112
J. Phys. Chem. 1980, 84, 112-115
v (crn3/rnole) Flgure 5. The volume dependence of fluidity in liquid cyclohexane.
differences, 0.02-0.03 A between the u’s obtained by the RHS analysis and those determined from Dymond’s expression, are of no physical significance. The u’s determined by using these two methods are in satisfactory agreement because the differences are merely reflecting the accuracy of both the data and the treatment used to obtain the hard-sphere diameters. The availability of the self-diffusion and viscosity data enables us to test the applicability of the Stokes-Einstein equation at the molecular level. The hydrodynamic Stokes-Einstein equation relates the self-diffusion coefficient to the shear viscosity q
D = kT/C.rraq
(4)
where a is the hydrodynamic radius and the other symbols have their usual meaning. The constant C is equal to 4 in the slipping boundary limit whereas it is 6 for the sticking boundary limit. Using the experimental D’s and 7’s given in Table I11 we found for the constant C a value of 3.6 f 0.3 for cyclohexane over the range of temperatures and densities studied. The result is not unexpected, be-
cause we have found that the Stokes-Einstein equation is valid in the slipping boundary limit for a variety of It is not surprising that the value of molecular 1iq~ids.l~ C is 3.6 f 0.3 and not exactly 4.0 as it is required by the slipping boundary condition. For example, C = 3.9 for , ~ 5.0 for tetperfluorocyclobutane,3 4.0 for b e n ~ e n eand ramethyl~ilane.~ The important result is that the product of D and q is constant within approximately 10% over a wide range of densities and temperatures. Such behavior was observed even for liquid methylcyclohexane16 which molecules have a pronounced nonspherical shape. Our results show that the transport properties of liquid cyclohexane are well described by the RHS model of liquids. Analogous behavior has been observed for liquid benzene whereas further enhancement of the nonspherical shape such as one finds in liquid methylcyclohexane already results in the failure of the RHS model to describe the transport properties in methylcyclohexane.16 Acknowledgment. This work was supported in part by the National Science Foundation under Grant NSF CHE 77-07621.
References and Notes (1) D. Chandler, J. Chem. Phys., 62, 1358 (1975). (2) J. DeZwaan and J. Jonas, J. Chem. Phys., 63, 4606 (1975). (3) R. J. Finney, M. Fury, and J. Jonas, J. Chem. Phys., 66, 760 (1977). (4) J. J. Van Loef, Physics, 958, 34 (1978). (5) H. J. Parkhurst, Jr., and J. Jonas, J. Chem. Phys., 63, 2698,2705 (1975). (6) M. A. McCool and L. A. Woolf, H&h TernpHigh Press., 4,85 (1972). (7) J. M. Hlldebrand, Proc. Natl. Acad. Sci. U.S.A., 72, 1970 (1975). (8) J. M. Hildebrand and R. M. Lameaux, Roc. Natl. Acad. Sci. U.S.A., 69, 3428 (1972). (9) R. C. Reid and T. K. Sherwocd, “Properties of Gases and Liquids”, 2nd ed, McGraw-Hill, New York, 1966. (10) D. E. O’Reilly, E. M. Peterson, and D. L. Hogenboom, J. Chem. phys., 57, 3969 (1972). (11) B. J. Alder, D. M. Gass, and T. E. Wainwright, J . Chem. Phys., 53, 3813 (1970). (12) J. H. Dymond, J. Chem. Phys., 80,969 (1974). (13) E. Wiihelrn, J . Chem. Phys., 58, 3558 (1973). (14) D. Chandler, J , Chem. Phys., 60,3508 (1974). (15) J. Jonas, Annu. Rev. Phys. Chem., 26, 167 (1975). (16) J. Jonas, D. Hasha, and S. G. Huang, J . Chem. Phys., 71, 3996 (1979).
Rapid Sedimentation under Gravity. Basic Theory and Experimental Demonstrations F. Galembeck,” P. R. Robllotta,+ E. A. Plnheiro,+I. Joekes, and N. Bernardes Institute of Chemistty, University of SSo Paulo, CP 20780, SSo Paulo, Brazil (Received May 8, 1979) Publication costs assisted by Gessy-Lever ( S ~ O Paulo)
Vertical concentration gradients are obtained when macromolecular solutions or peptized colloidal dispersions in tall dialysis cells, in contact with solvent through a semipermeable,vertical membrane, are allowed to approach equilibrium. This effect has not been previously reported in the literature and we found that it can be understood by considering the thermodynamic behavior of solutions under gravity. This phenomenon is distinct from normal sedimentation because (a) it depends primarily on the rate of solvent flow across the membrane and (b) solute concentration gradients may be obtained on a shorter time scale. The net result of this effect is that the usual kinetic restrictions on the approach to sedimentation equilibrium are by-passed and the feasibility of separation of a colloidal solute under normal gravity is demonstrated.
Introduction A n component fluid system of uniform composition (i-e., concentrations are constant throughout the system) subjected to gravity (or to a centrifugal field) is in a nonequilibrium state in the thermodynamic sense. SedimenI. Oceanogrfifico da USP, Ssio Paulo, Brazil. 0022-3654/80/2084-0112$01 .OO/O
tation of the denser component(s) occurs until the state of sedimentation equilibrium is reached. At this point the chemical potentials of all the components of the system are uniform and no flux of any component is observed inside the system. In a solution at constant P and T a t equilibrium under gravity, the activity of a component i varies with height according to the barometric equation1 0 1980 American
Chemical Society
The Journal of Physical Chemistty, Vol. 84, No. 1, 1980 113
Rapid Sedimenltation under Gravity
a(1n ai) -- -Mi(1 - ~ i P d g ah
RT
(1)
This equation predicts that most polymer solutions, if allowed to equilibrate, should develop considerable vertical gradients. For instance, the widely used chromatography marker, blue dextran, is a strongly colored macromolecular substance of mol wt E 2 X lo6, l j = 0.6 cm3 8-l. Its concentration in ideal aqueous solutions should vary with height with a gradient d(ln c)/dh = 3.2 m-l, that is, a solution cont,ainedin a l-m tall vertical column would show a change in concentration greater than 10-fold from top to bottom. ISveryday practice shows that no visible concentration giradients are detected in solutions of this or other polymers2 because approach to sedimentation equilibrium under normal gravity is a very slow process, as predicted by the classical theoretical work of Mason and Weavere3 Equation 1 and Mason and Weaver's theory apply to peptized hydrophobic colloidal dispersions as well4 and these are also frequently found in the laboratory and in the environment as systems far from sedimentation equilibrium. Until now, the attainment of sedimentation equilibria under pavity has only been observed in very short columns (
