DONA. ALBRIGHT AND J. W. WILLIAMS
2780
radiation-produced D atoms and hot T atoms, produced by the He3(n,p)Tprocess, in their reaction with CH3Br. The radiation-produced D atoms react almost exclusively by the displacement reaction (14), CH3D being the only significant product. The hot T atoms, on the other hand, react in several ways, including H abstraction, halogen abstraction, as well as halogen displacement, these various reactions having similar specific rates.14 The formation of GO in the radiolysis of the mixtures
containing CH3Br is a surprising result. However, this product is not, apparently, formed by the reactions of D atoms or electrons, and it has not been studied in any detail. It probably results from the reactions of OD radicals, but this matter requires further investigation.
Acknowledgments. We thank Professor J. J. Weiss for his support. The work was assisted financially by the Atomic Energy Research Establishment, Harwell.
Sedimentation Equilibria in Polydisperse Nonideal Solutions'
by Don A. Albright and J. W. Williams Department of Chemistry, Uninersity of Wisconsin, Madison, Wisconsin (ReceiEed December 13, 1966)
For polydisperse solutes in nonideal solutions at sedimentation equilibrium, it is widely believed that the theory indicates no reliable molecular weight average can be extracted from the experimental observations, and, as a corollary, one is restricted to operations a t the Flory temperature. It now appears that the thermodynamic equations which describe the sedimentation equilibrium in such systems can be put into a form such that a series of experiments performed at several low concentrations and a t several low rotor speeds will provide the information necessary not only to compute M , but also to evaluate the light-scattering second virial coefficient. Experimental tests with two polystyrenes in toluene and with a polyisobutylene in cyclohexane now have been completed and are here described. I n the three cases the weight-average molecular weights and the lightscattering second virial coefficients computed from the photographic records of the sedimentation equilibrium experiments are found to be in good agreement with data of the literature obtained by the more traditional approaches of the osmotic pressure and lightscattering determinations.
As a method for the determination of thermodynamic properties in solutions of organic high polymers, the sedimentation equilibrium experiment has a number of advantages, yet, in general, it has not found the same general acceptance in polymer chemistry that it has received a t the hands of the protein physical chemist. There are several reasons for this situation, with probably the most important one being the fact that the theoretical equations for the description of the sedimentation equilibrium in polydisperse, concentraT h e Journal of Physical Chemistry
tion-dependent systems are relatively complex. At any fixed point in the cell, the distribution of molecular weights will differ from that in the original solution. Furthermore, the correction for solute-solvent interaction will be dependent both on the total concentration and on the particular distribution of species a t the given point. Typical of this situation is the conclu(1) Presented a t the 152nd National Meeting of the American Chemical Society, New York, N. Y . , Sept 11-10, 1960.
SEUIMESTATIOS EQUILIBRIA
IN POLYDISPERSE ?;ONIDEAL
sion of JIandelkern, et C L ~ . ,to ~ the effect that interpretable measurements with organic high-polymer solutions can be made only a t the Flory temperature, a judgment which if true would present a most discouraging outlook. Others have felt the experimental difficulties which go with the determination are so great that it is unlikely any greitt reliance can be assigned t o the results. However, the now widespread use of interferometric methods to determine the concentration distribution in the cell, and mechanical improvements in the ultracentrifuge itself, have done much to dispel these fears. In this report me shall attempt to show how sedimentation equilibrium studies may supplement lightscattering and osmotic pressure measurements as a means of investigating the thermodynamic properties of polymer solutions. For this purpose me have made extensive use of reports by Fujita3&and by Osterhoudt and Rilliams,3b which have indicated that a series of sedimentation equilibrium experiments for a polymer in a good solvent completed at several low centrifugal field strengths and at several low concentrations ought to permit the evaluation of the weight-average molecular weight and of the light-scattering second virial coefficient. We are here concerned with the performance and interpretation of such experiments for two different polymers-polystyrene (two samples) and polyisobutylene (one sample)-dissolved in toluene and cyclohexane, respectively, both of them being “good” solvents at 20‘. For each of these three systems, both weight-average molecular weight and second virial coefficient data were independently available, though no prior knowledge of them was used to obtain the results we norv present.
Theory Suppose a polymer consists of coniponents which differ in molecular weight. Then, the equations which describe the sedimentation equilibrium are4
dc, dr
=
(1 - bpo)u2(r22 2RT
7.12)
it can be shown that
This expression describes the concentration gradient of a typical polymeric species a t sedimentation equilibrium. It appears as eq 5.103 in the Fujita monograph. Now, with the substitution of two approximations (which become exact as cn. 0 )
into this equation, followed by series expansions, we obtain
+
rl
+
~ ~ ~ z C i x C k o l ~ t ((fi/lvk)) B
