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Central molecular potentials, combination rules and properties of gases and gas mixtures
CHEMICAL
PHYSICS
LETTERS
1 (1967)
224-226.
CENTRAL MOLECULAR AND PROPERTIES
NORTH -HOLLAND
PUBLISHING COMPANY,
AMSTERDAM
POTENTIALS, COMBINATION RULES OF GASES AND) GAS MIXTURES
S. C. SAXENA * and B. P. MATHUR** Department
of Pkxsics.
Unircrsit_v of Rajasthan.
Received
Jaipur.
Rajastknn.
India
17 July 1967
A discussion is presented of the potential parameters determined from diffusion data and it is concluded that the unlike molecular interactions are well zpproximated on the basis of combination rules especially when polyatomic molecules are involved.
It has been quite common to correlate and predict the different equilibrium and nonequilibrium billk properties of gases and gas mixtures on the theory of monatomic spherically symmetric gas molecules undergoing elastic collisions and in terms of simple central intermolecular potentials. This is because the mathematical theory of nonspherical molecules, ;aking into account inelastic collisions, has just begun to be developed and specific numerical calculation of properties is still not a straightforward job. The two spherically symmetric potentials which are very often used are the Lennard-Jones (12-6) and modified Buckingham exponential-six. The potential parameters being determined from the temperature dependence of viscosity for the pure gases. The unlike potential parameters are then determined from the corresponding pure parameters characterizing the like interactions through the use of combination rules. In principle, the unlike parameters in analogy with like parameters, can also be determined from the temperature dependence of any property of the gaseous mixture. In table 1, we list the two Lennard-Jones parameters E/& and 0, as determined from the temperature dependence of the diffusion co&%cient [1,2] and from the conventional combination rules viz., geometric mean rule for ~12 and arithmetic mean rule for 012. Here E is the depth of +he potential energy well, u the molecular diameter and k the Holtzmann constant. The pure parameters are of Hirschfelder, Curtiss .
* Now at Purdue ** Now at Haven,
Thermophysical Properties Research Center, University, Lafayette, Indiana, USA Department of Physics, Yale University, New Connecticut, USA.
Table 1 Unlike potential parameters for the Lennard-Jones (12-6) potential.
Gas pair He-Ar He-H2 Ar-H2 CH4-02 H2-02 co-03 N2-CG2 02-cop
Combination rules Refs. [3]-[4] E/k (OK)
34.99 31.29 66.58 13i.B 66.82 118.7 131.9 156.0
dl
2.98 3.14 3.17 3.51 3.17 3.49 3.84 3.70
Experimental Refs. [l]-[2] E/k (OK)
Cdl
125
2.587
69 155 182 152 91 154 213
2.87 2.76 3.367 2.825 3.480 3.52 3.365
and Bird [3,4]. The experimental [1,2] as well as calculated from the combination rules of Mason and Rice [5] potential parameters for the modified exp-six potential are recorded in table 2. The potential parameters for pure gases used in each case are also of Mason and Rice [6]. For this potential combination rules have also been suggested by Mason [7], and Srivastava and Srivastava [ 131.The latter two sets of combination rules do not lead to values which are appreciably different from those reported in table 2. It will be seen from these two tables that in general the experimental values are in poor agreement with the computed potential parameters. Further, for the exp-six potential it has not been possible to make a unique choice of parameters on the basis of diffusion data alone. In fact, the authors choose arbitrarily the values of o! as 12, 14 and 17 in all cases except He-N2 and Ar-H2 and determine the corresponding values of e/i? and Ym. This indicates as if the choice
CENTRAL Table 2 Unlike potential parameters for the exp-six Gas pair
Combination rules refs. [5]-[i]
potential.
POTENTIALS
Percentage
225
deviation.
orctical
Experimental refs. [l]-[2]
~~
-
MOLECULAR
Table 3 Dthear -Dexpt)
Dexptl and experimental
Gas (Y
E/k(oK)
r,(i)
He-Ar
13.21
33.4
3.498
He-N2 Ar-A2
14.86 13.97
27.43 69.7
3.63 3.57
CH4-02
15.43
He-02
15.55
co-02
17
of a is somewhat
153.1
67.83
125.4
3.91
3.55
3.83
immaterial
QI
e/&OK)
12 14 17 17 12 14 12 14 17 12 1-t 17 12 l-1 17
88 114 150 85 112 142 145 170 220 117 143 183 -10 76 110
rm(&
pair
3.17 2.971 2.777 3.069 3.37 3.17 4.061 3.855 3.622 3.121 3.234 3.032 4.525 4.04 3.733
Ar-He
or alternatively
diffusion data by themselves are not sufficient to determine the three parameters of the exp-six potential. An analogous conclusion was drawn by Weissmann, Saxena 2nd Mason [9] while discussing the diffusion 2nd thermal diffusion data. They [9] found that diffusion data were reproduced by theory though thermal diffusion data were not. Values of the diffusion coefficients (0) were calculated from the first approximation e-xpression on the basis of the potential parameters of tables 1 and 2 obtained from the combination rules. In all the cases and for both the potentials calculated values are always smaller +%xnthe observed values. To give an idea of the quantitative agreement we computed the percentage deviations at temperatures 400, 700 and 1OOOoK 2nd these are reported in table 3. Diffusion coefficients are also weakly dependent upon the composition and the numerical agreement will improve if this is also considered. In the case of Ar-He system on the average values for the expsix potential will improve by sn so that the calcuJated and experimental values will differ by only about 4%. As the trend of theoretical values being always smaller than the experimental values is quite universal for all the eight systems, many speculations are possible. Whether or not this is due to some experimental error in the point source measurements is not known [lo]. There are no other direct measurements of diffusion coefficients at such temperatures. It is an interesting point to be resolved by the experts of diffusion measurements at high temperatures in a conclusive fashion.
He-N2
Ar-HZ 02-H2
02-‘334 02-co 02-co2 N2-CO2
Potential
400
L-J (12-6) Exp -6 L-J (12-6) Exp -6 L-J (12-6) Exo -6 L-J (i2-6) RXD -6 L-J (i2-6) Exp -6 L-J (12-6) Exp -6 L-J (12-6) L-J (i2-6)
- 8 - 8 - 10 - 9 - 10 - 7 - 7 - 8 - 13 0 -6 - 8 - 8 - 1.I
x LOO. of the the-
values
of D.
Temp.(OK) 706 - 9 - 6 - 6 - 9 - z-4 - LO 1 - 5 - 16 - 3 -5 - 8 - 15 - 16
1000 =-ii - i - 6
I
-
-
1; 12
8 11 I.8 7 5 7 15 1G
Unfortunately, the parameters determined from the experimental diffusion data also do not reproduce the other properties satisfactorily. We quote here the results for the system Ar-He, where the Chapman-Enskog theory applies and one expects better results. The second virial coefficient as obtained from the diffusion ~;arameters is always smaller than the experimental values. The best agreement is obtzineci for the exp-six potential corresponding to a! = 12, but here also the disagreement is 2s Zarge 2s 20 cc/mole on the 2ver2ge in the temperature range of experiment21 measurements. The combination rules of Mason and Rice lead to vakxes which are smaller than the experimentat values by 3.2 cc/mole on the average. The thermal diffusion data is very poorly reproduced by the diffusion parameters 2nd the exp-six potential. Even the qualitative trend of temperature variation is not reproduced and the disagreements range between 10 and 25%. On the other hand the combination rule parameters reproduce the data satisfactorily. Thus we may conclude as follows: (a) the temper2ture dependence of the diffusion data alone is not enough to determine the molecuIar interactions, (b) the large difference between the values of the unlike parameters determined from the diffusion data 2nd those obtained from the COXIbination rules is due to the fact given in (a), [cc) the conventional combination rules for the L-J (12-6) 2nd exp-six potentials seem to be competent to correlate the diffusion data 2s well 2s other properties, and consequently (d) the combination rules should still be trusted to determine
22~
S, C, S A X E N A nnd B, P, ~ A T H U K .
:~p.~xtmate!Yi~:the.lmolecii~
.
itnteractloris even
~.h~!~lY~tOmfc"rnolecules:..are t~,olved,T h e a u t h o r s a r e t h a n k f u l to t h e D e p a r t m e n t
:
:
..'
[2J A- A; W~s~enberg and c o - w o r k e r s , J . C h e m . Pt~ys. 29 (Is58) : t l z o , 1147:1bI~. sl (19~9) s19: Ibid. :~2
(xseo) 4:36; tb~d. 36 (1962) 3499, of
atomic energy, Bombay, which supported this w o r k . S , C . S . i s p l e a s e d to a c k n o w l e d g e t h e d i s c u s s i o n s on the s u b l e c t w i t h D r . A. A. W e s t e n b e r g . REFERENCES [1] R. E. Wal k er. L. Monchick. A. A. W e s t e n b e r g and S. Favin, P h y s i c a l C h e m i s t r y in A e r o d y n a m i c s snd Space Flight ( P e r g a m o n P r e s s , Oxford, 1961) p. 221.
[3] J . O. HirsvhfeI,~er, C. F, C u r ~ t s s and R . B. B~rd, The M o l e c u l a r T h e o r y of Gases and Liquids (John Wiley and Sons. Inc., New York, 1964). [4] S. c . Saxena and B. P. Mathur, Rev. Mod. Phys. 38 (1966) 380. [5] E. _%.Mason and W. E. Ri ce, J. Chem. Phys. 22 (1954) 522. [6] E..%. Mason and W. E. Ri ce, J . Chem. Phys. 22 (1954) 843. [7] E..%. Mason. J . Chem. Phys. 23 (1955) 49. [8] B . N . S r i v a s t a v a and K. P. S r i v a s t a v a , J . Chem. Phys. 2~ (1956) 1275. [9] S.%Veissman, S. C. Saxena and E . A . Mason, Phys. Fluids 3 (1960) 510. [10] Pr-[vate c o m m u n i c a t i o n f r o m Dr. A. A . W e s t e n h e r g .
PHYSICS
LETTERS
1 (1967)
224-226.
CENTRAL MOLECULAR AND PROPERTIES
NORTH -HOLLAND
PUBLISHING COMPANY,
AMSTERDAM
POTENTIALS, COMBINATION RULES OF GASES AND) GAS MIXTURES
S. C. SAXENA * and B. P. MATHUR** Department
of Pkxsics.
Unircrsit_v of Rajasthan.
Received
Jaipur.
Rajastknn.
India
17 July 1967
A discussion is presented of the potential parameters determined from diffusion data and it is concluded that the unlike molecular interactions are well zpproximated on the basis of combination rules especially when polyatomic molecules are involved.
It has been quite common to correlate and predict the different equilibrium and nonequilibrium billk properties of gases and gas mixtures on the theory of monatomic spherically symmetric gas molecules undergoing elastic collisions and in terms of simple central intermolecular potentials. This is because the mathematical theory of nonspherical molecules, ;aking into account inelastic collisions, has just begun to be developed and specific numerical calculation of properties is still not a straightforward job. The two spherically symmetric potentials which are very often used are the Lennard-Jones (12-6) and modified Buckingham exponential-six. The potential parameters being determined from the temperature dependence of viscosity for the pure gases. The unlike potential parameters are then determined from the corresponding pure parameters characterizing the like interactions through the use of combination rules. In principle, the unlike parameters in analogy with like parameters, can also be determined from the temperature dependence of any property of the gaseous mixture. In table 1, we list the two Lennard-Jones parameters E/& and 0, as determined from the temperature dependence of the diffusion co&%cient [1,2] and from the conventional combination rules viz., geometric mean rule for ~12 and arithmetic mean rule for 012. Here E is the depth of +he potential energy well, u the molecular diameter and k the Holtzmann constant. The pure parameters are of Hirschfelder, Curtiss .
* Now at Purdue ** Now at Haven,
Thermophysical Properties Research Center, University, Lafayette, Indiana, USA Department of Physics, Yale University, New Connecticut, USA.
Table 1 Unlike potential parameters for the Lennard-Jones (12-6) potential.
Gas pair He-Ar He-H2 Ar-H2 CH4-02 H2-02 co-03 N2-CG2 02-cop
Combination rules Refs. [3]-[4] E/k (OK)
34.99 31.29 66.58 13i.B 66.82 118.7 131.9 156.0
dl
2.98 3.14 3.17 3.51 3.17 3.49 3.84 3.70
Experimental Refs. [l]-[2] E/k (OK)
Cdl
125
2.587
69 155 182 152 91 154 213
2.87 2.76 3.367 2.825 3.480 3.52 3.365
and Bird [3,4]. The experimental [1,2] as well as calculated from the combination rules of Mason and Rice [5] potential parameters for the modified exp-six potential are recorded in table 2. The potential parameters for pure gases used in each case are also of Mason and Rice [6]. For this potential combination rules have also been suggested by Mason [7], and Srivastava and Srivastava [ 131.The latter two sets of combination rules do not lead to values which are appreciably different from those reported in table 2. It will be seen from these two tables that in general the experimental values are in poor agreement with the computed potential parameters. Further, for the exp-six potential it has not been possible to make a unique choice of parameters on the basis of diffusion data alone. In fact, the authors choose arbitrarily the values of o! as 12, 14 and 17 in all cases except He-N2 and Ar-H2 and determine the corresponding values of e/i? and Ym. This indicates as if the choice
CENTRAL Table 2 Unlike potential parameters for the exp-six Gas pair
Combination rules refs. [5]-[i]
potential.
POTENTIALS
Percentage
225
deviation.
orctical
Experimental refs. [l]-[2]
~~
-
MOLECULAR
Table 3 Dthear -Dexpt)
Dexptl and experimental
Gas (Y
E/k(oK)
r,(i)
He-Ar
13.21
33.4
3.498
He-N2 Ar-A2
14.86 13.97
27.43 69.7
3.63 3.57
CH4-02
15.43
He-02
15.55
co-02
17
of a is somewhat
153.1
67.83
125.4
3.91
3.55
3.83
immaterial
QI
e/&OK)
12 14 17 17 12 14 12 14 17 12 1-t 17 12 l-1 17
88 114 150 85 112 142 145 170 220 117 143 183 -10 76 110
rm(&
pair
3.17 2.971 2.777 3.069 3.37 3.17 4.061 3.855 3.622 3.121 3.234 3.032 4.525 4.04 3.733
Ar-He
or alternatively
diffusion data by themselves are not sufficient to determine the three parameters of the exp-six potential. An analogous conclusion was drawn by Weissmann, Saxena 2nd Mason [9] while discussing the diffusion 2nd thermal diffusion data. They [9] found that diffusion data were reproduced by theory though thermal diffusion data were not. Values of the diffusion coefficients (0) were calculated from the first approximation e-xpression on the basis of the potential parameters of tables 1 and 2 obtained from the combination rules. In all the cases and for both the potentials calculated values are always smaller +%xnthe observed values. To give an idea of the quantitative agreement we computed the percentage deviations at temperatures 400, 700 and 1OOOoK 2nd these are reported in table 3. Diffusion coefficients are also weakly dependent upon the composition and the numerical agreement will improve if this is also considered. In the case of Ar-He system on the average values for the expsix potential will improve by sn so that the calcuJated and experimental values will differ by only about 4%. As the trend of theoretical values being always smaller than the experimental values is quite universal for all the eight systems, many speculations are possible. Whether or not this is due to some experimental error in the point source measurements is not known [lo]. There are no other direct measurements of diffusion coefficients at such temperatures. It is an interesting point to be resolved by the experts of diffusion measurements at high temperatures in a conclusive fashion.
He-N2
Ar-HZ 02-H2
02-‘334 02-co 02-co2 N2-CO2
Potential
400
L-J (12-6) Exp -6 L-J (12-6) Exp -6 L-J (12-6) Exo -6 L-J (i2-6) RXD -6 L-J (i2-6) Exp -6 L-J (12-6) Exp -6 L-J (12-6) L-J (i2-6)
- 8 - 8 - 10 - 9 - 10 - 7 - 7 - 8 - 13 0 -6 - 8 - 8 - 1.I
x LOO. of the the-
values
of D.
Temp.(OK) 706 - 9 - 6 - 6 - 9 - z-4 - LO 1 - 5 - 16 - 3 -5 - 8 - 15 - 16
1000 =-ii - i - 6
I
-
-
1; 12
8 11 I.8 7 5 7 15 1G
Unfortunately, the parameters determined from the experimental diffusion data also do not reproduce the other properties satisfactorily. We quote here the results for the system Ar-He, where the Chapman-Enskog theory applies and one expects better results. The second virial coefficient as obtained from the diffusion ~;arameters is always smaller than the experimental values. The best agreement is obtzineci for the exp-six potential corresponding to a! = 12, but here also the disagreement is 2s Zarge 2s 20 cc/mole on the 2ver2ge in the temperature range of experiment21 measurements. The combination rules of Mason and Rice lead to vakxes which are smaller than the experimentat values by 3.2 cc/mole on the average. The thermal diffusion data is very poorly reproduced by the diffusion parameters 2nd the exp-six potential. Even the qualitative trend of temperature variation is not reproduced and the disagreements range between 10 and 25%. On the other hand the combination rule parameters reproduce the data satisfactorily. Thus we may conclude as follows: (a) the temper2ture dependence of the diffusion data alone is not enough to determine the molecuIar interactions, (b) the large difference between the values of the unlike parameters determined from the diffusion data 2nd those obtained from the COXIbination rules is due to the fact given in (a), [cc) the conventional combination rules for the L-J (12-6) 2nd exp-six potentials seem to be competent to correlate the diffusion data 2s well 2s other properties, and consequently (d) the combination rules should still be trusted to determine
22~
S, C, S A X E N A nnd B, P, ~ A T H U K .
:~p.~xtmate!Yi~:the.lmolecii~
.
itnteractloris even
~.h~!~lY~tOmfc"rnolecules:..are t~,olved,T h e a u t h o r s a r e t h a n k f u l to t h e D e p a r t m e n t
:
:
..'
[2J A- A; W~s~enberg and c o - w o r k e r s , J . C h e m . Pt~ys. 29 (Is58) : t l z o , 1147:1bI~. sl (19~9) s19: Ibid. :~2
(xseo) 4:36; tb~d. 36 (1962) 3499, of
atomic energy, Bombay, which supported this w o r k . S , C . S . i s p l e a s e d to a c k n o w l e d g e t h e d i s c u s s i o n s on the s u b l e c t w i t h D r . A. A. W e s t e n b e r g . REFERENCES [1] R. E. Wal k er. L. Monchick. A. A. W e s t e n b e r g and S. Favin, P h y s i c a l C h e m i s t r y in A e r o d y n a m i c s snd Space Flight ( P e r g a m o n P r e s s , Oxford, 1961) p. 221.
[3] J . O. HirsvhfeI,~er, C. F, C u r ~ t s s and R . B. B~rd, The M o l e c u l a r T h e o r y of Gases and Liquids (John Wiley and Sons. Inc., New York, 1964). [4] S. c . Saxena and B. P. Mathur, Rev. Mod. Phys. 38 (1966) 380. [5] E. _%.Mason and W. E. Ri ce, J. Chem. Phys. 22 (1954) 522. [6] E..%. Mason and W. E. Ri ce, J . Chem. Phys. 22 (1954) 843. [7] E..%. Mason. J . Chem. Phys. 23 (1955) 49. [8] B . N . S r i v a s t a v a and K. P. S r i v a s t a v a , J . Chem. Phys. 2~ (1956) 1275. [9] S.%Veissman, S. C. Saxena and E . A . Mason, Phys. Fluids 3 (1960) 510. [10] Pr-[vate c o m m u n i c a t i o n f r o m Dr. A. A . W e s t e n h e r g .