540
HUGH 31. HCLBURT Departments of Chemistry ant1 Chemical E n g i n e e r i n g , The CuLhoEic r n i v e r s i t y of 21 tiien’cn, Tvashington 17, D.C. Recciuetl Scpfenibcr 6 , 1948
Khile one of the outstanding triumphs of the early 1Iaxn.ellian kinetic theor>of gases n-as the prediction that the viscosity shoultl be independent of the pressure, it has long been recognized that this conclusion is rigorously tenable onl>for an ideal gas at pressures such that the mean free path is small compared to the size of the container. The more exact theory developed by Enskog and Chapman (1) has not modified this conclusion. F,n.~l;og,however, n-:IS ahle to modify the fundamental theory for a gas of hard .spheres at moderate densities ( 2 ) . -In extension of Ensliog’s methods to the molecular models which give best results at lox- pressures presents forniitlable practical clifficulties. In vielv of this it is proposed in this paper to adapt the ,&freepath” approach of kinetic theory to molecules Tvhich are centers of attractive and repidsire forces. The equililirium properties of such molecules are Twll linon-n (71, a n d this knon.1eilge can be used in an approximate theory of transpo1.t properties. In the free path theory, it is supposed that a gas in the non-uniform stnte (i.e.. in idiich there exist gradients of temperature, mas.. 7-elocity. density, and concentration) has the same distribution of nioleculai, velocities as it IT-oulclhave at equilibrium in the absence of these gradients. If, further, the molectiles are supposed of finite size, the number of collisions per second per unit volume c m he calculated, n-hich, divided into the mean moleciilar velocity, defines :I ‘mean free path” hetn-een collisions. -%long this path the nioleciile is supposed to he free of any forces arising from other molecules in the gas and hence carries Ivit2-i it the properties characteristic of the site of its last collision. It is this last assumption in particular which is violated when the gas density becomes high enough so that an appreciable fraction of the molemlar trajectory is sutjject t o forces arising from other molecules. In equilibrium theory, the same difficulty is met in the theory of the equation of state, but it is in this case readily solved, at least for densities intherange \\-here third and higher virial coefficients are unimportant. It is shown (7that the imperfect gas behaves like an ideal gas in a someivhat diminished volume. cnlled the “free volume.” The precise definition of the free volume is some-\\-liat (id hoc (6) but, once defined, all the thermodynamic properties n-hich hare a pres,m‘e dependence can be related to it. It is here proposed that the proper free path to use for an imperfect gas is proportional to its free volume. On this ha expression is derived in which the pressure dependence of the viscosity is related to the equation of state. The usnal statement of the derivation of the coefficient of viscosity is easily 1 Presented a t the 112th Meeting of the Americmi C h e m i c d Society, which Sen- 3-ork City, September, 1947.
WLLSIicld
in
VISCOSITT O F COMPRESSED GASES
511
modified to introduce the free volume. Figure 1 represents the potential energy of a molecule in a gas as a function of the distance travelled by its center since any specified collision. For a rigid sphere, the potential energy is represented by the clotted line. The abscissa, A, is the free path of the hard sphere, rvhile A , is the portion of the trajectory that can be considered to be free of intermolecular force-. The standard derivation (3) leads to the formula qo = +rnizCX
R-here HZ II
= =
7 =
1
=
mass of a molecule number of molecules per cubic centimeter (Sl:T/a?n)''' = the average speed of the molecules mean free path.
r
FIG.1. Potentia1 energy of a
gas molecule
versus distance since its last collision
In a dense gas, the above analysis applies to transport over the distance n figure 1, since molecular interaction is taken as zero over the mean free path. \\-e may correct equation 1 by Tvriting n,; for 7 1 , xhere nj is the number of molewles per cubic centimeter in the region of negligible molecular interaction. Because of the molecular attractions, these nj molecules have the mean forward nomentnm of moleciile. a t the origin in figure 1. Jloreover, this momentum vi11 be coniniunicated to another molecule when the first has travelled a distance i f , rather than the distance X required in a rare gas. The number of molecules )er cubic centimeter d i i c h traJ-el the distance X, per 3econd is n j E A' !, each of i-hich tranbports momentum over a distance X. The rate of momentum trans)ort dou-nn-ard through a ,+.en plane is thus
hibtracting a similar espression for the upwird flus of momentum give3 the net rac t ion
542
HCGH X I . HIULBTRT
where 7“ is the viscosity in the ideal gas state at low density. The parameters 1 1 , and A, must non- he related to other molecular properties. The concept of a free path depends upon supposing the molecules to present a finite target area, T U ? , which can be defined for a given molecular model in terms of the temperature and the parameters of intermolecular force. At low densities, it is assumed that A is the distance travelled by the center of the molecule. xo’h should then represent the volume swept out by each molecule between collisions and is clearly proportional to the molecular volume of the gas. On the other hand, au2Af is proportional to the free volume and we may write
The free volume is given by ( 5 )
whenever the average mutual potential energy of molecular interaction is a function of the density only. Integrating this from low densities at which
v, = v,
I t should be noted that In (V,, 1’) = ( S - S”),’R,where So is the entropy of an ideal gas a t 1 atm. and S that of the actual gas at a given pressure. The integral in equation i can be evaluated from an equation of state. The factor n f ,72 accounts for the fact that only those molecules which are “free”, i.e., whose centers lie in the region of negligible molecular interaction, are in fact transporting momentum. The number of such “free” molecules a t eqiiilibrium will be given by Ilf = ,le-”! R 1 (8) where E’ is the energy of the minimum in the intermolecular potential. This mill be related to the difkrence in enthalpy between an ideal gas and the real gas a t elevated pressure, H: - H T , since this difference also arises solely trom intermolecular forces. One should not expect, honerer, that the entire enthalpy of the intermolecular forces operates in limiting momentum transport, for home molecular pairs migrate together and contribute, t o L: first approximation, the full quota of momentum transported by two unassociated molecule>. One might expect some fraction of the total enthalpy to be operative, and, in fact, :t larger fraction the 11eaker the intermolecular forces and the lo\\-er the denbity, since the molec.iile~spend less time in association \I-hen thew forces are n eali.
VISCOSITY OF COMPRESSED GASES
543
TTe write then, finally,
FIG.2. Viscosity ratio for nitrogen at 25°C.; d a t a of Michels and Gibson (8)
FIG.3. Viscosity ratios for carbon dioxide ( T R = 1.02) arid cthylcnc
where
(Y
( T I R
= 1.10)
is espected t o vary from substance to sul)stan~e,:md, somewhat, iyith
N o - H , diminishing as H o - H increases for the same substance. In figures 2 and 3 these results are compared with actual data. The theoretical curves were cwmputed using the Beattie-Bridgeman equation of state. For nitrogen (figtire 2'1 o = 0.45 and is constant up to reduced pressures of 9. -4t still higher pres-
544
HUGH M. HULBVRT
sures, the calculated results are too lon-. Hon-ever, for reduced pressures above 10 we find the free volume smaller than the excluded volume. A\t such high densities momentum transfer cannot be tthe only dissipative mechanism and some liquid-like viscosity must come in. In figure 3, calculated curves for carbon dioxide (a = 0.227) and ethylene (a = 0.15) are plotted. The a-values are averages over the pressure range, actual values decreasing somen-hat with pressure in each case. At higher pressures than shown, calculated results are too low, as for nitrogen. Bgain, the deviation sets in a t the pressure where T’, = +V and one would expect liquid-like viscosity to appear. Comings and Egly (4’) have proposed on empirical grounds that viscosity is a function of reduced temperature and pressure only. The present theory accounts only partially for the success of this correlation, since a need not be and is not the same for all substances. Hon-ever, a comparison of calculated values of a with values of In (17/V;l shows that, roughly,
a(HO- H-) _- 1 ln,Y RT 3 1; Hence a useful rough rule appears to he 11/70 = (T-,/l’,)2
(1 1)
The difficulty of ohtaining precise viscosities esperimentally a t high pressures makes it impossible to set the limits of accuracy of this expression with assurance. but it appears to have the same usefulness as the empirical correlation referred t o above for viscosity ratios below 1.6. SUJIJIhRY
An expression for the viscosity bf an imperfect gas at, elevated pressures has been derived ahd shown to be applibble in predicting viscosity ratios below 1.6, using t’he equations of state of the gds. REFEltESCES ( 1 ) CBAPAIAN A N D Cow-r.rsc;: The JfathetuaticaE Theor!/ u j S(tu-Cnijorrn Gases. University Press, Cambridge (19:39). (a) Reference I,pp. 273-94. (3) Reference 1, pp. 100 ff. (4) COMINGS AND EGLT:Ind. Eny. Chem. 32, 714-18 (1940); Chem. Met. Eny. 63, No. 3 . 1 1 5 (1945). ( 5 ) FOWLER A N D GCGGESHIXJI : Statistical Tliermon‘gnamics, p. 331. University Press, Cambridge (1939). (GI FRASK:J. Chem. Phys. 13,178432 (19451. (71 MAYERAND MATER:Statistical Mechanics, pp. 27i ff. John Wiley ~lriil SOIIS.In?., Yew York (1940). (8) MICHELA N D GIBSOS:Proc. Roy. SOC.(London) A134, 258 (1932).
