimental dynamic viscosities. The model dynamic viscosity values are compared with the experimental ones by means of the relative standard deviation (MRSD)
MRSD )
(6)
(
where T is the absolute temperature (K), TR is a reference temperature, taken as 298.15 K, and the adjustable parameters PA1 , PB1 , PB2 , PB3 , and PA4 are given in Table 1 for each system. Equations 7 and 8 are equivalent to truncated forms of the general temperature expression suggested by Holmes and Mesmer8 and used by Pitzer and coworkers9-11 for the ion-interaction model parameters. The model adjustable parameters have been fitted by means of literature experimental viscosity data (see Table 1). The objective function used in the determination of model parameters was
(5)
and
(7)
[ ( 1
ND
∑
ND i)1
)]
exp ηcal i - ηi
ηiexp
2 1/2
(10)
and the overall mean relative standard deviation is defined as follows:
MRSD )
1 Nsyst
Nsyst
∑ MRSDk
(11)
k)1
where Nsyst is the number of systems considered. Table 1 shows the fitting results for the different systems that have been studied. It can be seen in Table 1 that the studied temperature range was, for most of the systems, in the interval from 10 to 55 °C, with the exception of the NaCl + methanol system, for which the available literature data was in the temperature range of -35 to 35 °C. The studied concentration ranges are
Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002 5111 Table 1. Correlation Results and Model Parameters at 0.1 MPaa Solvent: Water electrolyte
temperature range (°C)
concentration range (M)
NSYST
ND
P1A (P1B) (L/mol)
P2B (L K/mol)
P3B (L/mol)
P4A (L/mol K)
MRSD (%)
NaBr BaCl2 CaCl2 CoCl2 CrCl3 CuCl2 KCl LaCl3 MgCl2 NaCl NiCl2 SrCl2 Cd(NO3)2 NaNO3 K2SO4 Li2SO4 MgSO4 Na2SO4
15-55 15-55 15-55 20-50 20-50 20-50 12.5-42.5 15-55 15-55 20-50 20-50 15-55 15-55 15-55 12.5-42.5 10-40 15-42.5 15-55
0.05-6.50 0.05-1.50 0.50-5.00 0.1-3.0 0.1-2.0 0.1-4.0 0.001-3.0 0.05-2.70 0.0025-4.5 0.1-4.0 0.1-4.0 0.05-2.5 0.05-4.00 0.05-6.50 0.0025-0.5 0.00084-3.0 0.0005-0.20 0.05-1.5
9 7 7 7 7 7 7 7 8 7 7 7 9 7 8 8 6 7
90 (12) 57 (12, 13) 56 (12, 13) 49 (13) 62 (13) 63 (13) 84 (13, 14) 56 (12) 104 (12, 13, 14) 63 (13, 14) 56 (13) 49 (12, 13) 81 (12) 63 (12) 57 (14) 102 (13, 14) 60 (14) 62 (12, 14)
0.0395 (0.5089) 0.1778 (0.1752) 0.2239 (0.1662) 0.34989 (0.15723) 0.84091 (0.11514) 0.36811 (0.11157) 0.00157 (0.2465) 0.7159 (0.0990) 0.3430 (0.1795) 0.0552 (0.4609) 0.35349 (0.15998) 0.2436 (0.1412) 0.2387 (0.1601) 0.0478 (0.4373) 0.1293 (0.1974) 0.3934 (0.1808) 0.8240 (0.4213) 0.2917 (0.1993)
3723.9386 672.9918 346.3001 569.68181 134.19666 811.80847 162581.8936 140.1388 207.2082 6810.9672 648.42634 107.6043 481.9701 2214.7622 -14071.0433 282.0491 1027.4982 2950.4830
10.5715 2.0667 0.9132 1.72935 0.36805 2.46304 499.1796 0.3574 0.4956 20.9986 1.92735 0.2487 1.3853 6.1334 -43.7442 0.6948 3.5350 9.1092
0.0008 0.0023 0.0029 0.00082 0.00209 0.0004 0.00153 0.0066 0.0023 0.0006 0.00123 0.0026 0.0018 0.0007 0.0021 0.0009 0.0009 0.0016
0.45 0.45 1.72 1.67 2.57 1.64 0.24 1.84 1.48 0.77 1.62 0.92 0.68 0.93 0.09 1.06 0.17 0.82 1.03
MRSD Solvent: Methanol electrolyte NaCl KI
temperature range (°C)
concentration range (M)
NSYST
ND
P1A (P1B) (L/mol)
P2B (L K/mol)
P3B (L/mol)
P4A (L/mol K)
MRSD (%)
-35-35 25-50
0.005-1.150 0.00080-0.0640
8 6
50 (15) 50 (15, 16)
0.7378 (3.9819) 0.58047 (0.83913)
-28297.4308 -3832.636
-123.1775 -11.2088
0.0017 0.00065
0.30 0.22
MRSD
0.26
a N syst, number of systems; ND, number of experimental data points (number in parentheses is the reference for the literature experimental data); MRSD, mean relative standard deviation), and MRSD is the overall average mean relative standard deviation.
Figure 1. Temperature dependence of the A(T) parameter, by means of eq 7, for the Cd(NO3)2 + water system (s) and for the LaCl3 + water system (--).
strongly dependent on the studied system. For example, for the MgSO4 + water system, the concentration range of the available literature data goes from 0.0005 to 0.2 M, whereas for the NaBr + water system, the concentration range goes from 0.05 to 6.5 M. Figures 1 and 2 show the temperature dependence of A(T) and B(T), respectively, for two different systems: Cd(NO3)2 + water and LaCl3 + water, at 0.1 MPa. It is interesting that linear temperature dependence for A(T) (eq 7) and the expression obtained for B(T) (eq 8) are particular cases of the expression proposed by Holmes and Mesmer.8 It has been found that further parametrization does not lead to any improvement in the correlation capabilities of the model. It can been seen from Table 1 that the best correlation results are obtained for the system K2SO4 + water
Figure 2. Temperature dependence of the B(T) parameter, by means of eq 8, for the Cd(NO3)2 + water system (s) and for the LaCl3 + water system (--).
(MRSD ) 0.09%), and worst results are obtained for the system CrCl3 + water (MRSD ) 2.57%). Figures 3-6 show some typical results obtained from the model. Figure 3 shows the viscosity composition curves of the NaBr + water system at 0.1 MPa for five different temperatures (15, 25, 35, 45, and 55 °C). It can be seen from the Figure that the model correlates, with rather good agreement, the experimental data up to 6.5 M. Figures 4 and 5 show similar results for, respectively, Cd(NO3)2 + water and BaCl2 + water systems. Figure 6 shows the viscosity composition curves for the NaCl + methanol system at 0.1 MPa for five different temperatures (-35 , -5, 5, 15, and 35 °C) The correlation model results are quite satisfactory. The overall mean relative standard deviation for the present model is 0.98%, which corroborates the correla-
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Figure 3. Comparison between experimental and calculated dynamic viscosity for the binary system NaBr + water at 0.1 MPa and at different temperatures: (9) experimental values at 288.15 K from ref 12; (--) model calculated values at 288.15 K; (b) experimental values at 298.15 K from ref 12; (s) model calculated values at 298.15 K; (2) experimental values at 308.15 K from ref 12; (‚‚‚) model calculated values at 308.15 K; ([) experimental values at 318.15 K from ref 12; (- ‚ -) model calculated values at 318.15 K; (×) experimental values at 328.15 K from ref 12; and (- ‚ ‚ -) model calculated values at 328.15 K.
Figure 5. Comparison between experimental and calculated dynamic viscosity for the binary system BaCl2 + water at 0.1 MPa and at different temperatures: (9) experimental values at 288.15 K from ref 12; (--) model calculated values at 288.15 K; (b) experimental values at 298.15 K from refs 12 and 13; (s) model calculated values at 298.15 K; (2) experimental values at 308.15 K from refs 12 and 13; (‚‚‚) model calculated values at 308.15 K; ([) experimental values at 318.15 K from refs 12 and 13; (- ‚ -) model calculated values at 318.15 K; (×) experimental values at 328.15 K from ref 12; and (- ‚ ‚ -) model calculated values at 328.15 K.
Figure 4. Comparison between experimental and calculated dynamic viscosity for the binary system Cd(NO3)2 + water at 0.1 MPa and at different temperatures: (9) experimental values at 288.15 K from ref 12; (--) model calculated values at 288.15 K; (b) experimental values at 298.15 K from ref 12; (s) model calculated values at 298.15 K; (2) experimental values at 308.15 K from ref 12; (‚‚‚) model calculated values at 308.15 K; ([) experimental values at 318.15 K from ref 12; (- ‚ -) model calculated values at 318.15 K; (×) experimental values at 328.15 K from ref 12; and (- ‚ ‚ -) model calculated values at 328.15 K.
Figure 6. Comparison between experimental and calculated dynamic viscosity for the binary system NaCl + methanol at 0.1 MPa and at different temperatures: (9) experimental values at 288.15 K from ref 15; (--) model calculated values at 288.15 K; (b) experimental values at 298.15 K from ref 15; (s) model calculated values at 298.15 K; (2) experimental values at 308.15 K from ref 15; (‚‚‚) model calculated values at 308.15 K; ([) experimental values at 318.15 K from ref 15; (- ‚ -) model calculated values at 318.15 K; (×) experimental values at 328.15 K from ref 15; and (- ‚ ‚ -) model calculated values at 328.15 K.
tion capabilities of the model for aqueous and nonaqueous systems at 0.1 MPa and different temperatures.
systems for wide concentration and temperature ranges at 0.1 MPa. The value for the overall average mean relative standard deviation is 0.98%.
4. Conclusions An explicit temperature dependence has been introduced in the two adjustable parameters of a recently proposed model for calculating the viscosity of binary strong electrolyte solutions at 25 °C and 0.1 MPa.7 The adjustable parameters for empirical expressions for A(T) and B(T) have been fitted by means of literature experimental data. The correlation results obtained show that the modified model allows the calculation of dynamic viscosities of aqueous and nonaqueous binary strong electrolyte
Acknowledgment We are grateful to the Brazilian agencies FUJB, FUNDCCMN, CAPES, FINEP, CNPq, and FAPERJ for financial support. Nomenclature a ) distance of closest approach of the ionic species (m) A ) model parameter (L/mol) B ) model parameter (L/mol)
Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002 5113 B ) Jones-Dole’s parameter c ) molar concentration (mol/L) D ) solvent dieletric constant F ) objective function I ) ionic strength (mol/L) MRSD ) mean relative standard deviation MRSD ) overall average mean relative standard deviation Nsyst ) number of systems considered ND ) number of experimental data points NSOLU ) total number of solute species P ) pressure P1-P4 ) model adjustable parameters R ) gas constant (8.314510 J/mol K) T ) absolute temperature (K) Greek Letters ψ ) free energy per mole of solute η ) dynamic viscosity (mPa s) κ ) inverse of the Debye length (m-1) ν ) stoichiometric coefficient Subscripts + ) cation - ) anion 1 ) solvent component 2 ) electrolyte E ) excess property i ) solute species id ) ideal dilute solution k ) system considered R ) reference property Superscripts A ) model parameter A B ) model parameter B * ) activation property cal ) calculated value DH ) Debye-Hu¨ckel exp ) experimental value G ) Guggenheim
Literature Cited (1) Horvath, A. L. Handbook of Aqueous Electrolyte Solutions; Ellis Horwood Limited: Chichester, U.K., 1985. (2) Jenkins, H. D. B.; Marcus, Y. Viscosity BsCoefficients of Ions in Solution. Chem. Rev. 1995, 95, 2695.
(3) Carto´n, A.; Sobro´n, S. B.; Gerbole´s, J. I. Density, Viscosity, and Electrical Conductivity of Aqueous Solutions of Lithium Sulfate. J. Chem. Eng. Data 1995, 40, 987. (4) Mahiuddin, S.; Ismail, K. Temperature and Concentration Dependence of the Viscosity of Aqueous Sodium Thiosulphate Electrolytic Systems. Fluid Phase Equilib. 1996, 123, 231. (5) Pereira G.; Moreira, R.; Va´zquez, M. J.; Chenlo, F. Kinematic Viscosity Prediction for Aqueous Solutions with Various Solutes. Chem. Eng. J. 2001, 81, 35. (6) Kumar, A. A Simple Correlation for Estimating Viscosities of Solutions of Salts in Aqueous, Nonaqueous and Mixed-Solvents Applicable to High-Concentration, Temperature and Pressure. Can. J. Chem. Eng. 1993, 71, 948. (7) Esteves, M. J. C.; Cardoso, M. J. E. de M.; Barcia, O. E. A Debye-Hu¨ckel Model for Calculating the Viscosity of Binary Strong Electrolyte Solutions. Ind. Eng. Chem. Res. 2001, 40, 5021. (8) Holmes, H. F.; Mesmer, R. E. Thermodynamics Properties of Aqueous Solutions of the Alkali Metal Chlorides to 250 °C. J. Phys. Chem. 1983, 87, 1242. (9) Silvester, L. F.; Pitzer, K. S. Thermodynamics of Electrolytes. 8. High-Temperature Properties, Including Enthalpy and Heat Capacity, with Application to Sodium Chloride. J. Phys. Chem. 1977, 81, 1822. (10) Pitzer, K. S. Thermodynamics, 3rd ed.; McGraw-Hill: New York, 1995. (11) Activity Coefficients in Electrolyte Solutions, 2nd ed.; Pitzer, K. S., Ed.; CRC Press: Boca Raton, FL, 1991. (12) Isono, T. Density, Viscosity, and Electrolytic Conductivity of Concentrated Aqueous Electrolyte Solutions at Several Temperatures. Alkaline-Earth Chlorides, LaCl3, Na2SO4, NaNO3, NaBr, KNO3, KBr, and Cd(NO3)2. J. Chem. Eng. Data 1984, 29, 45. (13) Afzal, M.; Saleem, M.; Mahmood, T. Temperature and Concentration Dependence of Viscosity of Aqueous Electrolytes from 20 to 50 °C. Chlorides of Na+, K+, Mg2+, Ca2+, Ba2+, Sr2+, Co2+, Ni2+, Cu2+, and Cr3+. J. Chem. Eng. Data 1989, 34, 339. (14) Lobo, V. M. M. Electrolyte Solutions: Literature Data on Thermodynamic and Transport Properties; Coimbra Editora: Coimbra, Portugal, 1984; Vol. I. (15) Barthel, J.; Neueder, R.; Meier, R. Electrolyte Data Collection, Part 3: Viscosity of Nonaqueous Solutions, I: Alcohol Solutions; Chemical Data Series; Dechema: Frankfurt, 1977; Vol. XII. (16) Jones, G.; Fornwalt, H. J. The Viscosity of Solutions of Salts on Methanol, J. Am. Chem. Soc. 1935, 57, 2041. (17) CRC Handbook of Chemistry and Physics, 76th ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 1995.
Received for review April 5, 2002 Revised manuscript received July 22, 2002 Accepted July 24, 2002 IE020260K
