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Theoretical evaluation of viscosity of binary molten electrolytes
Ekclrvchim;ca Ae*lz. Vol. 29. No. 3. pp. 403405, Ptinkd in Great Brim,“.
THEORETICAL
1984.
w134.586184
krgamon
EVALUATION OF VISCOSITY MOLTEN ELECTROLYTES J. D.
Department of Chemistry,
53.004-0.00 Ihss
Ltd.
OF BINARY
PANDEY and USHA GUPTA
of Allahabad,
University
Allahabad
211002,
India
(Received16 May 1983) Abstract-Significant structure them-y (SST) of Eyring has been employed to catcutatk theoraticatly the viscosity of binary molten electrolytes on the basis of rigid sphere model. For KCl-NaCI and KCl-KBr systems the agreement between theoretical and experimentalvalues is found to be satisfactory.
I. INTRODUCIION Molten salts, especially alkali halides and their binary and multi-component mixtures are recently of interest in engineering applications at high temperatures, such as nuclear reactions and thermal energy storage equipment etc. In this context numerous efforts have been made to study their properties in both engineering and physical aspects. Successful attempts[l-31 have been made to apply significant structure theory (SST) to molten alkali halides and their binary mixtures in order to deduce various thermodynamic properties. Recently[4] this theory has been extended to evaluate the sound velocity of molten alkali halides. Some attempts to predict viscosity for molten alkali halides have, in fact, been made; either theoretically or empiricallyE2, S-lo]. However these previous schemes cannot come up to a practical and reliable predictive method especially for the mixture viscosity. Recently[ll], the principle of corresponding states has been applied to the viscosity of alkali halide single salts and their binary mixtures. Ree et nl.[lZ] derived an equation for the viscosity of rigid sphere utilising significant structure theory. It appears from the literature that no attempt has, so far, been made to apply this equation of rigid sphere to molten electrolytes. The aim of the present paper is to evaluate theoretically the viscosity in some molten alkali halides and their binary mixtures using the viscosity equation for a rigid sphere[12] from significant structure theory. The theoretical results are compared with the experimental findings[ 13, 143.
Here, the subscripts, s and 8, indicate that the quantity belong to solid-like and the gas-like molecules, respectively. In the last equality of Equation (l), it is assumed that x, equals the fraction of the solid-like molecules, V,/ V, and xr, = ( V- V,)/ V. The term qe in Equation (1) is taken as equal to the rigid sphere viscosity at infinite dilution and is equal to[17]:
where, d is the molecular diameter. The quantity, Q in (1) is calculated in accordance with Eyring’s procedure[17] as[ 15, 161:
where 1i, & and Aj are lattice parameters, L is the equilibrium distance, 0, is the angle between the direction ofjumping and k’ is the jumping frequency of the molecule to the ith neighbouring hole for every position i. Since the sites are randomly distributed over a solid angle, we may take the average of cos Bi and cos* Bi. Thus, ~cose, = 0
II. THEORETICAL The significant structure theory of liquids has implicit within it, a general theory of transport properties. Since, there are solid-like and gas-like degrees of freedom in the liquid, both must be taken into account in calculating the viscosity. If a fraction x, of the shear plane is covered by solid-like molecules and the remaining fraction, xs, by the gas-like molecules, then the viscosity, q, which is the ratio of shear stress, f, to rate of strain, 3, is[lS, 163:
Ccos’
et = z y,“(1/3)
where, Z is the number of nearest neighbours and Z( V- V, )/ V is the number of holes in the immediate vicinity of a molecule. If we admit the flow occurs through ion-pairs, then, Such that.
Al m=7 403
1
J. D. PANDEY AND USHA GUITA
404
where, a is the distance between two nearest neighbours. From the above equation, it is possible to correlate molecular dimensions of Equation (3) to V, as fouows,
Substituting
9 = (xmkT)“2NLf 2(V-
V
-=’ ;2
(2) and (10) into (1) leads to, a’KW(4 exp - (V- v,)2kT)
K)K
(11)
+
N’
In (4), li’ is written in accordance rate theory[17] as,
with Eyring’s reaction
After introducing
the rigid sphere 9W
In the above viscosity,
= 0
(12) (13)
equation,
& = v, where
2(V-
L,
Q’K WJ (4 exp - (V- V,)2kT
V,)K
L, is free-length
(10)
and equal t;r /aJ.
Table 1.
NaCl-KC1 100
65.15 41.00 20.75 0
loo 80 60 50 20 0
Temperature
AND
DISCUSSION
For the theoretical evaluation of viscosity from Equation (15), two binary systems of molten salt, NaCMCCl and KCl-KBr systems were taken. The computation of values of V, at different compositions is
%Xpt. (mp)
q theor. (mp)
1200 1220 1200 1220 1200 1220 1200 1220 1200 1220 1200 1220
0.951 0.852 0.888 0.886 0.840 0.809 0.809 0.782 0.786 0.764 0.822 0.814
0.907 0.848 0.857 0.837 0.853 0.833 0.854 0.833 0.886 0.861 0.929 0.900
1073 1173 1073 1173 1073 1173 1073 1173 1073 1173 1073 1173
1.093 0.849 1.078 0.845 1.064 0.841 1.057 0.840 1.034 0.834 1.021 0.831
1.152 0.740 1.118 0.948 1.022 0.938 1.012 0.810 1.015 0.851 0.975 0.816
(KI
for
v- v,
A per cent
System
84.77
KCl-KBr
RESULTS
(14)
Theoretical and experimental viscosities of molten electrolytes and their binary mixtures
Mol per cent of NaCl
Mol per of KC1
III.
1.
one gets the expression
q = (nmkT)“” N
where, v, and a, are the free volume and.the free area, respectively and J(T) is partition function for the internal degree of freedom. Hence, substituting (4F(9) into (3), ql is written as,
condition,
L, = 2(a -4 K= where K is the transmission coefficient, and F and F * are the partition functions of particles in normal and activated states respectively, and are given by[ 18-201,
V (zmkT)“‘N
1
-0.7 - 1.1 - 3.4 5.4 1.1 2.9 5.3 6.1 11.2 11.2 11.4 9.5
system
cent 5.1 -
12.8 3.6 10.8 - 3.8 10.2 - 4.2 3.5 -1.8 - 1.9 - 4.4 - 1.7
s
Viscosity of binary molten electrolytes based on the expression
of this variable as a function of the interatomic distance[3], using literature values [21] of V, for respective pure molten salts. The other necessary paratieters needed for this computation were taken from the work of Eyring er al. (~$31) and others[22,23]. Theoretical computed values of viscosity for the above mentioned binary molten electrolytes were reported, along with the experimental values of viscosity in Table 1. A perusal of Table 1, reveals that theoretical values of viscosity are very close to the experimental values in the case of NaCl and KBr. The per cent deviations are -0.7 at 1200 K and - 1.1 at 1220 K in NaCl system. The maximum deviation in KC1 system is not more than 13 “/o, From the results of Table 1, it appears that the theoretical values of viscosity from SST in both the binary mixtures under the present investigation, show satisfactory agreement with the experimental values. In KCl-KBr system, the deviation between theoretical and experimental values is around 24% at 1073 K, whereas at 1173 K, it shows a deviation of not more that the deviation from than 117”. It is apparent Equation (15) in this system increases by increasing the temperature. In the NaCl-KCl system the range of deviation between theoretical and experimental values is 1.1-11.2°/0 at 12OOK and 2.9-11.2% at higher temperature. In general, it may be concluded that, for both the systems under present investigation, viscosity decreases with rise in temperature, which is in agreement with the variation shown by experimental values. Thus Equation (IS), derived from SST, can be suitably applied to molten electrolytes and their binary mixtures for the evaluation of viscosity.
REFERENCES 1. R. Vilcu and C. Misdolea,
J. them. Phys. 63, 906 (1967).
405
2. H. Eyring, V. G. Gerrard, T. Ree and W. Lu, J. them. Phys. 49, 797 (1968). 3. R. Vilcu and C. Mtsdolea, J. them. Phys. 45, 3414 (1966). 4. J. D. Pandey, Electrochim. Arta 37, 1097 (1982). 5. G. Ciccotti. G. Jacucci and I. R. McDonald. Phvs. _ Rev. A-13, 426 (1976).
6. I. G. Muraulescu and C. Misdolea, Reuw roum. Chim. 22, 1433 (19fi)*
D. Cartianu, Rrwe wum. Phys. 23, 393 (1978). 8. M. S. John. G. Clemend and E. R. Van Alstsdaien. J. .,ohvs. Chem. 72, 4155 (1963).
9. J. J. van Loef, Z. Nazurf: A-31, 967 (1976). 10. R. E. Yong and J. P. O’Connel, Ind. Eng. Chrm. Fundam, 10, 418 (1971). 11. A. Yashivki and A. Naaashima. J. them. Phvs. 75. 3977 (1981). 12. T. S. Ree, T. Ree and H. Eyring, J. phys. Chem. 68,326Z (1964). 13. I. G. Murgulescu and S. Zuca, Revue roum. China. 10,129
(1965). 14. G. J. Janz and R. P. T. Tomkins, J. phys. them. Ref: Data 4, 871 (1971). 15. T. S. Ree. T. Ree and H. Evrinz. Proc. nutn Acad. Sci. U.S.A. 48; 501 (1962). -’ 16. H. Ewing and T. Ree. ibid. 47. 526 (1961). 17. S. CGapr&n and T. G. Cow&-&The Mathe&wticrrlTheory of Non-uniform Gases Chap. 10. Cambridge University Press. Cambridae. (1939). 18. D. Henderson, 7. c&m. Whys. 39, 1857 (1963). 19. T. S. Ree, T. Ree and H. Evring. ibid. 41, 524 (1964). 20. R. H. Wentroff, R. J. Buehle.;, J. -d. Hirschfelder gnd c. F. Curt&, ibid. 18, 1484 (1950). 21. V. H. Schink and F. Sauerwald, Z. anorg. A&. Chem. 287, 313 (1956). 22. S. Sternberg and V. Vasilescu, Revue roum. Chim. f&l187 (1967). 23. E. R. Van Arstsdalen and I. S. Yaffe, J. phys. Chem. 59,118 (1955).
THEORETICAL
1984.
w134.586184
krgamon
EVALUATION OF VISCOSITY MOLTEN ELECTROLYTES J. D.
Department of Chemistry,
53.004-0.00 Ihss
Ltd.
OF BINARY
PANDEY and USHA GUPTA
of Allahabad,
University
Allahabad
211002,
India
(Received16 May 1983) Abstract-Significant structure them-y (SST) of Eyring has been employed to catcutatk theoraticatly the viscosity of binary molten electrolytes on the basis of rigid sphere model. For KCl-NaCI and KCl-KBr systems the agreement between theoretical and experimentalvalues is found to be satisfactory.
I. INTRODUCIION Molten salts, especially alkali halides and their binary and multi-component mixtures are recently of interest in engineering applications at high temperatures, such as nuclear reactions and thermal energy storage equipment etc. In this context numerous efforts have been made to study their properties in both engineering and physical aspects. Successful attempts[l-31 have been made to apply significant structure theory (SST) to molten alkali halides and their binary mixtures in order to deduce various thermodynamic properties. Recently[4] this theory has been extended to evaluate the sound velocity of molten alkali halides. Some attempts to predict viscosity for molten alkali halides have, in fact, been made; either theoretically or empiricallyE2, S-lo]. However these previous schemes cannot come up to a practical and reliable predictive method especially for the mixture viscosity. Recently[ll], the principle of corresponding states has been applied to the viscosity of alkali halide single salts and their binary mixtures. Ree et nl.[lZ] derived an equation for the viscosity of rigid sphere utilising significant structure theory. It appears from the literature that no attempt has, so far, been made to apply this equation of rigid sphere to molten electrolytes. The aim of the present paper is to evaluate theoretically the viscosity in some molten alkali halides and their binary mixtures using the viscosity equation for a rigid sphere[12] from significant structure theory. The theoretical results are compared with the experimental findings[ 13, 143.
Here, the subscripts, s and 8, indicate that the quantity belong to solid-like and the gas-like molecules, respectively. In the last equality of Equation (l), it is assumed that x, equals the fraction of the solid-like molecules, V,/ V, and xr, = ( V- V,)/ V. The term qe in Equation (1) is taken as equal to the rigid sphere viscosity at infinite dilution and is equal to[17]:
where, d is the molecular diameter. The quantity, Q in (1) is calculated in accordance with Eyring’s procedure[17] as[ 15, 161:
where 1i, & and Aj are lattice parameters, L is the equilibrium distance, 0, is the angle between the direction ofjumping and k’ is the jumping frequency of the molecule to the ith neighbouring hole for every position i. Since the sites are randomly distributed over a solid angle, we may take the average of cos Bi and cos* Bi. Thus, ~cose, = 0
II. THEORETICAL The significant structure theory of liquids has implicit within it, a general theory of transport properties. Since, there are solid-like and gas-like degrees of freedom in the liquid, both must be taken into account in calculating the viscosity. If a fraction x, of the shear plane is covered by solid-like molecules and the remaining fraction, xs, by the gas-like molecules, then the viscosity, q, which is the ratio of shear stress, f, to rate of strain, 3, is[lS, 163:
Ccos’
et = z y,“(1/3)
where, Z is the number of nearest neighbours and Z( V- V, )/ V is the number of holes in the immediate vicinity of a molecule. If we admit the flow occurs through ion-pairs, then, Such that.
Al m=7 403
1
J. D. PANDEY AND USHA GUITA
404
where, a is the distance between two nearest neighbours. From the above equation, it is possible to correlate molecular dimensions of Equation (3) to V, as fouows,
Substituting
9 = (xmkT)“2NLf 2(V-
V
-=’ ;2
(2) and (10) into (1) leads to, a’KW(4 exp - (V- v,)2kT)
K)K
(11)
+
N’
In (4), li’ is written in accordance rate theory[17] as,
with Eyring’s reaction
After introducing
the rigid sphere 9W
In the above viscosity,
= 0
(12) (13)
equation,
& = v, where
2(V-
L,
Q’K WJ (4 exp - (V- V,)2kT
V,)K
L, is free-length
(10)
and equal t;r /aJ.
Table 1.
NaCl-KC1 100
65.15 41.00 20.75 0
loo 80 60 50 20 0
Temperature
AND
DISCUSSION
For the theoretical evaluation of viscosity from Equation (15), two binary systems of molten salt, NaCMCCl and KCl-KBr systems were taken. The computation of values of V, at different compositions is
%Xpt. (mp)
q theor. (mp)
1200 1220 1200 1220 1200 1220 1200 1220 1200 1220 1200 1220
0.951 0.852 0.888 0.886 0.840 0.809 0.809 0.782 0.786 0.764 0.822 0.814
0.907 0.848 0.857 0.837 0.853 0.833 0.854 0.833 0.886 0.861 0.929 0.900
1073 1173 1073 1173 1073 1173 1073 1173 1073 1173 1073 1173
1.093 0.849 1.078 0.845 1.064 0.841 1.057 0.840 1.034 0.834 1.021 0.831
1.152 0.740 1.118 0.948 1.022 0.938 1.012 0.810 1.015 0.851 0.975 0.816
(KI
for
v- v,
A per cent
System
84.77
KCl-KBr
RESULTS
(14)
Theoretical and experimental viscosities of molten electrolytes and their binary mixtures
Mol per cent of NaCl
Mol per of KC1
III.
1.
one gets the expression
q = (nmkT)“” N
where, v, and a, are the free volume and.the free area, respectively and J(T) is partition function for the internal degree of freedom. Hence, substituting (4F(9) into (3), ql is written as,
condition,
L, = 2(a -4 K= where K is the transmission coefficient, and F and F * are the partition functions of particles in normal and activated states respectively, and are given by[ 18-201,
V (zmkT)“‘N
1
-0.7 - 1.1 - 3.4 5.4 1.1 2.9 5.3 6.1 11.2 11.2 11.4 9.5
system
cent 5.1 -
12.8 3.6 10.8 - 3.8 10.2 - 4.2 3.5 -1.8 - 1.9 - 4.4 - 1.7
s
Viscosity of binary molten electrolytes based on the expression
of this variable as a function of the interatomic distance[3], using literature values [21] of V, for respective pure molten salts. The other necessary paratieters needed for this computation were taken from the work of Eyring er al. (~$31) and others[22,23]. Theoretical computed values of viscosity for the above mentioned binary molten electrolytes were reported, along with the experimental values of viscosity in Table 1. A perusal of Table 1, reveals that theoretical values of viscosity are very close to the experimental values in the case of NaCl and KBr. The per cent deviations are -0.7 at 1200 K and - 1.1 at 1220 K in NaCl system. The maximum deviation in KC1 system is not more than 13 “/o, From the results of Table 1, it appears that the theoretical values of viscosity from SST in both the binary mixtures under the present investigation, show satisfactory agreement with the experimental values. In KCl-KBr system, the deviation between theoretical and experimental values is around 24% at 1073 K, whereas at 1173 K, it shows a deviation of not more that the deviation from than 117”. It is apparent Equation (15) in this system increases by increasing the temperature. In the NaCl-KCl system the range of deviation between theoretical and experimental values is 1.1-11.2°/0 at 12OOK and 2.9-11.2% at higher temperature. In general, it may be concluded that, for both the systems under present investigation, viscosity decreases with rise in temperature, which is in agreement with the variation shown by experimental values. Thus Equation (IS), derived from SST, can be suitably applied to molten electrolytes and their binary mixtures for the evaluation of viscosity.
REFERENCES 1. R. Vilcu and C. Misdolea,
J. them. Phys. 63, 906 (1967).
405
2. H. Eyring, V. G. Gerrard, T. Ree and W. Lu, J. them. Phys. 49, 797 (1968). 3. R. Vilcu and C. Mtsdolea, J. them. Phys. 45, 3414 (1966). 4. J. D. Pandey, Electrochim. Arta 37, 1097 (1982). 5. G. Ciccotti. G. Jacucci and I. R. McDonald. Phvs. _ Rev. A-13, 426 (1976).
6. I. G. Muraulescu and C. Misdolea, Reuw roum. Chim. 22, 1433 (19fi)*
D. Cartianu, Rrwe wum. Phys. 23, 393 (1978). 8. M. S. John. G. Clemend and E. R. Van Alstsdaien. J. .,ohvs. Chem. 72, 4155 (1963).
9. J. J. van Loef, Z. Nazurf: A-31, 967 (1976). 10. R. E. Yong and J. P. O’Connel, Ind. Eng. Chrm. Fundam, 10, 418 (1971). 11. A. Yashivki and A. Naaashima. J. them. Phvs. 75. 3977 (1981). 12. T. S. Ree, T. Ree and H. Eyring, J. phys. Chem. 68,326Z (1964). 13. I. G. Murgulescu and S. Zuca, Revue roum. China. 10,129
(1965). 14. G. J. Janz and R. P. T. Tomkins, J. phys. them. Ref: Data 4, 871 (1971). 15. T. S. Ree. T. Ree and H. Evrinz. Proc. nutn Acad. Sci. U.S.A. 48; 501 (1962). -’ 16. H. Ewing and T. Ree. ibid. 47. 526 (1961). 17. S. CGapr&n and T. G. Cow&-&The Mathe&wticrrlTheory of Non-uniform Gases Chap. 10. Cambridge University Press. Cambridae. (1939). 18. D. Henderson, 7. c&m. Whys. 39, 1857 (1963). 19. T. S. Ree, T. Ree and H. Evring. ibid. 41, 524 (1964). 20. R. H. Wentroff, R. J. Buehle.;, J. -d. Hirschfelder gnd c. F. Curt&, ibid. 18, 1484 (1950). 21. V. H. Schink and F. Sauerwald, Z. anorg. A&. Chem. 287, 313 (1956). 22. S. Sternberg and V. Vasilescu, Revue roum. Chim. f&l187 (1967). 23. E. R. Van Arstsdalen and I. S. Yaffe, J. phys. Chem. 59,118 (1955).