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Dimerization and the Initial Pressure Dependence of the Viscosity of Polar Gases
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Dimerization and the Initial Pressure Dependence of the Viscosity of Polar Gases Yashwant Singh, S. K. Deb, and A. K. Barua Citation: The Journal of Chemical Physics 46, 4036 (1967); doi: 10.1063/1.1840483 View online: http://dx.doi.org/10.1063/1.1840483 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/46/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Pressure dependence of viscosity J. Chem. Phys. 122, 074511 (2005); 10.1063/1.1851510 Initial Pressure Dependence of the Viscosity of Polar Gases J. Chem. Phys. 54, 4841 (1971); 10.1063/1.1674760 Averaged Potentials and the Viscosity of Dilute Polar Gases J. Chem. Phys. 50, 4718 (1969); 10.1063/1.1670961 Temperature and Pressure Dependence of the Viscosity of Gases Am. J. Phys. 33, 835 (1965); 10.1119/1.1970996 Initial Pressure Dependence of Thermal Conductivity and Viscosity J. Chem. Phys. 31, 1545 (1959); 10.1063/1.1730650 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.174.255.116 On: Thu, 27 Nov 2014 17:11:41
Transcript of Dimerization and the Initial Pressure Dependence of the Viscosity of Polar Gases
Dimerization and the Initial Pressure Dependence of the Viscosity of Polar GasesYashwant Singh, S. K. Deb, and A. K. Barua Citation: The Journal of Chemical Physics 46, 4036 (1967); doi: 10.1063/1.1840483 View online: http://dx.doi.org/10.1063/1.1840483 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/46/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Pressure dependence of viscosity J. Chem. Phys. 122, 074511 (2005); 10.1063/1.1851510 Initial Pressure Dependence of the Viscosity of Polar Gases J. Chem. Phys. 54, 4841 (1971); 10.1063/1.1674760 Averaged Potentials and the Viscosity of Dilute Polar Gases J. Chem. Phys. 50, 4718 (1969); 10.1063/1.1670961 Temperature and Pressure Dependence of the Viscosity of Gases Am. J. Phys. 33, 835 (1965); 10.1119/1.1970996 Initial Pressure Dependence of Thermal Conductivity and Viscosity J. Chem. Phys. 31, 1545 (1959); 10.1063/1.1730650
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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 46, NUMBER 10 15 MAY 1967
Dimerization and the Initial Pressure Dependence of the Viscosity of Polar Gases
YASHWANT SINGH, S. K. DEB, AND A. K. BARUA
Indian Association for the Cultivation of Science, Calcutta-32, India
(Received 9 January 1967)
The contributions of bound and metastably bound molecules to the second virial coefficient of polar gases have been calculated by assuming equal probability for all the relative orientations of the interacting dipoles. The results have been utilized to explain the initial pressure dependence of the viscosity of polar gases. The results obtained for steam and ammonia are quite satisfactory.
I. INTRODUCTION
I T is possible to explain the pressure dependence of the transport coefficients of gases from a knowledge
of the equilibrium constants for cluster formation and collisional transfer. In order to explain the initial pressures dependence it is sufficient to consider dimerization only.
Stogryn and Hirschfelder l have considered the initial pressure dependence of the thermal conductivity and viscosity of nonpolar gases on realistic potential models. They have found that in the case of thermal conductivity at low temperatures, molecular association is more important than collisional transfer while at higher temperatures collisional transfer predominates. For viscosity the initial pressure dependence is primarily due to collisional transfer. This problem has subsequently been treated by Kim and Ross2 by considering the presence of quasidimers in addition to stable and metastable dimers. However, for polar gases, due to long-range dipole-dipole forces, association is expected to playa much more significant role than for nonpolar gases (except for helium3,4 only). Highly polar substances like steam and ammonia show a negative pressure coefficient of viscosity.5-7 Barua and Das Gupta8 ,9 have shown that the apparently anomalous behavior of ammonia and steam is due to association, and the sign of the pressure coefficient depends on the relative effects of association and collisional transfer. Their results are, however, qualitative, as they had to employ the semiempirical method of Hirschfelder, McClure, and Weekslo to obtain the
[D. E. Stogryn and J. O. Hirschfelder, J. Chern. Phys. 31, 1531 (1959).
2 S. K. Kim and J. Ross, J. Chern. Phys. 42,263 (1965). 3 G. F. Flynn, R. V. Hanks, N. A. Lemaire, and J. Ross, J.
Chern. Phys. 38, 154 (1963). 4 P. K. Chakraborti and A. K. Barua, Appl. Sci. Res. 14, 294
(1965) . 5 J. Kestin and H. E. Wang, Physica 26, 575 (1960). 6 H. Iwaski, J. Kestin, and A. Nagashima, J. Chern. Phys. 40,
2988 (1964). 7 L. T. Carmichael, H. H. Reamer, and B. H. Saga, J. Chern.
Eng. Data8,400 (1963). 8 A. K. Barua and A. Das Gupta, Trans. Faraday Soc. 59,
2243 (1963). 9 A. Das Gupta and A. K. Barua, J. Chern. Phys. 42, 2849
(1965). 10 J. O. Hirschfelder, F. T. McClure, and 1. F. Weeks, J. Chern.
Phys. 10, 201 (1942).
percentage of dimers at any particular temperature and pressure. Recently, Barua, Chakraborti, and Saran,l1 and Saran, Singh, and Barual ? have evaluated Bb ( T) and Bm( T), the contributions of bound and metastably bound molecules to the second virial coefficient, respectively, for polar gases by assuming the dipoles to be in the head-to-tail position. The results of Barua et al,u have subsequently been refined by Singh, Saran, and Barua.13 However, for transport properties at not too low temperatures, it is probably justified14 ,15 to assume equal probability for all the relative orientations of the interacting dipoles. The equilibrium constant for dimerization is related to Bb (T) and Bm (T) and hence it is necessary to evaluate them by putting equal weight to all the relative orientations of the dipoles. In this paper our purpose has been twofold, viz.:
(1) Evaluation of Bb(T) and Bm(T) by assuming equal probability for the relative orientations of the interacting dipoles.
(2) Consideration of the initial pressure dependence of viscosity of polar gases.
II. EQUlLmRIUM CONSTANT FOR DIMER COMPLEX
The interaction energy between two polar molecules is usually represented by the Stockmayer potential which can be written asl6
<I>(r) =4e[(0/r)1L (o/r)L A (o/r)3J, (1)
A = (JL2/4eo3)g(t'h, (h, cf» =!JL*2g(OI, O2, cf», (2)
g(fJr, O2 , cf» =2 COSOI cos02-sinOI sin02 coscf>, (3)
where JL is the dipole moment of the interacting molecules (the dipole being assumed to be point dipoles embedded at the center of the molecule), and 81, 82 the angles of inclination of the dipoles to the line joining
11 A. K. Barua, P. K. Chakraborti, and A. Saran, Mol. Phys. 9,9 (1965).
12 A. Saran, Y. Singh, and A. K. Barua, J. Phys. Soc. Japan 22, 77 (1967).
13 Y. Singh, A. Saran, and A. K. Barua (unpublished results). 14 L. Monchick and E. A. Mason, J. Chern. Phys. 35, 1676
(1961) . 15 A. K. Pal and A. K. Barua (unpublished results). 16 W. H. Stock mayer, J. Chern. Phys. 9,398 (1941).
4036
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VISCOSITY OF POLAR GASES 4037
the molecules. For ,u---t0 Eq. (1) reduces to the well-known Lennard-Jones (12: 6) potentialfor nonpolar molecules.
For the nonspherical monomers the equilibrium constant for the dimerization reaction
can be defined as
nl, nz being the number of moles of monomers and dimers, respectively, QI, Qz being the corresponding partition functions. The factor 'Y takes into account the possibility of hindered rotation when two nonspherical monomers form a dimer. Thus the partition function of a dimer may be written as
(5)
where Qd is the partition function a dimer would have if the monomers were spherically symmetric; Qrot and Qvib are the rotational and vibrational partition functions.
The fact that polar molecules have an angle-dependent potential and the probability of rotational energy transfer is relatively high makes the evaluation of Qdimer very difficult. This is particularly so as 'Y cannot be calculated accurately. Qualitatively, it may only be stated that it is unity for spherical monomers and less than unity for nonspherical molecules. However, Monchick and Masonl4 have recently treated the
transport properties of polar gases by assuming (a) a fixed relative orientation of the interacting dipoles during a collision, and (b) that all relative orientations of the dipoles are equally probable.
With the above approximations, the calculations of either the collision integrals or the contributions of bound and metastable-bound molecules to the second virial coefficient are no more difficult than those for central forces except for the introduction of an additional parameter. For spherically symmetric intermolecular potentials, the equilibrium constant for dimerization can be written as
K(T) =nzV /nlz= - (Bb+Bm) = -Bd. (6)
It has been shown by Ghoshl7 that, like nonpolar gases, for polar gases as well the lifetime for the dissociation of metastable double molecules is usually longer than the average time between collisions. Thus the metastablebound molecules behave effectively as stable double molecules. Bb ( T) and Bm ( T) can be expressed asl8
Bb(T) =-NNQZb/V, (7)
Bm (T)=-NA6Q2m/V, (8)
Bd(T) =Bb(T)+Bm(T), (8')
QZb and Q2m being the corresponding partition functions, N Avogadro's number, V the volume, and A2= hZ/27rmkT.
Substituting the expression for QZb as obtained by Hill,18 Eq. (7) can be reduced tol
r' [¢(r)]{ [3 ¢(r)]/ } Bb(T) = -27rN ld r2 exp - kT r"2 - kT rm dr, (9)
where d is the distance of closest approach when the initial kinetic energy K and the impact parameter b are zero. Similarly, by substituting the expression for Q2m in Eq. (8) we can obtain,!
[2N7rI/2 ] lk' jrhCk) [ Kb?]1/2 ( K)
Bm(T) =-2 (kT)3/2 r2 K-¢(r)- --;2 exp - kT drdK, orICk)
(10)
where the upper limit Kc in Eq. (10) is the solution of the equation rf(K) =rh(K). The significance of these terms are clear from Fig. 1 in which the effective potential, defined by
<Peff(r, L) =<p(r)+(L/rZ),
L=Kbz,
(11)
(12)
is plotted against r for a particular value of L which is less than the value of Lc for which <Pelf has an inflection point (Fig. 1 of Ref. 12). The value of <pe!! at its inflection point is known as the critical energy Kc.
The evaluation of Bb(T) and Bm(T) from Eqs. (9) and (10) for fixed relative orientations of the interacting dipoles have been given in detail elsewhere.1l- 13 Here we have extended the tables reported in Refs. 12 and 13 by considering the repulsive orientations of the dipoles.
For A < 0, the relative orientation of the dipoles is repulsive and there are two repulsive portions in the interaction potential, one at long range and the other at short range. It can easily be shown by solving the two equations ¢(rm) =0 and d¢(rm)/drm=O simultaneously that for A~-0.38 the minimum of the potential-energy curve becomes zero. By solving the equations d¢/dr=O and d2¢/dr2 =0 simultaneously, it may also be shown that for A::::; - (8/27)1/2~-0.54, the potential has no minimum at all. The implications of these results are that for A::::; -0.38 and for A::::; -0.54 bound and metastably bound double molecules, respectively, cease to exist (Fig. 2). As Bd*(Bd*=Bd/bo, bo=j7rN(3) decreases steeply with the
17 A. Ghosh (unpublished results). 1ST. L. Hill, J. Chern. Phys. 23,617 (1955); Statistical Me
chanics (McGraw-Hill Book Co., New York, 1956), Chap. 5.
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4038 SINGH, DEB, AND BARUA
TABLE 1. Unaveraged value of -Bd*(T*).
T* -0.2 0 0.2 0.4
0.4 1.48005 13.01619 50.95428 0.5 0.96779 8.07744 26.21254 75.14650 0.6 0.69917 5.66783 16.62543 41.43659 0.7 0.53527 4.27326 11. 78049 26.94502 0.8 0.42662 3.37822 8.93237 19.30879 0.9 0.35034 2.76181 7.08833 14.73005 1.0 0.29435 2.31479 5.81104 11. 73733 1.25 0.20468 1.60666 3.89351 7.50531 1.50 0.15282 1.20048 2.84117 5.34392 1. 75 0.11965 0.94133 2.21608 4.06047 2.00 0.09696 0.76383 1. 76407 3.22258 2.50 0.06845 0.54077 1.23175 2.21541 3.00 0.05158 0.40843 0.92207 1.64362 4.00 0.03264 0.26319 0.58808 1.03696 5.00 0.02353 0.18760 0.41640 0.72979
10.00 0.00821 0.06575 0.14435 0.25015
0.6
100.27985 57.95300 38.44422 27.81875 21.33722 12.86125 8.86812 6.59824 5.16334 3.48930 2.55907 1.59670 1.11487 0.37772
0.8 1.0 1.2 1.4 1.6
128.21685 77.08374 51. 03989 94.80436 37.94839 65.07536 115.17711 21.14036 33.06397 52.18664 80.46711 133.59272 13.93184 20.75082 30.75084 44.14442 67.10868 10.08405 14.58916 20.82050 28.67577 41.20573 7.74917 11.00130 15.31940 20.56158 28.46265 5.11490 7.09946 9.60011 12.50876 16.55813 3.70435 5.07254 6.75149 8.66827 11.17727 1. 27523 3.07455 4.01905 5.07758 6.37878 1.57834 2.11855 2.74315 3.44008 4.26324 0.52743 0.69706 0.89165 1.10238 1.33932
III. CALCULATION OF MOLE FRACTIONS OF DlMERS
larger negative value of A (e.g., it is of the order of 10--5 for A = -0.4) we have evaluated it up to A = -0.2 and for T* ranging from 0.4 to 10. The values of Bd* (inclusive of the results of Refs. 12 and 13) are given in Table I. The average value of Bd* by putting equal probability for all the relative orientations can be obtained from the following equations14
:
From Eq. (6), the mole fraction of the dimers can be written as
The integrals in Eq. (12) were evaluated by using one-interval, five-point Gaussian integration for each of the variables COS01, C0502, and cp. It is found that the accuracy for (Bd*(Amax, T*), where Amax=!.u*2, is about 1 % in extreme cases (i.e., for larger values of Amax and lower values of T*) and is much better in other cases. Amax is the maximum value of A as obtained from Eq. (2). The averaged values of Bd* as a function of Amax and T* are recorded in Table II.
~ ... z ...
w _______ L<:Lc
I
Or-~~------~--~---------r
FIG. 1. Typical effective potential energy curve for a value of L<Lc .
X2d= X2b+X2m = -bo(Bb *+ Bm *) (n/V) (14)
where X2b and X2m are, respectively, the mole fractions of bound and metastably bound dimers, n is the number of
1·5,-----------------------------------,
1'0
0'5
• 1'-
FIG. 2. Potential-energy curves for several values of A.
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VISCOSITY OF POLAR GASES 4039
TABLE II. Averaged value of Bd*.
T* 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
0.4 -13.0162 -16.0423 0.5 - 8.0774 - 8.8962 -11.6181 0.6 - 5.6678 - 6.1026 - 7.8601 -10.9783 0.7 - 4.2733 - 4.6413 - 5.6224 - 7.5639 -10.6526 0.8 - 3.3782 - 3.6074 - 4.2394 - 5.5864 - 7.7711 0.9 - 2.7618 - 2.9207 - 3.4318 - 4.3846 - 5.9374 -8.1531 -10.5864 1.0 - 2.3148 - 2.4385 - 2.8184 - 3.5492 - 4.7435 -6.3204 - 8.6048 1.25 - 1.6067 - 1.6880 - 1.9090 - 2.3446 - 2.9528 -3.8976 - 5.0402 -6.4098 -8.2993 1.50 - 1.2005 - 1.2171 - 1.3761 - 1.6944 - 2.1385 -2.7015 - 3.3835 -4.2378 -5.2785 1. 75 - 0.9413 - 0.9591 - 1.0646 - 1.2989 - 1.6395 -2.0371 - 2.5237 -3.0961 -3.6873 2.00 - 0.7638 - 0.8321 - 0.9629 - 1.0000 - 1.3313 -1.6473 - 1.9968 -2.3744 -2.9077 2.50 - 0.5478 - 0.5525 - 0.5897 - 0.7012 - 0.8782 -1.0819 - 1.3105 -1.5704 -1.8731 3.00 - 0.4084 - 0.4206 - 0.4248 - 0.4987 - 0.6348 -0.7845 - 0.9506 -1.1376 -1.3504 4.00 - 0.2632 - 0.2753 - 0.3065 - 0.3504 - 0.4253 -0.5113 - 0.6128 -0.7249 -0.8438 5.00 - 0.1876 - 0.2007 - 0.2135 - 0.2356 - 0.2999 -0.3516 - 0.4251 -0.4979 -0.5677
10.00 - 0.0658 - 0.0682 - 0.0727 - 0.0832 - 0.1014 -0.1202 - 0.1430 -0.1676 -0.1923
moles, and bo=i·llNua. In Table III we have given the results of sample calculation Xu for NHa, CHaCl and CHCla. The force constants for these substances were taken from Ref. 14. Hirschfelder, McClure, and WeekslO have suggested a semiempirical method for calculation of the molefractions of dimers. According to this method
IV. INITIAL PRESSURE DEPENDENCE OF VISCOSITY
where ba is a constant depending on the intermolecular forces and B (T) is the experimental second virial coefficient. For NHa, ba was obtained by fitting the experimental data, and for all other gases by an empirical relation given bylO
(16)
where Ve , the critical volume, should be used. It may be seen from Table III that the agreement between the values of X2d calculated from Eqs. (6) and (15) is fairly good particularly in view of the empirical nature of Eq. (15).
The initial pressure dependence of the viscosity of a gas can be represented asl
(17)
where 'I/o is the viscosity at zero pressure and 'l/p the viscosity at pressure p. The initial pressure dependence of viscosity is determined by the effect of dimerization and collisional transfer. Clusters larger than dimers contribute to the quadratic and higher terms in Eq. (17) and can thus be neglected. By considering the gas to be a mixture of monomers and dimers 'l/p can be written asl
(18)
where ?1dm is the viscosity of the dimer-monomer mixture and 1]e represents the collisional transfer. For 1]e Stogryn and Hirschfelderl have obtained an ex-
TABLE III. Mole fraction of dimers.
Force parameters used X2
.Ik From semiempiri-
Temp Pressure cal relation of Eq. Substance 0' (0 A) (OK) Amax (OK) (atm) (15) This work
Ammonia 3.150 358 0.7 423 1 0.0048 0.0034 5 0.0228 0.0167
10 0.0437 0.0324 25 0.0973 0.0741
Chloroform 5.310 355 0.07 255 1 0.0387 0.0354 5 0.1509 0.1405
10 0.2411 0.2272 25 0.3897 0.3735
Methyl chloride 3.940 414 0.5 273 1 0.0185 0.0248 5 0.0992 0.1043
10 0.1414 0.1765 25 0.2616 0.3098
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4040 SINGH, DEB, AND BARUA
TABLE IV. Experimental and calculated values of RTa/bo for steam and ammonia.
Force parameters for monomer-dimer interaction
Gas U12 (0 Al
Steam 3.15 668 1.44
Ammonia 3.65 473 0.85
pression, by extending the Enskog theory,19 which may be expressed as
1/c=-io(B( T) + T[dB(T) jdTJ h11 (njV) , (19)
where B (T) is the second virial coefficient at temperature T and 1/11 is the viscosity of the monomer at zero pressure. 1/dm can be expressed in the ChapmanEnskog theory2o as
(20)
where A12* is a function of the collision integrals and have been tabulated for the polar gases by Monchick and Mason. 14 The subscript 12 indicates monomerdimer interaction.
Substituting the values of 1/c and 1/dm from Eqs. (19) and (20) in Eq. (17) the following expression for the pressure coefficient a can be obtained in terms of reduced quantities:
RTa/bo= - Bd*(0.5113+0.2481[( 6D21jD11)
-4.5455 (Dn/D21) Jl +0.17S0[B*+ T*(dB*/dT*) J, (21)
where
(D21/D11) =!v'J (0'11/0'12)2 (g11(1,1) * /g12(1,1)*), (22)
the g's are the collision integrals and the D's the diffusion coefficients. The evaluation of the right-hand side of Eq. (21) requires a knowledge of the force parameters for dimer-monomer interaction, viz., ~12jk,
19 S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge University Press, New York, 1952).
20 J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Liquids and Gases (John Wiley & Sons, Inc., New York, 1954).
T(OCl (RTa/bolexptl (RTa/bol. a1c
170 -1.60 -6.44 200 -2.63 -6.18 230 -0.83 -5.85 270 -0.19 -5.64
20 -3.73 -3.85 30 -2.16 -3.64
0'12, and (Amaxh2' At present this problem cannot be tackled rigorously and the force parameters are generally estimated by assuming the two molecules which form the dimer to be at equilibrium distance from each other and a monomer moves at the same distance from the nearest molecule in the dimer'! The force parameters for dimer-monomer interaction for steam and ammonia have been obtained by Barua et at. 8•9 and the values are given in Column 2 of Table IV. The calculated values of RTa/co are given in Column 4 of the same table. For the sake of comparison we have chosen the experimental viscosity data of Kestin and Wang5 for superheated steam and those of Iwasaki, Kestin, and N agashima6 for ammonia. The experimental data were fitted to a cubic equation in pressure at a particular temperature and was obtained from it. The experimental values of RTa/bo are given in Column 5 of Table IV.
It may be seen from Table IV that, in agreement with the experimental values, the calculated values of RTa/bo are negative. For ammonia the quantitative agreement between experiment and theory is also satisfactory. These results should be considered as gratifying particularly in view of the strongly polar nature of steam and ammonia. In fact, the agreement with experimental values is as good as that obtained by Stogryn and Hirschfelder1 for nonpolar gases. For steam a part of the disagreement between experimental and calculated values of RTa/bo may be due to experimental uncertainties as it is difficult to measure the viscosity of steam very precisely. Further improvement in the calculated values of a can also be made by having a better averaging for dipole-dipole and dimer-monomer interactions.
ACKNOWLEDGMENT
The authors are grateful to Professor B. N. Srivastava for his kind interest.
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