Ind. Eng. Chem. Res. 2001, 40, 5021-5028
5021
A Debye-Hu 1 ckel Model for Calculating the Viscosity of Binary Strong Electrolyte Solutions 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, cep 21949-900, Rio de Janeiro, RJ, Brazil
In this article we present a new model for correlating dynamic viscosity of binary strong electrolyte solutions. The proposed model is based on Eyring’s absolute rate theory and the Debye-Hu¨ckel model for calculating the excess (electrostatic) free energy of activation of the viscous flow. In the present model we consider that the free energy of activation of the viscous flow as being the same as the appropriate thermodynamic free energy used for calculating equilibrium properties of the electrolyte solution. Modifications of Eyring’s absolute rate theory must be performed to take into account the solvent as a continuous medium, as considered in the Debye-Hu¨ckel theory. This is accomplished by means of the osmotic-pressure framework for solutions. In this framework one adopts a thermodynamic free energy, which is considered as a function of the absolute temperature, pressure, number of moles of the solute species, and chemical potential of the solvent. The proposed model contains two adjustable parameters that have been fitted by means of experimental viscosity data of the literature. The total number of 21 binary electrolyte systems (at 0.1 MPa and 25 °C) with different solvents (water, methanol, ethanol, and 1-butanol) have been studied. The calculated viscosity values are in good agreement with the experimental ones. The overall average mean relative standard deviation is 0.52%. 1. Introduction There is a considerable large number of empirical and semiempirical models, in the literature, for calculating and correlating the concentration dependence of the viscosity of binary aqueous strong electrolyte solutions.1 At present, one of the most used equations is the one proposed by Jones and Dole in 1929, which can be written as follows,1-3
η/η1 ) 1 + Ac21/2 + Bc2
(1)
where η and η1 are, respectively, the solution and pure solvent dynamic viscosities (at same temperature and pressure) and c2 is the molar concentration of the electrolyte. The coefficient A of eq 1 can be calculated as a function of the equivalence conductance of the ions at infinite dilution, as proposed by Falkenhagen and coworkers.1,3-6 The coefficient B is fitted by means of experimental viscosity data.3,7-9 The Jones-Dole equation (eq 1) is, in general, valid only for electrolyte concentration up to 0.1 M.1 To apply this equation to more concentrated electrolyte solutions, one can add to it empirical terms, taking into account higher degrees of the electrolyte concentration.1,3,7-9 A recent and comprehensive review of the application of the Jones-Dole equation for binary, aqueous and nonaqueous, electrolyte solutions as well as the determination of B coefficients of the ions is given by Jenkins and Marcus.3 Chandra and Bagchi10,11 recently remarked that the microscopic model of the concentration dependence of transport properties (e.g., conductivity and viscosity) of * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: (55) (21) 2562.7172. Fax: (55) (21) 2562.7265.
nondiluted electrolyte solutions is an open research subject. The modeling of the concentration dependence of the viscosity of an electrolyte solution can be considered as a rather difficult task due, among other possible effects, to the so-called “anomalous concentration dependence”.10,11 That is, for some electrolyte systems (e.g., NaCl + H2O or LiCl + H2O) the solution viscosity increases monotonically with the electrolyte concentration. For other types of electrolyte systems (e.g., KCl + H2O or KBr + H2O), the solution viscosity increases, up to a maximum value, at very low electrolyte concentration values, then it decreases, reaching a minimum value, and afterward it increases monotonically for higher electrolyte concentrations. Chandra and Bagchi,10,11 by means of the mode coupling theory, have developed a new model for calculating the viscosity of electrolyte solutions. They have also demonstrated that their new model reduces to the Falkenhagen expression in the limit of low electrolyte concentrations. Chandra and Bagchi10,11 show two figures (for 1:1 and 2:2 electrolyte aqueous solutions) comparing the results obtained by their model and the ones obtained by the Falkenhagen equation. The absolute rate theory, of Eyring and co-workers,12-15 which can be used for calculating the viscosity of liquids12-15 (pure and mixtures) has also been applied, by different authors, for calculating the viscosity of electrolyte solutions.1 In a series of articles Good16-19 and Good and Ingham20 have calculated, based on Eyring’s equation, the enthalpy, entropy, and Gibbs free energy of activation of the viscous flow for a series of different binary electrolyte systems. On the basis of these values, they have tried to interpret, by means of ionic hydration and the concept of structure-breaking and structure-making mechanisms of the solvent structure, the viscosity of electrolyte solutions. More recently,
10.1021/ie010392y CCC: $20.00 © 2001 American Chemical Society Published on Web 10/06/2001
5022
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001
It is possible to rewrite the original Eyring’s equation for the solution viscosity as follows (see Appendix), NSOLU
ηΠ ) Np exp(∆ψ*/RT) Figure 1. A pure solvent compartment in osmotic equilibrium with a solution compartment. The two compartments are separated by a semipermeable membrane that allows the interchange of the solvent molecules. T is the absolute temperature, µ1 is the solvent chemical potential, P1 is the pressure in the solvent compartment, P is the pressure in the solution compartment, Π is the osmotic pressure, V is the volume of the solution compartment, η1 is the pure solvent dynamic viscosity, η is the solution dynamic viscosity, and ηΠ is the solute contribution to the solution dynamic viscosity.
Abraham and Abraham21 have tried to use the absolute rate theory for modeling viscosity and electrical conductance of multicomponent electrolyte aqueous solutions. They have studied the whole concentration range from dilute electrolyte solutions up to the fused salts. The molar Gibbs free energy of activation of the viscous flow is calculated as being a linear combination of the Gibbs free energy of activation of the fused salt and the apparent free energy of activation assigned to the water molecules linked to the salt. The purpose of this article is to present a viscosity model for strong electrolyte solutions based on Eyring’s absolute rate theory12-15 and the Debye-Hu¨ckel model.22-27 The Debye-Hu¨ckel theory is adopted for calculating the excess (electrostatic) free energy of activation of an electrolyte solution by taking into account the solvent as a continuous medium. The proposed new model considers that the free energy of activation of the viscous flow is the same as the thermodynamic (equilibrium) free energy of the electrolyte solution.15 One must bear in mind that appropriate modifications of Eyring’s absolute rate theory must be performed to consider the solvent as a continuous medium, as considered in the Debye-Hu¨ckel theory. This is accomplished by means of the so-called osmoticpressure framework for solutions.28,29 The organization of the rest of this article is as follows. In section 2, the modifications of Eyring’s absolute rate theory for viscosity calculations of electrolyte solutions are discussed. In section 3 we present the DebyeHu¨ckel model for viscosity calculations. In section 4 we present and discuss the calculations results by comparing them with literature viscosity data. Finally, in section 5 we summarize our conclusions. 2. A New Model for Calculating Viscosity of Binary Electrolyte Solutions The proposed model is based on the absolute rate theory of Eyring and co-workers12-14 and the thermodynamic framework for solutions based on the MacMillan-Mayer level by means of the so-called osmotic equilibrium formulation.28 By taking into account a multisolute solution in osmotic equilibrium with a pure solvent compartment, one could write (see Figure 1),
η ) η1 + ηΠ
(2)
where η and η1 are, respectively, the solution and pure solvent dynamic viscosities and ηΠ is the solute contribution to the solution dynamic viscosity by considering the solvent as a continuous medium.
∑ i)1
ci
(3)
where ci is the molar concentration of the solute species i, NSOLU is the total number of solute species, N is the Avogadro number, p is Plank’s constant, ∆ψ* is the free energy of activation for the flow process, per mole of solute, R is the gas constant, and T is the solution absolute temperature. In the so-called osmotic equilibrium thermodynamic framework (continuous solvent), one can write the appropriate thermodynamic potential for a solution as follows,28
Ψ ) Ψ(T, P, µ1, ni)
(4)
where T is the absolute temperature, P is the solution pressure, ni is the number of moles of the solute species i, and µ1 is the chemical potential for the solvent species 1. We can also define the intensive (per mole of solute) thermodynamic potential as28
ψ)
Ψ
(5)
NSOLU
∑ i)1
ni
It is important to remark that the generalized osmoticequilibrium framework (i.e., for any number of solvent and solute species) can been found elsewhere.28 The basic assumption of the present model for viscosity calculation is to consider the free energy of activation (per mole of solute) for the flow process to be same as the ψ, the thermodynamic (i.e., equilibrium) free energy (per mole of solute) of the solution in the osmoticequilibrium framework.15 Therefore, by considering the ∆ψ* as a sum of an ideal dilute solution contribution and a correction, an excess contribution, one can write * ∆ψ* ) ∆ψ* id + ψE
(6)
where ∆ψ* id is the ideal dilute solution free energy contribution (at same T, P, ni, and µ1 as the real solution) and ψ* E is the excess free energy of the electrolyte solution. It is important to remark that the ideal dilute solution obeys the Van’t Hoff equation.28 By substituting eq 3 into eq 2, one obtains NSOLU *
η ) η1 + Np exp(∆ψ /RT)
∑ i)1
ci
(7)
Analogously, one could write the dynamical viscosity of the ideal solution, at the same T, P, µ1, and ni as NSOLU
ηid ) η1 + Np
exp(∆ψ* id/RT)
∑ i)1
ci,id
(8)
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 5023
By dividing eq 8 by eq 7, one obtains the following:
η - η1
( )
ηid - η1
)
NSOLU
∑ i)1
ci
NSOLU
∑ i)1
ci,id
exp
(
)
∆ψ* - ∆ψ* id RT
( ) NSOLU
)
∑ i)1
ci
NSOLU
∑ i)1
exp
ci,id
( ) ψ* E
thickness of the ionic atmosphere), k is the Boltzmann constant, V ′ is the volume of solution (in m3), and a is the distance (in m) of the closest approach of the ionic species (which is the same for all pairs of ions in the so-called restricted primitive model). One can rewrite eqs 11 and 12 by defining the ionic strength of an electrolyte solution as follows, NION
RT
(9)
The equation above represents the relative deviation from the ideal solution viscosity behavior due to the interaction among the solute particles by taking into account the solvent as a continuous medium. In the proposed model, the ideal dilute solution viscosity is calculated by means of the following expression,
I)
1
∑
nizi2
(14)
2 i)1 V
with
ni ) si/N
(15)
where V is the volume of the solution (in liters) and ni is the number of moles of species i. Thus, the ratio ni/V represents the molar concentration of species i. Substituting eqs 14 and 15 into eq 12, one arrives at
NSOLU
ηid ) η1(1 + A
∑ i)1
ci,id)
(10)
where A is taken as an empirical adjustable parameter. This equation corresponds to the linear concentration dependence term of the Jones-Dole model (see eq 1). This expression takes into account the short-range ionsolvent interactions in dilute electrolyte solutions, as discussed by Chandra and Baghi.10 3. The Debye-Hu 1 ckel Model for Viscosity To model the viscosity of dilute strong electrolyte solutions, the Debye-Hu¨ckel theory22,23 has been adopted for calculating the excess free energy of activation due to electrostatic (long-range) interactions of the ionic species, in a continuous solvent. By performing a charging process of a hypothetical electrolyte solution in which all the ions, considered as hard spheres without charge, are charged simultaneously to their final charge values (Debye Charging Process), one obtains total excess (electrostatic) free energy of the electrolyte solution, which in S.I. units can be written as follows,23-25
) Ψ*,DH E
NION
-e (
12π0D
sizi2)κτ(κa) ∑ i)1
(11)
κ2 )
) Ψ*,DH E
(
∑
(12)
0DkT i)1 V ′
1/2
VI3/2 τ(κa) D3/2T1/2
(17)
by considering the numerical value of the universal constants (see Nomenclature section), one finally obtains
κ)
50.29 × 1010 I1/2 (DT)1/2
(18)
and
) Ψ*,DH E
-4.6581 × 107 3/2 VI τ(κa) D3/2T1/2
(19)
To apply the proposed model for calculating the dynamic viscosity of an electrolyte solution, we need the expression for the excess (electrostatic) free energy of solution per mole of the solute species
2
NIONs z i i
)
-e2 2 × 103 N3 6π 03k
ψ*,DH ) E e2
(16)
and if one uses eqs 14, 15, and 16 in eq 11, one obtains
with
κ2 )
2 × 103 e2NI 0DkT
-4.6581 × 107V 3/2 I τ(κa) n D3/2T1/2
(20)
with NION
and
[
τ(κa) ) 3(κa)-3 ln(1 + κa) - κa +
]
n)
2
(κa) 2
(13)
where e is the elementary charge of the electron, 0 is the vacuum permittivity, D is the dielectric constant of the continuous medium (solvent), NION is the total number of ionic species, si is the number of particles of the ionic species i, zi is the charge of the ionic species i, κ is the inverse of the Debye length (inverse of the
∑ i)1
ni
(21)
where n is the total number of moles of the ionic species (assuming the total dissociation of the electrolyte). For a binary (one electrolyte plus one solvent) solution we can write
n2 n n+ + n- ν+n2 + ν-n2 ) ) ) (ν+ + ν-) ) νc2 (22) V V V V
5024
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001
Table 1. Correlation Results and Model Parameters at 25 °C and Pressure of 0.l MPaa concentration range (M)
electrolyte
ND
B (L/mol)
A (L/mol)
MRSD (%)
BaCl2 K2SO4 Li2SO4 LiCl LiNO3 MgCl2 MgSO4 NaCl KCl KBr NaBr NaI NaNO3 Na2SO4 Ba(NO3)2 CaCl2 MRSD
0.00500-1.42500 0.00050-0.50000 0.00084-0.17309 0.08900-8.16840 0.01734-5.34950 0.00250-1.90700 0.00050-0.20000 0.00200-2.02100 0.00100-2.96208 0.00100-3.74927 0.49600-6.52300 0.02540-2.94700 0.04980-6.24600 0.00050-1.88724 0.04970-0.34250 0.04980-4.99000
Solvent: Water 10 (9, 33) 11 (9) 8 (9) 19 (8) 16 (9) 13 (9, 33) 10 (9) 10 (9) 24 (9) 15 (9) 16 (9, 33) 14 (9) 9 (33) 18 (9) 7 (8) 8 (33)
0.1707 0.2163 0.5258 0.3014 0.6642 0.1970 0.4201 0.4816 2.7980 -1.1170 0.5059 1.0279 0.4369 0.1888 0.4102 0.1660
0.1821 0.1316 0.2858 0.1061 0.0592 0.3266 0.8242 0.0601 0.0002 -0.0451 0.0398 0.0136 0.0479 0.2991 0.1021 0.2247
0.27 (2.20) 0.03 (0.49) 0.02 (0.42) 0.64 (9.25) 0.87 (9.23) 0.90 (3.73) 0.14 (0.39) 0.033 (1.05) 0.12 (0.62) 0.54 0.40 0.05 0.99 0.53 0.04 1.88 0.47
KBr KCl KI NH4Cl LiCl CsCl NaCl NaI MRSD
0.00041-0.11890 0.00048-0.04002 0.00025-0.60320 0.00050-0.35116 0.00105-0.45260 0.00142-0.10000 0.00131-1.14630 0.00079-0.48636
Solvent: Methanol 12 (34) 2.6520 9 (34) 1.7770 14 (34) 1.0470 16 (34) 1.5300 21 (7) 2.8620 17 (7) -0.2546 18 (7) 1.1550 8 (7) 1.4380
0.5491 0.5834 0.5650 0.5304 0.7493 0.4949 1.1098 0.6856
0.014 (0.64) 0.01 (0.14) 0.14 (3.47) 0.05 (1.52) 0.26 0.11 1.70 0.21 0.31
NaI LiNO3 MRSD
0.00070-0.33811 0.25030-3.20600
Solvent: Ethanol 15 (7) 7 (7)
1.0948 2.3310
0.15 4.38 2.12
LiI NH4I MRSD
0.00080-0.01593 0.00066-0.01327
1.0552 1.2461
0.19 0.16 0.17
2.7040 0.9155
Solvent: 1-Butanol 9 (7) 101.370 9 (7) 111.100
a
ND is the number of experimental data points (number in parentheses is the reference for the literature experimental data); MRSD is the mean relative standard deviation (number in parentheses is the deviation obtained by means of the Jones-Dole model3); MRSD is the overall average mean relative standard deviation.
where ν ) ν+ + ν-, and the ionic strength as
I)
1 1 (n z 2 + n-z-2) ) (ν n z 2 + ν-n2z-2) ) 2V + + 2V + 2 + n2 c2 (ν+z+2 + ν-z-2) ) (ν+z+2 + ν-z-2) ) 2V 2 νc2 |z+z-| (23) 2
where n+ and n- are, respectively, the cation and anion mole numbers and ν+ and ν- are the cation and anion stoichiometric coefficients for a given electrolyte. Therefore, one could rewrite eq 20 as follows:
) ψ*,DH E
-4.6581 × 107 I3/2 τ(κa) νc2 D3/2T1/2
(24)
Because of the electrolyte concentration range limitations of the Debye-Hu¨ckel theory, one could also try to introduce an empirical correction in eq 24 to extend its applicability to higher electrolyte concentrations. One possible extension could be the following, based on the Guggenheim correction of the Debye-Hu¨ckel mean ionic activity coefficient expression,23 *,DH + Ψ*,G Ψ* E ) ΨE E
(25)
where Ψ*,DH is given by eq 24 and Ψ*,G is given by the E E following expression,
ψ*,G ) E
RTBI2 νc2
(26)
where B is taken as an empirical adjustable parameter. 4. Results and Discussion The proposed model has been used for correlating viscosity data for 21 binary strong electrolyte systems at 0.1 MPa and 25 °C in several solvents: water, methanol, ethanol, and 1-butanol. For each electrolyte, the distance of closest approach of the ions, a, has been taken as being the arithmetic mean of their crystal radios.32 The adjustable model parameters, A and B, have been fitted by means of experimental viscosity data taken from the literature (see Table 1). The objective function used in the determination of model parameters was ND
F)
(ηcal. - ηiexp.)2 ∑ i i)1
(27)
where ND is the number of experimental data points
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 5025
Figure 2. Comparison between experimental and calculated (η - η1) for the binary system LiCl + water at 298.15 K and 0.1 MPa. (b) Experimental values from ref 8; (- - -) Debye-Hu¨ckel term (eq 24); (-) Debye-Hu¨ckel and Guggenheim terms (eq 25).
Figure 3. Comparison between experimental and calculated viscosity for the binary system NaNO3 + water at 298.15 K and 0.1 MPa. (b) Experimental values from ref 33; (-) model calculated values.
and ηcal. and ηexp. are, respectively, the calculated and i i experimental values of the dynamic viscosity. The model dynamic viscosity values are compared with the experimental ones by means of the mean relative standard deviation (MRSD)
MRSD )
( ( 1
ND
∑
NDi)1
))
ηcal. - ηiexp. i ηexp. i
2 1/2
(28)
and the overall average mean relative standard deviation is defined as follows,
MRSD )
1
Nsyst
∑ MRSDk
Nsyst k)1
(29)
where Nsyst is the number of systems. Table 1 shows the fitting results for the different systems that have been studied. The values of A and B parameters obtained, for each system, are also shown. As can be seen from Table 1, the worst correlation results are obtained for the system LiNO3 + ethanol. Table 1 also presents a comparison of the present model results with the ones obtained by using the Jones-Dole model recently reviewed by Jenkin and Marcus.3 It must be stressed that all available experimental viscosity data, for the whole concentration range, for each binary system, has been used for fitting the JonesDole B parameter and A and B the parameters of the proposed model. It can been seen from Table 1 that for all studied systems the present model gives significantly smaller deviations than those obtained by the Jones-Dole model.3 Figures 2-6 show some typical results obtained with the present model. Figure 2 compares the contributions of the DebyeHu¨ckel (eq 24) and Guggenheim (eq 26) terms for correlating the difference between solution and the pure solvent viscosities (η - η1) for the LiCl + water system at 298.15 K and 0.1 MPa. It can be seen that in the low concentration range the Debye-Hu¨ckel contribution
Figure 4. Comparison between experimental and calculated viscosity for the binary system KCl + water at 298.15 K and 0.1 MPa. (b) Experimental values from ref 9; (-) model calculated values.
works quite well but for higher concentration (up to 8.0 M) the Guggenheim term corrects for large deviation between the experimental data and the Debye-Hu¨ckel calculated values. Figure 3 shows the viscosity composition curve of the NaNO3 + water at 298.15 K and 0.1 MPa. It can be seen that the model correlates with reasonable good agreement of the experimental data up to the concentration of 6.25 M. Figure 4 shows that the proposed model does not correlate properly the complex behavior of the experimental viscosity data for the KCl + water system at 298.15 K and 0.1 MPa. Similar model behavior occurs for the KBr + water system at 298.15 K and 0.1 MPa. Figures 5 and 6 show the calculated and experimental values of the viscosity of nonaqueous binary electrolyte systems. Figure 5 shows the results for the KI + methanol system at 298.15 K and 0.1 MPa, and Figure 6 shows the results for the LiI + 1-butanol system at 298.15 K and 0.1 MPa.
5026
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001
is based on a modification of Eyring’s absolute rate theory for viscous flow of a liquid solution. The appropriate free energy expression is obtained by means of the Debye-Hu¨ckel theory for dilute electrolyte solutions. The solvent is considered as being a continuous medium in which the ionic species (charged hard spheres) interact. The correlation of the viscosity data is applicable for different binary electrolyte systems, for the whole concentration range, at 0.1 MPa and 25 °C. The value for the overall average mean relative standard deviation is 0.52%. Nomenclature
Figure 5. Comparison between experimental and calculated viscosity for the binary system KI + methanol at 298.15 K and 0.1 MPa. (b) Experimental values from ref 34; (-) model calculated values.
Figure 6. Comparison between experimental and calculated viscosity for the binary system LiI + 1-butanol at 298.15 K and 0.1 MPa. (b) Experimental values from ref 7; (-) model calculated values.
The overall average mean relative standard deviation for the present model is 0.52%, which shows that the proposed model can adequately correlate the viscosities of different binary aqueous and nonaqueous electrolyte systems at 0.1 MPa and 25 °C. It is important to remark that the present model can be readily extended for viscosity calculations of a multisolvent electrolyte solution. In that case one can use a model for calculating the viscosity of the solutefree, solvent mixture15 and an adequate mixing rule for the dielectric constant of the solvent mixture.27,31 For mixed electrolyte solutions one has to adopt a kind of mixing rule for calculating the multielectrolyte A and B parameters by means of the ones obtained for the different binary electrolyte systems.22
a ) distance of closest approach of the ionic species (m) A ) adjustable parameter (L/mol) A ) Jones-Dole parameter B ) adjustable parameter (L/mol) B ) Jones-Dole parameter c ) molar concentration (mol/L) D ) solvent dieletric constant e ) elementary charge of the electron (1.60217733 × 10-19 C) F ) objective function ∆G* 0 ) free energy of activation of the liquid molecules in a stationary fluid ∆G* ) free energy of activation of a fluid under stress p ) Planck’s constant (6.6260755 × 10-34 J‚s) I ) ionic strength (mol/L) k ) Boltzmann’s constant (1.380658 × 10-23 J/K) MRSD ) mean relative standard deviation MRSD ) overall average mean relative standard deviation n ) total number of moles N ) Avogadro’s number (6.0221367 × 1023 mol-1) ND ) number of experimental data points NION ) total number of ionic species NSOLU ) total number of solute species P ) pressure (MPa) R ) gas constant (8.314510 J/mol‚K) s ) number of particles T ) absolute temperature (K) V ) volume of solution (L) V ′ ) volume of solution (m3) z ) charge of the ionic species Greek Letters R ) the distance to the nearest vacant site in given fluid layer ∆ψ* ) free energy of activation for flow process, per mole of solute ∆Ψ* ) free energy of activation for flow process δ ) distance between two adjacents layers of the fluid 0 ) vacuum permittivity (8.854187817 × 10-12 F/m) η ) dynamic viscosity (mPa‚s) ηΠ ) solute contribution to the solution dynamic viscosity κ ) inverse of the Debye length (m-1) µ ) chemical potential ν ) stoichiometric coefficient Π ) osmotic pressure τyx ) applied shear stress ψ ) thermodynamic potential per mole of solute Ψ ) thermodynamic potential Subscripts
5. Conclusions A new model for correlating the viscosity of binary electrolyte solution is presented. The proposed model
+ ) cation - ) anion 1 ) solvent component 2 ) solute component
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 5027 E ) excess property i ) solute species id ) ideal dilute solution Π ) solute contribution to the solution property
mole of solute (in J/mol) and ci is the molar concentration of the solute species i. By means of eq A3 and following an analogous derivation as the one proposed by Bird et al.13 for pure fluids, one finally obtains eq 3
Superscripts * ) activation property cal. ) calculated value DH ) Debye-Hu¨ckel exp. ) experimental value G ) Guggenheim
where ηΠ is in mPa‚s.
Appendix: Derivation of the ηπ Term
Acknowledgment
This Appendix presents the application of Eyring’s absolute rate theory12 for calculating the solute contribution to the dynamic viscosity of a solution by taking into account the solvent (or a mixture of solvents) as a continuous medium. The derivation will follow the one given by Bird et al.13 for pure liquids. Therefore, only the main differences are discussed here. It is possible to define, for a pure liquid, a free energy of activation for the rearrangements of the liquid molecules in a stationary fluid (∆G* 0 ). This potential free energy barrier is distorted for the fluid under flow because of the applied shear stress. For a fluid flowing in the x direction, one could write13
The authors are grateful to the Brazilian agencies FUJB, CAPES, FINEP, CNPq, and FAPERJ for financial support.
-∆G* ) -∆G* 0 (
NSOLU
ηΠ ) Np exp(∆ψ*/RT)
( )( ) R τyxV ′ δ 2
(A1)
where ∆G* is the activation free energy for the rearrangements of the molecules, of a fluid under stress; R is the distance to the nearest vacant site in a given fluid layer; δ is a distance between two adjacents layers of the liquid; τyx is the applied shear stress; and V′ is the fluid volume in m3. The second term of the right-handed side of eq A1, represents the work done on the molecules to move them up the potential free energy barrier. The plus and minus signs correspond, respectively, for the molecules moving along or against the direction of the flow. In an analogous way, for a liquid solution, it would be possible to define a free energy of activation for the rearrangements of the solute particles in a continuous solvent medium. The appropriate thermodynamic free energy for considering the solute as particles and the solvent as a continuous medium (Ψ) is discussed elsewhere.4 Therefore, eq A1 can be recasted as follows,
-∆Ψ* ) -∆Ψ* 0 (
( )( ) R τyxV ′ δ 2
(A2)
where ∆Ψ* and ∆Ψ* 0 are, respectively, the activation free energy for the rearrangements of the solute particles for the solution under stress and without applied stress (solution at rest). The second term of the righthanded side of eq A2 represents the work on the solute particles (in a continuous solvent medium) to move them to the top of the potential free energy barrier. By dividing eq A2 by the total number of moles of the solute species, one obtains *
-∆ψ )
-∆ψ* 0
( )(
-3 NSOLU R τyx × 10 ( ci)-1 ( δ 2 i)1
∑
)
(A3)
where ∆ψ* and ∆ψ* 0 are the activation free energy per
∑ i)1
ci
(3)
Literature Cited (1) Horvath A. L. Handbook of Aqueous Electrolyte Solutions; Ellis Horwood Limited: Chichester, 1985. (2) Jones, G.; Dole, M. The Viscosity of Aqueous Solutions of Strong Electrolytes with Special Reference to Barium Chloride. J. Am. Chem. Soc. 1929, 51, 2950. (3) Jenkin, H. D. B.; Marcus, Y. Viscosity B-Coefficients of Ions in Solution. Chem. Rev. 1995, 95, 2695. (4) Falkenhagen, H.; Vermon, E. L. Das quantitiative Grenzgesetz der inneren Reibung beliebiger einfacher starker Elektrolyte. Phys. Z. 1932, 33, 140. (5) Falkenhagen, H.; Vermon, E. L. The Viscosity of Strong Electrolyte Solutions according to Electrostatic Theory. Philos. Mag. 1932, 14, 537. (6) Falkenhagen, H.; Dole, M. Die innere Reibung von Elektrolytischen Lo¨sungen und ihre Deutung nach der Debyeschen Theorie. Phys. Z. 1929, 30, 611. (7) Barthel, J.; Neueder, R.; Meier, R. Electrolyte Data Collection; Part 3: Viscosity of Nonaqueous Solutions, I: Alcohol Solutions, Chemical Data Series, Vol. XII; Dechema: Frankfurt, 1977. (8) Lobo, V. M. M. Electrolyte Solutions: Literature Data on Thermodynamic and Transport Properties; Coimbra Editora: Coimbra, 1981; Vol. II. (9) Lobo, V. M. M. Electrolyte Solutions: Literature Data on Thermodynamic and Transport Properties; Coimbra Editora: Coimbra, 1984; Vol. I. (10) Chandra, A.; Bagchi, B. Ionic Contribution to the Viscosity of Dilute Electrolyte Solutions: Towards a Microscopic Theory. J. Chem. Phys. 2000, 113, 3226. (11) Chandra, A.; Bagchi, B. Beyond the Classical Transport Laws of Electrochemistry: New Microscopic Approach to Ionic Conductance and Viscosity. J. Phys. Chem. B 2000, 104, 9067. (12) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Process; McGraw-Hill: New York, 1941. (13) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons: New York, 1960. (14) Hischfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley & Sons: New York, 1954. (15) Martins, R. J.; Cardoso, M. J. E. de M.; Barcia, O. E. Excess Gibbs Free Energy Model for Calculating the Viscosity of Binary Liquid Mixtures. Ind. Eng. Chem. Res. 2000, 39, 849. (16) Good, W. The Effect of Solute Concentration on Fluidity and Structure in Aqueous Solutions of Electrolytes-I. Alkali-Metal and Ammonium Halides. Electrochim. Acta 1964, 9, 203. (17) Good, W. The Effect of Solute Concentration on Fluidity and Structure in Aqueous Solutions of Electrolytes-II. Alkaline Earth Chlorides, Cupric Chloride and Ferric Chloride. Electrochim. Acta 1965, 10, 1. (18) Good, W. The Effect of Solute Concentration on Fluidity and Structure in Aqueous Solutions of Electrolytes-III. Nitrates and Chlorates. Electrochim. Acta 1966, 11, 759. (19) Good, W. The Effect of Solute Concentration on Fluidity and Structure in Aqueous Solutions of Electrolytes-IV. Sulphates and Carbonates. Electrochim. Acta 1966, 11, 767. (20) Good, W.; Ingham, D. B. Enthalpy-Entropy Relationships in the Fluid Kinetics of Aqueous Electrolyte Solutions Containing Positively Hydrated Ions. Electrochim. Acta 1975, 20, 57.
5028
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001
(21) Abraham, M.; Abraham, M.-C. Transport Properties and Hole Theory in the Concentration Range from Fused Salts to Dilute Aqueous Solutions. Electrochim. Acta 1988, 33, 967. (22) Activity Coefficients in Eletroclyte Solution, 2nd ed.; Pitzer, K. S., Ed.; CRC Press: Boca Raton, FL, 1991. (23) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworth: London, 1965. (24) Fowler, R. H.; Guggenheim, E. A. Statistical Thermodynamics; Cambridge University Press: Cambridge, 1949. (25) Davidson, N. Statistical Mechanics; McGraw-Hill: New York, 1962. (26) McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1976. (27) Cardoso, M. J. E. de M. Calculation of Phase and Chemical Equilibria in Electrolyte Systems; Danmarks Tekniske Hojskole: Lyngby, Denmark, 1988. (28) Cardoso, M. J. E. de M.; Medeiros, J. L. A Thermodynamic Framework for Solutions Based on the Osmotic Equilibrium Concept-1. General Formulation. Pure Appl. Chem. 1994, 66, 383. (29) Cardoso, M. J. E. de M.; O’Connell, J. P. Activity Coefficients in Mixed Electrolyte Solutions. Fluid Phase Equilibr. 1987, 33, 315.
(30) CRC Handbook of Chemistry and Physics, 76th ed.; Lide, D. R., Editor-in-chief; CRC Press Inc.: Boca Raton, FL, 1995. (31) Hasted, J. B. Dielectric Properties. In Water A Comprehensive Treatise; Franks, F., Ed.; Plenum Press: New York, 1973; Vol. 2. (32) Pauling, L. General Chemistry; Dover Publications: New York, 1998. (33) 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. (34) Jones, G.; Fornwalt, H. J. The Viscosity of Solutions of Salts on Methanol. J. Am. Chem. Soc. 1935, 57, 2041.
Received for review May 1, 2001 Revised manuscript received August 28, 2001 Accepted August 30, 2001 IE010392Y
