Ind. Eng. Chem. Fundam. 1083, 22,
283-292
283
Nonideality of Single and Mixed Electrolyte Solutions Up to Moderately High Concentrations; Theory Based on Debye-Huckel Radial Distribution Function Kazuo Kondo and Charles A. Eckert' Department of Chemical Englneerlng, University of Illlnols, Urbana 6 180 1
Expressions are developed from statistlcai mechanics for the activity and osmotic pressures for electrolyte solutions with an arbitrary number of solutes. These are applied to solutions at 25 O C up to an ionic strength of 3. The estimated molal activity and osmotic coefficientsfor both single and mixed electrolytes are functions of a unique parameter; this parameter, d + - , has a significant physical meaning.
Introduction The nonideality of electrolyte solutions is a key to understanding various phenomena in the environment, biological systems, and industrial processes. Because of the complex nature of the solvent effect and of the mathematical difficulty in the treatment of long-range Coulombic forces, the theoretical treatment of the equilibrium properties of electrolyte solutions is not fully developed, especially compared with the current understanding of nonelectrolyte solutions. Theories currently available can in general be classified into two categories statistical mechanical theory and semiempirical theory. The major developments of the statistical mechanical approach have been accomplished by the integral equations of the hypernetted-chain (HNC) and of the Percus-Yevick (P-Y). Blum and Hoye (1975,1977) solved the P-Y equation with the primitive model and obtained an analytical solution. Because of the intrinsic restrictions involved, it is difficult to include the solvent effect properly or to extend the treatment to mixed electrolytes. Triolo et al. (1977, 1978) proposed either a density-dependent hard-sphere radius or a density-dependent dielectric constant with two adjustable parameters. However, these approaches seem to have inconsistencies when compared with the ideas proposed by Ramanathan and Friedman (1971). Rasaiah (1968, 1970) applied the HNC equation to aqueous solutions of the alkali halides in the McMillianMayer (M-M) system. The osmotic coefficients of these systems were calculated introducing the depth of the square-well potential as an adjustable parameter and setting the hard sphere radii ai; equal to the crystal radii. This was applied up to the molarity of 1, and the sign and magnitude of the parameter d+- seems to reflect the solvation effect around each ion. However, because of the extent of computation time required to obtain a numerical solution, this method is not well-suited for practical purposes. Hirata and Arakawa (1975) adopted the mean spherical approximations of Waisman and Lebowitz (1970,1972) as a reference system, and they introduced the square-well potential for the solvent effect as a perturbation. The same treatment was also applied to the osmotic coefficients of single electrolytes in the M-M system up to a molarity of 1. This approach seems to lose some of the original advantages of the mean spherical approximation, such as its simple analytical form, since the equation for the radial distribution function of the reference system is very complex. Note also that this method has the limitation of ai = aj and z, = z;.
Bich, Ebeling, and Krienke (Ebeling et al., 1971, Bich et al., 1976) used the pressure equation introducing three adjustable parameters for the depth of the square-well potential, d++, d-- and d+-. This method was applied to the osmotic coefficient of single and mixed electrolytes and to the activity coefficient of single electrolytes in the M-M system up to a molarity of 1 (Wichert et al., 1978; Ebeling et al., 1979). Unfortunately, the parameters dij reported for osmotic coefficients were not consistent with those for activity coefficients, and some parametric mixing rules were required for mixed electrolytes. Pitzer (1973) developed a semiempirical method based on the virial type expansion of Guggenheim. A complete table of parameters for 227 single electrolytes is available with rather small standard deviations (Pitzer and Mayoroga, 1973). However, to achieve comparable standard deviations for mixed electrolytes requires the introduction of two additional parameters (Pitzer and Kim, 1974). In general, statistical mechanical models require fewer adjustable parameters than do semiempirical models, and the parameters may have a more rigorous physical interpretation. However, the extension of Statistical mechanical theories, from those of single electrolytes to mixed electrolytes, may involve serious difficulties due to increasing mathematical complexity. On the other hand, the semiempirical models are frequently easily applicable to mixed electrolytes, but with some decrease in accuracy. In such cases, when larger numbers of adjustable parameters are involved, they usually have less physical meaning. Both the statistical mechanical and the semiempirical methods have their inherent disadvantages. Our approach is intermediate between these methods in order to compensate for the disadvantages of both. We develop simple but easily extended equations which are based on statistical mechanics. Fewer parameters with a more rigorous physical interpretation are used, compared to semiempirical theory. However, because of the simplicity of the equations, our method is easier to convert to other thermodynamic properties or to extend to mixed electrolytes. The interionic potential is the sum of the primitive model and the square-well potential. Rather than numerically solving the integral equation, we assume that the radial distribution function is given a priori by the Debye-Huckel radial distribution function. This radial distribution function agrees well with the results of the HNC or Monte Carlo methods up to modest concentrations of a few molar. A perturbation method is introduced to account for the solvent effect using a square-well potential. The pressure equation yields an osmotic coeffi-
Q196-4313/03/ 1Q22-Q283$Q1.5Q/Q0 1983 American Chemical Society
284
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1983
I
I
1
I
I:I
I
I
/ *
I
u
"82
05
I O
4 [-
mole-solute
1
Figure 1. The osmotic coefficient from the primitive model for a 1:l electrolyte solution where the cation and anion have either the same or different radii. Legend: solid curves and dotted curves are the results of HNC, a, = a- and aJa, = 3.6, respectively; (A) prediction of the present method for a, = a_; ( 0 )that for aJa, = 3.6; HNC results from Rasaiah (1970).
cient, and this is converted to the activity coefficients with the aid of the Helmholtz energy. These equations are applicable to a system with an arbitrary number of solutes and are applied to 22 1:l type and 21 2:l type single electrolytes in the Lewis-Randall system. Seventeen systems for the osmotic coefficients and 24 systems for the activity coefficients of mixed electrolytes were also tested. Assumptions Card and Valleau (Card and Valleau, 1970; Valleau and Card, 1972) used a Monte Carlo calculation to study the primitive model of 1:l type electrolyte solutions at 25 "C and a size parameter for the case of an anion and cation of the same radius. The calculated radial distribution function was compared with the integral equations of the HNC, P-Y, and also the Debye-Huckel radial distribution function (DHRD). The DHRD, which is of course the correct equation for the behavior of electrolyte solutions in the limit of infinite dilution, resulted in excellent agreement with the Monte Carlo results both at low concentrations and also at molarities up to 1 or 2.
I
1
I*O
2 .o mole-solute
Ic[ P-solution
1 3,O
I
Figure 2. The osmotic coefficient of the primitive model for the case of 1:1, 1:2, 1:3 electrolyte solutions. The anion and cation have the same radius, and their sum is 4.2 A. Legend: solid curves are the result of HNC; (A)prediction of the proposed method; HNC results from Rasaiah (1972). H.S,
3-
I
I
-
2Coulombic (unlike1
I -
-
0 -I
-
I![
r
Figure 3. An example of individual contributions to the interionic potential; ai, is the ionic radius and W is the diameter of the water molecule. Legend: H.S.; hard-sphere potential, S.W.; square-well potential; and Coulombic, Coulombic potential.
Coulombic and the hard-sphere terms are what constitute the primitive model, and a square-well term is introduced to take care of the solvent effects ZiZi@e2 exp(caij)
E(l
+ Kaij)
(2)
ZiZje2 uij(r)= Er + uij*(r)H.S. + uij*(r)S.W.
uij*(r)H,S,= Because of the extensive computation time required, little work has been done on a Monte Carlo calculation for the primitive model. h a i a h (1970,1972),however, used the HNC approach to obtain the osmotic coefficient for the case of an anion and cation of different radii (Figure 1) and that of multivalence electrolytes (Figure 2). Both are in good agreement with experiment up to an ionic strength of 1, and the results are always better at smaller ionic strengths. This DHRD is regarded as the reference radial distribution function to describe the contribution of the model, but it is not as accurate as a more rigorous statistical mechanical method, such as Monte Carlo or HNC. The interionic potential consists of the Coulombic term, the hard-sphere term, and the square-well term. The
=0 ui,*(r)s,w,= dij
=O
r < aij
m
(4)
(5)
r 2 uij aij
aij+w
where ai, is the hard-sphere diameter, d , is the depth or height of the square-well, and w is the diameter of the water molecule ( w = 2.76 A). The relative magnitudes of these potentials are shown in Figure 3. The hard-sphere and the square-well terms were both introduced to yield a simple analytical form. Following Ramanathan and Friedman (1971), both anion and cation have cospheres of thickness of one water molecule, and if these cospheres overlap, the mutual volume is assumed to return to its normal state, i.e., the
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1983
bulk solution. The square-well term accounts for the effect of the solvent around each ion. The squarewell potential has in it three parameters, d++, d-- and d,. Since the probability of ions of the same sign approaching each other is much less than those of opposite sign, due to the electrostatic forces between ions, we choose to reduce the number of adjustable parameters to one d++ = d-- = 0 (7)
285
-.-
(8)
d+- # 0
Because of this choice of parameter, no new mixing rule is required to extend our model to mixed electrolytes. If there were three, some kind of mixing rule would be required for the case of a common ion in different electrolytes. For example, the mixing rule for d-- would be required for the electrolyte mixture NaC1-LiC1. For univalent electrolytes,the hard-sphere radius aijwas set equal to the crystal radius throughout the present calculation. For multivalent electrolytes such as MX2, however, we adopt the apparent hard-sphere radius for the cation M, found equal to the crystal radius plus one water molecule in thickness. This method presents a physical picture of the cation M being rigidly hydrated by a hydration shell of one water molecule in thickness, which is not readily displaced in the interaction with other ions. The pressure equation connects these microscopic properties to the macroscopic property of the osmotic coefficient. As was mentioned above, the DHRD includes the primitive model contribution. Hence the osmotic coefficient can be obtained by perturbation theory, choosing the reference system as the primitive model and the perturbed system as square-well contribution. 4 - 1 = (4 - 110 + (4 - 1)1 =
Here uij(r)ois the primitive model potential and uij*(& is the square-well potential. The proposed model was first examined in terms of the McMillian-Mayer (M-M) system, and in Figure 4 the parameters d+- are compared with those from several statistical mechanical models. These were obtained by fitting the osmotic coefficient up to a molarity of 1,except for the results of Bich et al. (1976) and Wiechert et al. (1978), where d + B values were obtained from an extrapolation to infinite dilution to avoid the use of three parameters. The abscissa represents the results of Rasaiah (1970), based on the more rigorous HNC method. The parameters of the proposed model were found to be closest to those of Rasaiah. A few discrepancies for lithum as a cation were caused by the large perturbation term. Equations for Nonideality Substituting eq 1, 4, 5, and 6, the pressure equation yields the osmotic coefficient
."-I,O
0
d+-p by
I .o Rosoioh
2,O
Figure 4. Comparison of the parameter (d+J) with three other approaches in the McMillian-Mayer system. The 45' line repreaenta the parameter obtained from HNC by Rasaiah (1970). Other data are presented for comparison: Hirata and Arakawa (1975) and Wiechert and Ebeling (1978).
The first term in eq 10 is the Coulombic term. Using the condition of electrical neutrality
(11) g..n
m
V exp(-nrr) n! rn-l
(9ij(r)D-H- 1)r = C
n=l
(12)
Each term in the series goes to zero at large separations, so that the integration in eq 11 always converges (see Appendix 2 for this integration). The second hard-sphere term takes the following form, since exp (-pU) is a heavy-side function (Barker et al., 1976)
With a heavy-side function of [l- exp(-pulj)] / [ 1 - exp(-@iij)], the square-well term can be written as (4 - 1)S.W. = 2a -CCcicj[l - exp(-pdij)](aij + w ) (14) 3~ i j In total, the osmotic coefficient is given as (4 - 1) (4 - 1)Coul + (4 - 1)H.S. + (4 - 1)S.W. =
+
2a -CCcicjai~g&ij) 3~ i j 4a - cicj[l - exp(-pdij)](aij 3c (+-) c =
+
+ w)3gij(aij+ w )
CCi i
(15) (16)
where is meant to be the sum of the anion-cation interactions only and ciis the number density. To convert eq 15 to the activity coefficient, we use Aex, the excess Helmholtz energy, since the proposed model is based on the McMillian-Mayer system (Friedman, 1972).
AT^^ = dr
.
I
V'
[P - Pd]dV
(17)
P = ckT4
(18)
P
(19)
= ckT
286
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1963
Table I. "he Significant Interactions for the Equation of the Activity Coefficient of Mixed Electrolytes case case - ion in
common + ion in
common no common ion
significant
electrolytes
interactions
M ( l ) X ( 2 ) , M(3)X(2)
1-2, 2-3
M ( l ) X ( 2 ) ,M ( l ) X ( 3 )
1-2, 1-3
M ( l ) X ( 2 ) , M(3)X(4)
1-2, 3-4 1-4, 2-3
e2ZiZj@ 1, = t
y , .=
The excess Helmholtz energy is related to the activity coefficient as ex.(
where yi is the activity coefficient of an ionic species i. The mean activity coefficient is lny* = (v+ In y+ + v- In y J / v (21) u = lJ+
+ u-
(22)
A lengthy derivation (see Appendix I) yields the equation for the mean activity coefficient of the electrolyte MX in the presence of n species of ions. In yhlx =
exp((aij-r)z) (1
+ ai;@
x [a&
- r(aija + l)]gij(r,t)dr dt -
gi, = exp[ - gexp(-zr) a y ]
(28)
-:)
[lij + aij] +
The significant interactions for the square-well contribution of eq 23, the last two terms, are given in Table I.
Applications and Discussion The statistical mechanical model is baaed on the theoretical system of M-M and the experimental results are tabulated in the Lewis-Randall (L-R) system (Friedman, 1972). The conversion between these two systems requires very accurate volume data, which are not available for every electrolyte, and this situation becomes much worse for the case of mixed electrolytes. Hence we assume that we may apply our statistical mechanical model to the L-R system. This is unavoidable if one wishes to explain the thermodynamic properties of the electrolyte solutions where volume data are not known. Both the molal activity and osmotic coefficient of the single electrolytes and their mixtures are explained in terms of a consistent, unique parameter d+-, and we consider this sufficient for the present purpose. The conversion required is between molarity and molality, since the statistical mechanical system has the concentration scale of molarity, but the experimental data are tabulated by molality. This conversion is discussed in Appendix 111. In figure 5, the parameters obtained from fitting the molal osmotic coefficient in the L-R system up to a molality of 3 are compared with the results of Rasaiah. There are some deviations from those in the M-M system, but the physical meanings of the solvent effects are still qualitatively correct. Because of the limitations both of the data fitting and of the assumption of pairwise additivity, we set the upper limit of the concentration as an ionic strength of 3. The osmotic and activity coefficients for single electrolytes were taken from Robinson and Stokes (1959), except for RbF and CsF (TiTien, 1963). The parameters @d+and the standard deviations for the single electrolytes are presented in Tables 11-IV, where parameters were obtained from fitting activity coefficient data. The parameter @d, is better evaluated from activity coefficient data, since the osmotic coefficient 4 always includes unity. In fact @d, values were also obtained from fitting osmotic coefficients, and the differences were not significant. In Tables I and 111, alkali halides and alkali earth halides are classified with the common anions, since larger changes of the size of crystal radii occur on the cation side than do on the anion side. As the size of the cations become larger, @d+-decreases monatonically except for fluorine as an anion. This corresponds to the current view of ion hydration, in the sense that smaller ions tend to hydrate
Ind. Eng. Chem. Fundam., Vol. 22,
No. 3, 1983 287
Table 11. The Parameters d+- and t h e Standard Deviations for t h e 1:1 w p e Electrolytes of Alkali Halides and Tetraalkylammonium Halide name
a+, A
a-, A
pd+-, Y
2.08 0.60 0.95 1.33 1.48 1.69 2.08 0.60 0.95 1.33 1.48 1.69 2.08 0.60 0.95 1.33 1.48 1.69 1.33 1.48 1.69 1.48
1.81 1.81 1.81 1.81 1.81 1.81 1.95 1.95 1.95 1.95 1.95 1.95 2.16 2.16 2.16 2.16 2.16 2.16 1.36 1.36 1.36 1.81
0.04691 1.092 0.1748 -0.09749 -0.1610 -0.2584 0.09590 1.060 0.1924 -0.1099 -0.2124 -0.2924 0.1132 1.159 0.1845 -0.1199 -0.2524 -0.3356 0.2153 0.2039 0.2131 -0.1298
HC1 LiCl NaCl KCI RbCl CSCl HBr LiBr NaBr KBr RbBr CsBr HI LiI NaI
KI RbI CSI KF RbF CsF NH,Cl
Table 111. The Parameters d+- and t h e Standard Deviations for t h e 1:1 Type Electrolytes for the Case of Alkali Earth Metals as Cations name
a+,A
a-,A
MgC4 CaC1, SrC1, BaCl, MgBr2 CaBr, SrBr, BaBr, MgI, CaI, SrI, BaI,
0.65 0.99 1.13 1.35 0.65 0.99 1.13 1.35 0.65 0.99 1.13 1.35
1.81 1.81 1.81 1.81 1.95 1.95 1.95 1.95 2.16 2.16 2.16 2.16
pd+-,r -0.1500 -0.2593 -0.3040 -0.3760 -0.0900 -0.2270 -0.2800 -0.3460 -0.0600 -0.1990 -0.2470 -0.2940
o(@) x 4 7 )
0 d+J
I .o by Rosoiah
lo2
o(B1um)x 10, O ( Y ) X10,
1.o 1.0 1.0 1.o 0.9 0.4 1.0 0.6 1.0 1.2 1.1 0.5 2.0 0.5 1.6 1.6 1.3 1.2 0.7
0.4 0.8 2.0 1.6 1.1 0.4 1.2 0.8 1.6 1.7
0.8 0.4
0.8 1.8 1.2 1.9 0.7 0.4 1.4 3.2 2.0 1.7
-
x
102
name
a+,A
0.7 0.4 0.3 0.4 1.4 1.1 0.4 0.4 1.3 0.9 0.7 1.4
0.3 0.5 0.3 0.2 0.9 0.7 0.6 0.3 1.2 0.7 0.5 0.7
MnCl, FeCl, coc1, CoBr, NiCl, CUCl, ZnC1, ZnBr, ZnI,
0.80 0.7 5 0.72 0.72 0.70 0.72 0.74 0.74 0.74
2 .o
Figure 5. Comparison of the parameter ( d d ) with the HNC result for the McMillian-Mayer system and the result for the Lewis-Randall system. The 4 5 O line represents the parameter obtained from HNC by Rasaiah (1970). Legend: ( 0 )as an anion; (A)Br- as an anion; (M) I- as an anion.
more strongly than the larger ones. In Table IV, the transition element halides are classified with the common cations, since larger changes of ion size can be observed on the anion side. In this case, Bd+- shows an opposite
0.7 1.0 2.0 1.7 1.2 0.5 2.2 1.0 1.6 1.7 1.4 0.6 1.7 2.1 1.4 1.9 1.0 0.7 1.4 1.9 1.1 1.6
Table IV. The Parameters d+- and the Standard Deviations for t h e 2 :1 Type Electrolytes in t h e Case of Transition Elements as Cations
10,
/ K + o ~ a cation
-1.0 -I -0
o(@)X
a-,A
1.81 1.81 1.81 1.95 1.81 1.81 1.81 1.95 2.16
pd+-,r -0.2430 -0.2170 -0.1940 -0.1240 -0.1850 -0.2760 -0.3300 -0.1900 -0.1000
4@) x 47)x 10,
10,
0.4
0.3 0.2 0.2 0.8 0.3 0.5 2.3 1.8 0.7
0.8 0.7 1.1 0.7 0.5 4.6 2.5 1.0
tendency, and as the anions become larger, the parameter @d+-tend to increase. Triolo et al. (1976)used the analytical solution of the mean spherical approximation to fit the osmotic coefficients. The radius of one of the ions was used as an adjustable parameter. Our standard deviations are comparable to their results, but B1u"s model was applied up to a molality of only 2, while the proposed model applies up to 3. The standard deviations of the predictions for mixed electrolytes and those for single electrolytes which make up the mixtures are listed in Tables V and VI. Osmotic coefficient data measured by the isopiestic method and activity coefficient data measured by the emf method have been used to test the proposed model for mixed electrolytes, since these experimental techniques are the most reliable and in the most general use. Comparison with Pitzer's Method The semiempirical model of Pitzer (Pitzer, 1973;Pitzer and Mayoroga, 1973),using three adjustable parameters, yields a standard deviation of less than f0.003 for most of the single electrolytes listed in Tables II-IV. This probably represents the limit of accuracy for fitting experimental data, as Rand and Spedding (1977)have s u m marized the osmotic coefficient data of CaC12,which are often used as a reference electrolyte. In comparing data from various sources, they estimate the uncertainty to be f0.3 70,or f0.003 on our scale.
288
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1983
Table V. The Standard Deviations of the Calculated Osmotic Coefficients for Mixed Electrolytes NaCl
* NaCl Na Br * KC1 KCl KCl CSCl CSCl CSCl NaCl NaCl KCl LiCl CSCl NaCl KC1 KCl
LiCl NaBr KBr KBr NaCl LiCl LiCl NaCl KCl CUCI,
2.0 2.0 1.6 1.6 1.6 1.6 0.4 0.4 0.4 2.0 2.0 1.6 0.8 0.4 2.0 1.6 1.6
coc1, SrC1, Baa, I3ac1, MnC1, CaC1, MgCl,
0.8 1.6 1.7 1.7 2.0 0.8 0.8 2.0 1.6 0.5 0.7 0.3 0.4 0.4 0.4 0.4 0.7
-
2.3 4.1 2.2 3.3 2.3 4.0 5.0 2.6 1.4 2.1 2.0 2.2 0.6 1.4 1.4 1.6 1.6
Y , dd
-
C C
0.9 ( 4 ) 1.4 (4.8) 4.5 (4.8) 10.0 ( 5 ) 2.7 ( 5 ) 0.3 ( 5 )
C
an Y 2, cc X
-
2
-
f
g
-
e
2.4 ( 4 ) 0.4 (5.5) 2.5 ( 5 )
U U
d bb
-
W
’ Values inChristenson, parentheses designate maximum molality. Chan, C.-Y.; Khoo, K. H. J. Ckem. SOC.,Faraday Trans. 1 1979,75, 1371. P. G.; Gieskes, J. M. J. Chem. Eng. Data 1971,16, 398. Covington, * Case of cation in common.
A. K.; Lilley, T. H.; Robinson, R. A. J. Pkys. Ckem. 1968, 72, 2759.
Downes, C. J. J. Chem. Eng. Data 1973,18, 412. 1974,6, 317. f Downes, C. J. J. Solution Chem. 1975,4,191. g Downes, C. J.; Pitzer, K. S. J. Solution Ckem. 1976,5, 389. Harned, H. S.; Schupp, 0. E., Jr. J. Am. Chem. SOC.1930,52,3892. Harned, H. S.; Copson, H. R. J. A m . Chem. SOC.1933,55,2206. j Harned, H. S.; Hamer, W. J. J. Am. Chem. SOC.1933, 55, 4496. Harned, H. S.; Gancy, A. B. J. Pkys. Chem. 1958,62, 627. Khoo, K. K.; Chan, C.-Y.; Lim, T. K. J. Solution Ckem. 1977,6, 651. Khoo, K. H.; Chan, C.-Y.; Lim, T. K. J. Solution Ckem. 1977,6, 855. Khoo, K. H.; Lim, T.-K.; Chan, C.-Y. J. Solution Chem. 1978, 7, 291. O Khoo, K. H.; Chan, C.-Y.; Lim, T. K. J. Chem. SOC.,Faraday Trans. 1 1978, 74, 837. Khoo, K. H.; Lim, T.-K.; Chan, C.-Y. J. Chem. SOC.,Faraday Trans. I 1978, 74, 2037. Q Khoo, K. H.; Lim, T.-K.; Chan, C.-Y. J. Solution Ckem. 1979,8,277. Khoo, K. H.; Lim, T.-K.; Chan, C.-Y. J. Ckem. SOC., Faraday Trans. 1 1979,75, 1067. Lanier, R. D. J. Pkys. Ckem. 1965,69,3992. t Lim, T.-K,; Khoo, K. H.; Chan, C.-Y. J. Solution Ckem. 1980,9, 785. Lindenbaum, S.; Rush, R. M.; Robinson, R. A. J. Chem. Thermodyn. 1972,4, 381. ” Macaskill, J. B.; Robinson, R. A.; Bates, R. G. J. Solution Ckem. 1977,6, 385. Padova, J.; Saad, D. J. Solution Chem. 1977,6, 57. Robinson, R. A. J. Am. Ckem. SOC.1952, 74, 6035. Y Robinson, R. A.; Lim, C. K. Trans. Faraday SOC.1953,49,1144. Robinson, R. A. Trans. Faraday SOC.1953,49, 1147. a a Robinson, R. A. J. Phys. Chem. 1961,65, 662. b b Robinson, R. A.; Covington, A. K. J. Res. Natl. Bur. Stand. Sect. A 1968,72A, 239. c c Robinson, R. A. J. Ckem. Thermodyn. 1973,5, 819. d d Robinson, R. A.; Wood, R. H.; Reilly P. J. J. Ckem. Tkermodyn. 1971,3, 461. e e Robinson, R. A.; Roy, R. N.; Bates, R. G. J. Solution Chem. 1974,3, 837. If Robinson, R. A.; Bates, R. G. Mar. Chem. 1978,6, 327. gg Roy, R. N.; Gibbons, J. J.; R o w e r , J. K.; Lee, G. A, J. Solution Chem. 1980,9, 535. h h Vance, J. E. J. Am. Chem. SOC.1933,55, 4518. I Z White, D. R., Jr.; Robinson, R. A , ; Bates, R. G. J. Solution Ckem. 1980,9, 457.
e Downes, C. J. J. Ckem. Tkermodyn.
‘
Table VI. The Standard Deviations of the Activity Coefficients of the Mixed Electrolytes In Y
* * * * * * *
*
*
* * *
* * *
“A X
OB X
OmixX
Y 0Pitze;X
A
B
102
102
102
10
HCl HCl HCl HC 1 HBr HBr HCl HC 1 HC1 HCl HC1 HCl HCl HBr HBr HBr KC1 KC1 KCl NaCl NaCl NaCl NaCl NaCl
LiCl NaCl KC 1 CSCl LiBr KBr NH,C1 MgClz CaC1, MnCl, COCl, NiCl, BaCl, Ca Br , BaBr, SrBr, MgC4 CaCl, Baa, MgC4 MgC4 CaC1, SrC1, BaCI,
0.6 0.6 0.6 0.6 1.6 1.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 1.6 1.6
1.2 2.9 2.7 0.8 1.o 2.7 2.6 0.5
3.3 1.9 1.4 8.5 4.6 4.8 1.8 3.1 1.9 1.5 3.3 3.7 1.8 1.4 1.2
2.3 ( 5 ) 4.0 ( 3 ) 1.4 (3.5) 8.2 ( 3 ) 2.7 (2.5) 3.0 ( 3 )
1.6
2.7 2.7 2.7 2.9 2.9 2.9 2.9 2.9
1.o
0.6 0.4 0.6 0.5 1.3 0.6 1.2 0.5 1.0 0.5
0.5 0.5 1.0 0.8 0.5
1.3
2.9 3.0 3.0 4.0 4.5 4.0 3.9 3.6
-
0.8 ( 3 ) -
0.9 ( 3 )
-
-
-
-
OA x 102
O B X
102
lo2
0.7 0.7 0.7 0.7 2.2 2.2 0.7 0.7 0.7 0.7 0.7 0.7 0.7 2.2 2.2
1.0 2.0 1.7 0.5 1.0 1.7 1.6 0.3 0.5 0.3 0.2 0.3 0.2 0.7 0.3
3.4 1.4 1.3 6.0* 5.1 4.3* 1.7 2.4 1.5 1.3 2.7 3.0 1.2 1.1 0.9
2.2
1.7 1.7 1.7 2.0 2.0 2.0 2.0 2.0
0.6
0.3 0.5 0.2 0.3 0.3 0.5 0.3 0.2
* Either Y A or yg is listed in the source. Values in parentheses indicate maximum molality. H. S. Harned et al. f The references are given in the footnotes of Table V. The model presented here uses only a single parameter, and in the development from statistical mechanics it has a rather specific physical meaning. As might be expected,
DmkX
0.9
1.8 1.9 1.8 2.4 3.2 2.7 2.7 2.3
ref$
*
1
*
a, k k kk
u
j
ee m, ii 1
gg P n 0
r 4 t
b b b
ff
S S S
S
* Measurement done by
the fit of the pure electrolyte data by our method, generally better than f0.020,is not as precise as Pitzer’s method. Surely more parameters would enhance the fit, but the
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1983 289 Table VII. Parameters for the Density and Their Standard Deviations. name A , X 10 A , x 10, A , X 103 LiCl NaCl KC1 RbCl CSCl HBr LiBr NaBr KBr RbBr CsBr HI LiI NaI KI RbI CSI KF RbF CsF NH,C1 MgCl, CaCl, SrC1, BaCl, MgBr, CaBr, SrBr, BaBr, MgI, CaI, SrI, &I, COCl, CoBr, ZnC1, ZnBr, ZnI, MnCl, FeCl, NiCl, CUCl,
0.1753 0.2340 0.3865 0.4414 0.8802 1.208 0.5499 0.5904 0.7624 0.8153 1.257 1.656 0.8690 0.9345 1.078 1.169 1.562 1.935 0.4943 0.8834 1.280 0.1483 0.7321 0.8559 1.336 1.788 1.478 1.598 2.045 2.487 2.233 2.362 2.744 2.995 1.175 1.932 0.6627 1.883 2.526 1.050 1.084 1.201 1.224
-0.03394 -0.01927 0.00501 -0.02497 -0.4171 -0.03553 -0.07915 0.06160 -0.1042 -0.1972 -0.6234 -0.8927 -0.02404 -0.05036 -0.00954 -0.3916 -0.5278 -0.5773 -0.01997 -0.04079 -0.1586 0.00706 -0.2350 -0.1594 -0.1639 -0.6484 -0.4349 -0.3731 -0.6069 -0.9706 -0.8724 -0.8468 -0.2980 0.6061 -0.3868 -0.6726 -0.1049 -0.9620 0.05562 -0.5471 -0.00015 -0.5742 -0.6403
-0.04379 -0.1190 -0.2601 -0.297 2 0.1808 -0.8469 -0.1213 -0.4741 -0.2439 -0.1630 0.2597 0.3755 -0.3638 -0.4643 -0.6072 -0.1801 -0.2415 -0.6971 -0.3672 -0.3861 -0.3965 -0.1343 -0.1505 -0.4490 -1.051 -0.7114 -0.5594 -1.030 -0.8360 -1.534 -0.9333 -1.743 -4.318 0.3198 -0.6028 -0.9613 -0.9403 -0.3802 -0.9780 0.3289 -0.04479 0.3248 -0.02405
pw =
0.9970 g/cm3 at 25 “C 103 max m
x
0.21 0.36 0.55 0.69 0.05 1.85 0.26 0.80 0.55 0.81 0.05 0.07 0.95 1.15 1.83 0.77 1.55 1.86 0.98 1.65 1.36 0.59 1.15 1.11 1.67 0.13 0.93 1.54 1.47 .58 0.70 0.74 2.01 5.21 0.64 0.96 0.56 3.59 5.23 0.19 1.53 0.08 0.2414
t, “C
3.7 3.8 3.8 3.8 3.5 4.0 3.5 3.6 3.8 4.5 4.0 3.8 4.2 4.0 4.4 4.0 4.7 3.8 3.3 4.5 4.4 4.7 4.5 3.9 3.4 1.7 3.6 3.3 3.3 2.2 2.4 2.3 2.4 1.1 1.9 2.0 2.1 3.6 1.3 4.2 1.1
ref a a a a a a a a a a a
a a a a a a a
18.0 18.0 20.0
a a
b a a a a a a a a a a a
a a
20.0 18.0
c a a a
20.0
a
18.0
4.0 ~. .
b b b
1.9
a
“International Critical Table of Numerical Data Physics, Chemistry and Technology”, Vol. 3;National Research Council, McGraw-Hill Book Co.: New York, London. “Landolt-Bornstein”, Gruppe IV, Band 1, Teilb, Springer-Verlag, Berlin, Heidelberg, New York, 1977. Weast, R. C.; Astle, M. J. “CRC Handbook of Chemistry and Physics”, 59th ed.; CRC Press, Cleveland, 1978. a
relative simplicity of a single parameter improves the extension to mixed electrolyte solutions. The technique proposed here, though more rigorous in derivation, is now more difficult to use computationally than is Pitzer’s method. The major reason for this difficulty is that we have not yet been able to solve analytically the Coulombic terms in eq 15 or 23,and were forced to use a numerical iniegration. If an analytical form were developed for this integration, the proposed method would in fact become simpler to apply than Pitzer’s. As stated earlier, the conversions between the M-M system and the L-R system are not possible for all of the completely dissociated electrolyte solutions, and certainly not for the mixed electrolytes. Thus we had no choice but to apply our method to the L-R system, in spite of the fact that it is really more adaptable to the M-M system. More accurate and extensive density data would ameliorate this problem. In addition, a certain amount of uncertainty is introduced by the use of the Debye-Huckel radial distribution function as the reference function for systems at finite concentrations.
One great advantage of the present formulation is that the rather fundamental use of statistical mechanical techniques results in an excellent physical interpretation of the parameter ( d + J ) , certainly with more physical significance than Pitzer’s parameters. The solvent effect term does take care of the contribution of the residual of the primitive model, and the agreement of our parameters with those found more rigorously by Rasiah (Figure 4) is most encouraging. The greatest advantage of our method lies in the extension to mixed electrolytes without additional parameters. If the Pitzer method is so extended, a very large increase in the standard deviation of the predictions results, sometimes greater than a factor of 10 when compared to the single electrolyte calculations (Pitzer and Kim, 1974). Our proposed method gives uncertainties for single electrolytes generally better than *0.02, and for mixed electrolytes, usually better than i0.03,using only the pure electrolyte parameters. It should be noted that the systems marked with 1in Table VI were taken from the work of Harned and Owen (1958) and were measured in the early
290
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1983
1930’s. The molality of one of the solutes (A in Table V) was limited to 0.01 throughout their work. It seems that this limitation severely strains the theoretical model, since not only our model but also Pitzer’s model shows large standard deviations for these systems.
Summary Equations have been presented to describe the molal activity and osmotic coefficients in the L-R system for electrolyte solutions. The activity and osmotic coefficients of mixed electrolytes were successfully described in terms of only the parameters used for individual single electrolytes making up the mixture. The model presented would certainly be easier to apply if an analytical solution could be found for the Coulombic term, using physically reasonable parameters.
Acknowledgment This project has been financed in part with Federal funds as part of the program of the Advanced Environmental Control Technology Research Center, University of Illinois at U-C, which is supported under Cooperative Agreement CR 806819 with the Environmental Protection Agency. The contents do not necessarily reflect the views and policies of the Environmental Protection Agency, nor does the mention of trade names or commercial products constitute endorsement or recommendation for use. Additional funding was provided by the National Science Foundation. The valuable suggestions of Dr. K. R. Cox at the onset of this work are very much appreciated.
1
ex.(-
kj
Uij(1
+ KUij)
)+
lmjexp(-Kwt) eXP[ - bmj(l + KU,jt)
]dtt
Appendix I Derivation of the Activity Coefficient. From eq 15 and 17, one obtains the expression for the Helmholtz energy by introducing new variables Aex/ V’ =
+
(AexCoul AeXH,S,+AeXS,W,)/
V’ = ex.[
]
lij exp(K(uij- r ) t ) -‘l/ dr dt r(1 u i j ~ t ) 27T 47T -cccicjyij + --EECiCjbij3 x 3aK2 i I 3P i j lij exp(-Kwt) 1 exp(-dij~)~ t exp dt (A-1) bij(1 + UijKt)
-
+
S,
[
+ KUijt) + aij X (1 + KUijt)2 1, exp(-rwt) - bij(1 + KUijt)
w(1
]
dt (A-5)
In taking derivatives, note that K is a function of c.i Finally eq 23 is derived from eq 21. For the case of the single electrolytes, eq 23 reduces as
]
(A-3) where K’
= K(v9lI2
(A-4) L
To avoid confusion, the summation for the square-well
term is given as CiCj instead of C(+-), and at the final stage of the derivation, unnecessary terms are eliminated. Note that K is a function of volume, since ci = Ni/V. The activity coefficient of an individual ionic species m can be given by eq 20 and A-1 as
(A-6) Equation A-6 can also be derived by the method of Ebeling and Scherwinski (1979)
Ind. Eng. Chem. Fundam., Vol.
(A-7) xi =
(A-8)
Vi/V
Note that eq 21 is not necessary in this case, since the electrical neutrality is already involved in eq A-8.
Appendix I1 Evaluation of Coulombic Term. An appropriate numerical approach is required, since the integration of the Coulombic term in eq 15, 23, and A-6 does not have a simple analytical form. Thanks to the exponential integral, this can be done with good accuracy and reasonable computation time. These Coulombic terms can be summarized as three types of integrations; the DHRD can be expanded into a series
Thus these three integrations can be written as
(A-10)
Ji, r exp(-rK)gij(r) dr =
22, No. 3, 1983 291
where pw is the density of pure water, Al, A2, and A3 are the adjustable parameters, and m is the molality. For mixed electrolytes (A-16) Pmixed = PSA + PSB - Pw where p s and ~ p s are ~ the densities of the single electrolytes A and B, respectively. The parameters are listed in Table VI1 with their standard deviations. Once the density is known, one can easily convert from molality to molarity with conventional equations (Robinson and Stokes, 1959).
Nomenclature Ai = coefficient of density equation (eq A-15) Aex = excess Helmholtz energy Q = hard-sphere radius of restricted primitive model = hard-sphere radius of species i. Qij (Qi + Qj)/2 bij = see eq 24 ci = numerical density of species i c = see eq 16 d , = height or depth of square-well potential e = electron charge gij(r) = radial distribution function &(r) = see eq 26 k = Boltzmann constant 1 , = see eq 28 m = molality Ni = number of ith molecule in the volume of system P = osmotic pressure r = distance t = see eq A-2 T = absolute temperature u = see eq A-3 u..(r) = potential 9=volume w = diameter of a water molecule x i = see eq A-8 Yij = see eq 29 Zi = valency of ith ion
Greek Letters
As the distance become larger, the integrand in each term of the series goes to zero. Hence these integrations converge. With the aid of the regression formula below, these integrals reduce to the form of an exponential integral (Abramowitz, 1970; Moriguchi, 1956). exp(-ar)
/3 = l / k T yi = activity coefficient of ith species y a = mean activity coefficient (eq 21) ~ M =X activity coefficient of electrolyte MX e = dielectric constant = see eq 2 Oij = see eq 27 K = inverse Debye length (eq 3) ii = see eq 25 K' = see eq A-4 v = see eq 22 vi = number of moles ith ion formed from 1mol of electrolyte p = density (4 - 1) = osmotic coefficient
Literature Cited exp(-ar)
(m- l)!
dr (A-13)
At smaller distances, these series frequently converge more slowly. In such cases, one has to integrate numerically at smaller distances and then switch to a series representation. Details are presented by Kondo (1981).
Appendix I11 Density Equation. The density data for single electrolytes were fitted to the power series psingle = pw
+ Alm + Azm2+ A3m3
(A-15)
Abramwtiz, M.; Stegun, I.A. "Handbook of Mathematical Functlons"; Dover Publications, Inc.: New York, 1970. Barker. J. A,; Henderson, D. Rev. Mod. Phys. 1976, 4 8 , 587. Bich, Von E.; Ebellng, W.; Krlenke, H. 2.fhys. Chem. Leipzig 1976, 257, 549. Blum, L. Mol. Phys. 1975, 30, 1529. Blum, L.; Hoye. J. S. J. Phys. Chem. 1977, 8 1 , 1311. Card, D. N.; Valieau, J. P. J. Chem. W y s . 1970. 52, 6232. Ebellng, W.; Scherwinski, K. Rostocker Physikallsche Manwkrlpte, Heft 4, 15, 1979; Wllhelm-Pleck-Universitiit, Rostock, Sektion Physik. Friedman, H. L. J. Solutbn Chem. 1972. 1 , 387. Harned, H. S.; Owen, B. B. "The Physical Chemistry of Eiectrowlc Solutions", 3rd ed.;Reinhoid: New York. 1958. Hirata, F.; Arakawa, K. Bull. Chem. SOC.Jpn. 1975, 48, 2139. Kondo, K. Ph.D. Thesis, Unlverslty of Illlnols, Urbana, IL, 1981. Morlguchi, S.; Udagawa, K.; Ichlmatsu, S."Mathematical Formula I"; Iwanaml Zensho Pubiishlng Co. 221, Tokyo, 1956. Pitzer, K. S. J. Phys. Chem. 1973, 77, 268. Piker, K. S.; Mayoroga, G. J. Phys. Chem. 1973, 77, 2300. Pitzer, K. S.; Kim, J. J. J. Am. Chem. Soc.1974, 96, 5701. Ramanathan, P. S.; Friedman, H. L. J. Chem. Phys. 1971, 54, 1086. Rard, J. A.; Spedding, F. H. J. Chem. Eng. Data 1977, 22, 56, 180. Rasalah, J. C. J. Phys. Chem. 1986, 72, 3352.
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Ind. Eng.
Chem. Fundam. 1983, 22, 292-298
Rasaiah, J. C. J . Chem. Phys. 1970, 5 2 , 704. Rasaiah, J. C. J . Chem. Phys. 1972, 56, 3071. Robinson, R. A.; Stokes, R . H. "Electrolyte Solutions", 2nd ed.; Butterworths. London, 1959. TiTien. H. J . Phvs. Chem. 1983. 67. 532. Triolo, R.; Grigha, J. R.; Blum, L. J . Phys. Chem. 1976, 80, 1858. Triolo, R.; Blum, L.; Floriano, M. A. J . Phys. Chem. 1978, 82,1368. Triolo, R.; Blum, L.; Floriano, M. A. J . Chem. Phys. 1077, 6 7 , 5956. Valleau, J. P.; Card, D. N. J . Chem. Phys. 1972, 5 7 , 5457.
Waisman, E.; Lebowitz, J. L. J . Chem. Phys. 1070, 52. 4307. Waisman, E.; Lebowitz. J. L. J . Chem. Phys. 1972, 56, 3086. Wiechert, V. H.; Krienke, H.; Feistel, P.: Ebeling, W. 2.Phys. Chem. Leipzig 1978, 259, 1057.
Received for review September 14, 1981 Revised manuscript received December 28, 1982 Accepted January 19, 1983
Analysis of Sulfur and Nitrogen Pollutants in Three-phase Coal Combustion Effluent Samples Jeffrey R. Burklnshaw,'
L. Douglas Smoot, Paul 0. Hedman, and Angus U. Blackham
Combustion Laboratory, Brigham Young University, Provo, Utah 84602
Methods for analysis of sulfur and nitrogen pollutants sampled with water-quenched probes from pulverized coal combustors have been developed. Analyses are outlined for measurement of SO2, H2S, COS, CS2, and NO in the gas-phase sample, S O:-, S2-,CN-, and NH', in the liquid-phase sample, and total sulfur and nitrogen in the solid-phase sample. Permanent gas analysis and solid ultimate analysis methods are also presented. The relative error for these measurements was 6 to 8 % , based on extensive laboratory testing. These methods were applied to the analysis of three-phase (gas, solid, and liquid) samples from a laboratory-scale pulverized coal combustor in a series of three tests with two different coals. Results emphasized sulfur pollutant measurements. Sulfur oxides, and to a lesser extent, H2S, were the principal sulfur species formed while low levels of COS and CS2 were also detected. Most of the SO, and H2S were recovered in the liquid-phase sample. A mass balance for sulfur including that remaining in the char agreed to within 1% when the coal feed rate was determined from a carbon balance.
Introduction The processes that control within a coal combustor are highly speculative if the only test data available are the analyses of the initial reactants and the final products. Furthermore, optimization of the combustion process becomes almost totally empirical in nature. To provide greater insight and a sound theoretical foundation for combustion technology and modeling, local samples taken from within the reactor are essential. Such local samples may be obtained with a probe which water-quenches the reactants and products to provide a representative sample (Thurgood et al., 1980). The chemical analysis of the three-phase (gas, liquid, and solid) sample is then an essential endeavor. Price et al. (1983) reported a similar but less extensive work for the analysis of samples from a coal gasifier (see also Skinner et al., 1980). Several investigators have reported methods and results for the measurement of each of the pollutant species of interest in this study: SO2,H2S,COS, CS2, SO:-, S2-, CN-, NH,+, NO, N(solid), and S(so1id). Burkinshaw (1981) provides a summary of the independent work from 24 studies relating to experimental measurement of one or another of these pollutants. However, no studies have been reported wherein the determination of all of these species have been considered from a single sample. A strategy is outlined for the analysis of the three-phase sample obtained from a water-quenched probe in a laboratory-scale coal combustor. The relative error of each analytical method is established based on extensive laboratory testing (Burkinshaw, 1981). Experimental Facilities Reactor. The coal combustor is shown schematically in Figure 1. Coal is extrained in the central primary air
* Phillips Petroleum Co.. Bartlesville, OK. 0196-4313/83/1022-0292$01.50/0
stream and transported into the reactor where it mixes and reacts with the gas from the coaxial secondary air stream. The secondary air flow can be directed into the reactor parallel to the primary jet or can be given a swirling flow in order to control mixing rates and flame ignition. Local samples of combustion products (both gaseous and particulate) are removed from the reactor with a waterquenched sample probe. The reactor is of modular construction so that the probe section can be located at various axial positions. Then with a radial traversing mechanism on the probe, samples were obtained at various radial locations. Sample Probe. A schematic of the probe design is shown in Figure 2. Distilled water was passed through a closed manifold to the probe tip where it was sprayed into the sample duct, rapidly quenching the chemical reactions, cooling the char to a non-sticky form, and preventing char deposition on the probe walls. It is difficult to obtain a representative gas and char sample from a reactor with a sample probe, especially in a turbulent flow where large density gradients exist and where gases and particles may flow in different directions. However, an effort was made to sample isokinetically (i.e., with the gas velocity in the probe inlet the same as the gas velocity in flow-field just ahead of the probe). This was attempted by balancing the static pressure measured inside the probe tip (see Figure 2a) to the local pressure in the reactor. The sampling rate through the probe was varied until these two pressures were nearly balanced. Balancing was routinely accomplished in tests without swirl or when the probe was located near the bottom of the reactor. Balancing of sample probe pressures in regions of strongly swirling flow was more difficult. A V-shaped probe (Figure 2b) was used. This probe was rotated into a highly swirling flow without change in radial or axial position. Alignment of this V-shaped probe to the 'C 1983 American Chemical Society
