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K. E. VANHOLDE
Vol. 63
THE SEDIMENTATION VELOCITY OF OSMIUM TETROXIDE IN AQUEOUS SOLUTION BY K. E. VANHOLDE Department of Chemistry and Chemical Engineering, University of Illinois, Urbana, Ill. Received March 2 , 1069
The sedimentation velocity of osmium tetroxide ( 0 5 0 4 ) has been measured at a number of concentrations in aqueous solution. The results indicate that the sedimentation coefficient at infinite dilution is about 0.97 X sec. and increases slowly with increasing concentration. Use of this value of so together with Stokes’ law leads to a value for the molecular radius in surprisingly ood agreement with that obtained from known molecular dimensions. The implications of this and similar instances of agreement of the sedimentation and diffusion of very small neutral molecules with Stokes’ law are discussed briefly.
Introduction Sedimentation velocity measurements, while commonly used in the study of macromolecules, have but rarely been carried out with small molecules. There are a number of reasons why such investigations should be of interest. I n the first place, many low molecular weight substances can be obtained in a high degree of purity and with accurately known molecular weights. Thus critical tests of the phenomenological equations proposed to relate such transport processes as sedimentation and diffusion should be possible. Also, such data should provide evidence concerning the applicability of hydrodynamic theories such as Stokes’ law to molecules whose dimensions are comparable to those of the solvent molecules. The ideal material for such a study would be readily purified, of accurately known composition, molecular weight and molecular dimensions, soluble in water, but non-ionized. The heteropoly acids which have been investigated in this way by Baker, Lyons and SingeriV2 satisfy many of these requirements. However, their acidic nature requires that either the ionization be taken into account or that excess salt be added to the system. I n the latter case, one is dealing with a three component system, and the questions of possible interaction of flows should be examined. Osmium tetroxide (0~01) satisfies all of the requirements mentioned above. Although it is technically an acid, its ionization in aqueous solutions is very slight; the ionization constant (K, = 8 X determined by Yost and Whitea indicates that in a 1% solution the fraction ionized is less than 5 X 10”. While the molecular weight is rather low (254.20), the high density of the solid (4.91 g./ml. a t 22’) and the small molecular radius would suggest that the sedimentation coefficient, while not large, should be readily measurable. This paper describes the determination of the sedimentation coefficient of Os04 in aqueous solutions a t a number of concentrations. Diffusion coeficient and activity coefficient measurements will be described in a later communication. Experimental Materials and Preparation of Solutions.-Reagent
grade
(1) M. C. Baker, P. A. Lyona and S. J. Singer, J . Am. Chem. SOC., 17, 2011 (1955). (2) M. C. Baker, P. A. Lyons and S. J. Singer, THISJOURNAL, 69, 1074 (1956). (3) D. M. Yost and R. J. White, J . Am. Cham. SOC.,60,81 (1928).
osmium tetroxide was further purified by sublimation in The sample was held at 35-40’, and sublimed onto a surface kept at about 18”. In order to avoid handling large quantities of this very toxic material, small amounts were purified as needed. A sample as obtained melted a t 40.50°, the product of a single sublimation at 40.55’, and twice sublimed material also at 40.55’. The latter two values compare favorably with the melting points quoted by Sidgwick‘ (40.6”, 39.5’, 40.6’). Since the second sublimation resulted in no change in melting point, all samples after the first were purified by a single sublimation. Solutions were prepared using doubIy distilIed water. The volatility of solid os04 makes the accurate preparation of solutions by weight rather difficult, for if the solid is first weighed into a dry flask, saturation of the air in the flask with Os04 vapor will lead to an appreciable loss of solute, unless most of this 0 5 0 , vapor dissolves in the water which is subsequently added. To avoid this difficulty, the following procedure finally was adopted. The approximate required amount of water was first added to a weighed flask, keeping the neck dry and using a flask of suitable volume to be nearly filled. After the water was weighed, the approximate amount of Os04 was added and weighed. Additions of 050, or water could then be made t o yield the desired concentration. All weighings were corrected for air buoyancy. I t was found difficult to store these solutions (even in flasks with glass stoppers greased with fluorocarbon grease) without loss of 0 s 0 4 . For this reason solutions were always used as soon as possible after preparation. Density Measurements.-The densities of three solutions were measured in duplicate, and a single determination was made with a fourth. All measurements were performed at 25.000 0.002’, using pycnometers fashioned from 25-ml. erlenmeyer flasks. The pycnometer necks were formed from 1 mm. (i-d.) capillary tubing; the cross-sectional area of these necks had been determined by measuring the length of a weighed.mercurp thread in each. This area together with the height of the meniscus sbove a reference mark was used in the determination of the solution volume. The pycnometers were calibrated with doubly distilled water, and densities quoted are based upon the assumed density of this water (0.997044 g./cm.8). The results of the density measurements can be expressed by the equation p = 0.99704 0.00800~ c I 4.20 where p is the density in g./cm.8 and c is the concentration in g./100 cm.8. The above equation represents the data with an average deviation of &0.003%, which is of the order of magnitude of the reproducibility of duplicate measurements (0.002%). The solutions used in these measurements were prepared by placing a weighed portion-of Os04in an empty flask, and then adding water. Hence, it is probable that some loss of OsO, had occurred. In the unlikely event that all of the OsOl vapor was lost,, the concentrations would average about 1% too high. Sedimentation Experiments.-The sedimentation velocities were determined in a Sphco Model E ultracentrifuge, using a phase plate at the schlieren diaphragm. The temvacuo in the presence of calcium chloride.
+
(4) N. V. Sidgwick, “Chemical Elements and Their Compounds,”
Vol. 11, Clarendon Press, 1950, p. 1504.
c
SEDIMENTATION VELOCITYOF OSMIUMTETROXIDE IN AQUEOUSSOLUTION
Oct., 1959
1575
0.150
i0.100
4E .-
v
N
0.05
0 0
0.05
0.10
A R (cm.).
Fig. 1.-2 vs. AR curves for the experiment with solution I. The first photograph (0) was taken approximately 2 minutes after the rotor reached full speed. Successive photographs were taken at the following times (taking the time of the first picture as zero): a, 360 sec.; 0, 730 sec.; 8 , 1080 sec.; 0 , 1440 sec.
".
peratures of all experiments were between 24.5 and 25.1 Results were corrected to 25.0" by means of the relation S ~ ~ / S T - ~ T / ~ ZDensity S. corrections were negligible. Two experiments were required for the determination of the sedimentation coefficient at each concentration: a sedimentation velocity experiment and a synthetic boundary cell experiment. Because of the difficulty in storing solutions, these were always carried out on the same day. The sedimentation velocity experiments were carried out at a rotor speed of about 59,200 r.p.m. (determined for each experiment by the revolution counter) and the synthetic boundary experiments at about 13,400 r.p.m. In the sedimentation velocity experiments, the correct determination of the solvent base line proved to be of utmost importance. The ideal solution would have been the use of a double sector cell, but the only such cell available had an epoxy-resin centerpiece. It was feared that this material would be attacked by the OsOl solutions, since Os04 catalyzes the oxidation of organic materials. For this reason a pair of aluminum centerpiece cells were used, one containing water, the other solution. The base line was thus obtained on the same photograph as the refractive index gradient curve. It was found that this base line was accurately parallel to the gradient curve in the zero gradient region of the cell, and only slightly displaced from it. This displacement could be measured to f O . O O 1 inch. Since this procedure avoids the problems of the accurate alignment of two separate photographic plates during measurement, it is felt to be superior to the use of separate solution and solvent experiments, though inferior to the use of a double sector cell. The photographic plates were measured with a two-way comparator. It was found possible to reproduce measurements of the phase boundary image to within f0.0005 inches on the best plates, and 10.001inch on the worst.
used. It has been shown, however, that it is possible to calculate the sedimentation coefficient from the change of the refractive index gradient curve with time. As long as there exists in the cell a point rg where (bc(rp)/br)t= 0, we can calculate the sedimentation coefficients from the equation of Baldwin6 swat = -2.303 log
(r
- T.)*
&=
dc dr dr
+ 2r.
[I
- &I
t
(r
(1)
- ra) dc -- dr dr
-
Corae
(2).
where ra is the distance of the meniscus from the center of rotation, c is the concentration a t the point r and w is the angular velocity of the rotor. The integrals can be expressed in terms of the observed deflection Z of the phase boundary image on photographs of the sedimenting solution. If the specific rehctive index increment (&/bc) T,P is a con: stant, the integrals in equation 2 become (T
dc - r,)* dr - dr =
: AR)aZ dR (2) Gab cot & 1 dc ( r - r.) - dr (E) Gab cot :1 1
(
(3a)
fS
dr
1
=
fz
aRZ dR
@
(3b) Results A small molecule like osmium tetroxide hae a rela- In these equations AR = f A r , where R is the distively large diffusion coefficient and small sedi- tance along the cell on the photographic plate, and mentation coefficient. Consequently a sedimenting f is the cell magnification factor. G is the cylinder boundary will not be formed even a t the highest lens magnification factor, a is the cell thickness, b rotor speeds obtainable with the ultracentrifuge (6) R. L. Baldwin, Biochsm. J., 66, 644 (1953).
K. E. VAN HOLDE
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Vol. 63
the optical lever arm and # the phase plate angle. The initial concentration co can be expressed in terms of similar measurements of photographs of a boundary obtained in the synthetic boundary cell
e= f
(e)
1 $R:ZidR Ga‘b cot $’
(4)
where R1 and Rz are positions in the zero gradient region on either side of the boundary, and a‘, #’ and Z’ the values of a, $ and Z for the synthetic boundary experiment. The quantity Q then becomes
Q=
$2(AR)2ZdR + 2fr, $2 ARZ dR f*(a cot $/a‘ cot $’)
:$
2’dR
(5)
,
The integrals are evaluated readily by numerical integration, using 10-20 points and the trapezoidal rule. After Q has been calculated in this way, the sedimentation coefficient can be determined from the slope of a graph of (-log [l - &I) us. t. I n Fig. 1,the Z os. AR curves for experiment with solution I are shown. Figure 2 represents the (-log. [l - Q]) vs. t graph obtained from the data. The time of the first photograph has arbitrarily been taken as zero. The sensitivity of the result to the position of the base line is shown by the dotted line in Fig. 2, which was obtained by arbitrarily moving the base line 0.001 inch in a direction parallel to the meniscus trace. An error of 3% in s results from this. Thus, although it is felt that the base line position could be estimated to within 0.001 inch, this remains the single largest source of error in these experiments. The yoerror does not increase proportionately at lower concentrations, for it was found that with the higher phase plate angles used a t these lower concentrations, the displacement could be measured nearly as well as in the example above. The area under the synthetic boundary cell curves has been used to replace the quantity co in the calculations. This will be strictly justified only if (bnlbc) is a constant, independent of concentration. In order to test this, three of the synthetic boundary cell experiments were performed immediately after preparation of the solutions. The results of these experiments are summarized in Table
0.5
I11 were prepared by dilution of solution I. Because of the possibility of evaporation of Os04 during the handling of the small quantities of solution, the concentrations of these solutions were calculated from the synthetic boundary runs, using the average value of A / c cot #. The results of the sedimentation experiments are summarized in Table 11. A relation between s and co was obtained by least squares, using the s values before rounding off to 2 decimals, and weighting the data according to the estimated error. 826
X
= 0.97
- 0 . 0 4 1 ~+ 0 . 0 1 7 ~ ~
The data fit this equation with a standard deviation of 0.8%. The value of &5,0 is thus about 0.97 X sec. The concentration dependence of s is rather unusual, a t least in terms of what has generally been observed with macromolecules, in that s increases with co a t high concentrations. Preliminary measurements of the diffusion coefficients and activity coefficients of aqueous OsOI yield results which are consistent with the form of this observed concentration dependence of s. TABLEI1 SEDIMENTATION VELOCITY MEASUREMENTS
T
1.
CQ
TESTOF
TARLE I CONSTANCY OF (bn/bc)
Soh.
CQ
Soln.
(g./100 cm.8)
A“
IV V I
1.464 3.344 4.760
2.543 4.705 5.241
A/c X cot &
59.70 64.70 69.70
2.972 2.976 2.977
-
Av. 2.975 Area under synthetic boundary cell curve in arbitrary units. b Corrected for true 90’ position of phase plate.
The quantity A / c cot $ should be constant if (an/&)is constant. The results indicate that this is so, within experimental error.6 Solutions I1 and (6) Another possible source of error is a small “shoulder” sometimes observed near the meniscus at the beginning of a sedimentation velocity experiment (at rotor speed of about 8,000 r.p.m.), This did not appear to be a high molecular weight impurity, since it always flattened out into the base line before full speed was reached. If,
I
1.0 1.5 t X lO-S(sec.). Fig. 2.-The graph of (-log [l - Q])us. t for the experiment with solution I. The dotted line indicates the effect of arbitrarily shifting the base lines in the photographs so as to decrease each 2 value by 0.001 inch. 0
IV I1 I11 V I
(g./100 cm.3)
1.464 2.003 2.870 3.344 4.760
T (“(2.)
24.5 24.6 24.7 25.1 24.6
x
8T
10’3 (seo.)
0.95 .94 .99 1.03 1.15
x
Sll
10‘8 (sec.)
0.96 .94 .99 1.03 1.16
’
The dimensions and shape of the os04 molecule are quite accurately known. The oxygen atoms are tetrahedrally placed about the osmium atom; the Os-0 bond distance, from electron diffraction measurements’ is 1.66 A., which may be taken as a however, it was such, the sedimentation coefficients will be in erroi to the extent that this material WLLE”counted” in the synthetic boundary cell experiment and not in the sedimentation velocity experiment. The area under this “shoulder” wag never more than about 3% ot the area under the synthetic boundary curve. It seems more likely, from the behavior of this “shoulder,” that it arose from a small temperature difference in the solution near the meniscus. (7) L. 0. Brockway, Rev. Mod. Phys., 8 , 260 (1936).
.
Oct., 1959
SEDIMENTATION VELOCITYOF OSMIUMTETROXIDE IN AQUEOUS SOLUTION
minimum for the “effective” radius of the molecule. oAssuming that the radius of the oxygen is 0.55 A. (that of a doubly-bonded oxygen in organic compounds), we obtain a maximum radius of about 2.2 A. It is of interest to attempt to calculate this same quantity from the sedimentation data. According to Stokes’ law, the sedimentation coefficient of a spherical molecule a t infinite dilution will be given by
1577
form of Stokes’ law leads to the picture of the sedimenting entity being a solvated oso4molecule. This is not the only case in which the transport of very small molecules has shown surprising agreement with Stokes’ law. Baker, Lyons and Singer’ found this to be tirue for silicotungstic acid ( R E 6 A,), and Muller and Stokes8 for undissociated citric acid (R = 3.7 A.), The suggestion that solvation plays an important role is supported by the observation of Stokesg that the “no slip” Stokes’ law is fairly well obeyed by a number of substances in aqueous solution, whereas the diff usion of carbon tetrachloride in organic solvents is in better agreement with the theory postulating free slip a t the molecular surface.
where M and R are, respectively, the molecular weight and radius of the solute, go is the solute partial specific volume at infinite dilution, and po and 110, respectively, the density and viscosity of solvent. DISCUSSION The value of OD is 0.201 cme3/g.,according to the R. L. BALDWIN (Stanford University).-Would you care density measurements. Inserting the known val- to comment on the use of such measurements to test the ues in equation 6, we obtain for the radius the value Svedberg equation a t finite concentrations? of 2.1 A. K. E. VANHoLDE.-The equation referred to is It is rather surprising to obtain from Stokes’ law a result in such good agreement with the theoretical value. Stokes’ law is derived for the transport of a spherical particle in a continuous medium, and it is certain that water cannot be considered a con- which can be obtained for a binary system from the thermodynamics of irreversible processes. We have some of the tinuous medium for a molecule of 2 A. radius. data to test this equation, namely, D and s as functions of Closer examination reveals that the situation is concentration. Activity coefficient data are still needed; more complex than is indicated above. Stokes’ I had hoped to obtain these from sedimentation equilibrium law assumes that the layer of solvent adjacent to a measurements, but it turns out that the base-line problems which caused difficulty in the determination of S are even solute molecule moves with it. If this is to be in- more bothersome here. A double-sector cell is needed. terpreted a t all for a real medium, a solvation shell W. HELLER(Wayne State University).-If the unexpected of water molecules is indicated. On the other hand, concentration dependence of the sedimentation coefficient another form of Stokes’ law has been obtained by is due to aggregation, this should be checked easily by light postulating “slip” of the solvent past the moving scattering, since the specific refractive index increment particle. In this case, the coefficient 6 in the de- should be large. nominator of equation 6 is replaced by 4. This K. E. VANHOLDE.-I would hesitate to cite specifically would lead to a radius for the sedimenting molecule aggregation as the cause of this concentration dependence. of about 3.2 A., which again represents the approxi- However, light scat,tering studios would be informative. mate radius of an os04 molecule surrounded by a (8) G. T. A. Muller and R. H. Stokes, Trans. Faraday Soc.. 63, 642 solvation shell of thickness equal to one water (1957). molecule. Evidently, the assumption of either (9) R. H. Stokes, Australian J . Sci.. 19,35 (1957).
