EQUILIBRIUM ULTRACENTRIFUGATIONS OF CHARGED SYSTEMS
June, 1959
787
THE USE OF INTERFERENCE OPTICS I N EQUILIBRIUM ULTRACENTRIFUGATIONS OF CHARGED SYSTEMS’ BYJAMEBS. JOHNSON, GEORGESCAT CHARD^^ AND KURTA. KRAUS Contribution from the Oak Ridge National Laboratory, Chemistry Division, Oak Ridge, Tennessee Received November 8 , 1968
The use of interference optics for concentration measurements is discussed for equilibrium ultracentrifugations of charged solutes in supporting electrolytes. A procedure suitable for machine computation is presented. Experimental tests on three “known” systems (N&MoOr in 1 M NaC104, BiOCIOr in 1 M NaC104 and BaC12in 1 M HCl) are presented together with pertinent apparent volume and refractive index increment measurements. The results show that interference optics is much more accurate than schlieren optics and that use of wedged centerpieces allows simultaneous ultracentrifugation of several solutions without significant loss of accuracy. The importance of activity coefficient derivatives in the evaluation of degrees of polymerization and of charges by ultracentrifugation is demonstrated with a discussion of the BaClrHCl system, and the feasibility of studying complexlng reactions (average charge determination) is discussed.
Procedures2-ahave been described for interpretation of equilibrium ultracentrifugation of charged solutes in the presence of slightly sedimenting supporting electrolytes. These discussions dealt primarily with results obtained by schlieren optics, with which the concentration of solutes in the centrifugal field is followed by measurement of the refractive index gradient dn/dx as a function of radius z. I n recent years interference optical systems, which give differences in refractive index, have come into use,4 and it is the purpose of this note to discuss the interpretation of results obtained with this system. With interference optics, a double compartment cell is used: one (solution compartment) contains solvent, the solute whose molecular weight is to be determined, and supporting electrolyte; the other one (background or solvent compartment) contains solvent and supporting electrolyte, usually at approximately the same concentration as in the solution compartment. Both compartments are sector shaped, i.e., the side walls lie along radii, and both cover the same range of radius. Monochromatic light passes through both compartments and is recombined to give interference fringes. With the ultracentrifuge which we use (manufactured by Spinco Division, Beckman Instruments), these fringes are horizontal when the refractive index difference between the two compartments n* is independent of the radius. Essentially this pattern is observed at the start of centrifugation. As sedimentation proceeds and the refractive indices of the solution and background begin to vary with x, the fringes will curve. Between any two adjacent fringes in the horizontal (radial) direction, there will be a difference in n* equal to h/h, where X is the wave length of light used and h is the thickness of the solution in the direction of observation. If the difference n* between the refractive index of solution and background can be established at one
radius (e.g., by following the movement of fringes from the start of centrifugation, or by an integration procedure), the values of n* are known for the other radii. Some complications stemming from details of construction and from the use of a multiple-cell rotor will be discussed later. 1. Uncharged Solutes.-If the solute is uncharged, the values of n* may be converted to concentrations (c), if the refractive index increment lc = an/& is known. The slope S = d In c/d(x2) can then be obtained. The molecular weight M may be computed by the equation M =
2RT (1
-0p)d
(1)
where R is the gas constant; T,the a b s h t e temperature; 8, the partial specific volume; p, the density of the solution; and LJ, the angular velocity. Activity coefficients, 8, and p are assumed constant in equation 1. Frequently, IC will be essentially constant, and with sufficient accuracy, S will be given by d In n*/d(x2). If an integration procedure is used to establish concentrations (hence n*) at one radius, the condition which must be satisfied for sector shaped cells is given by the equationb
iaw
cz d s = (c0/2)(zw2 - za2)
(2)
the initial concentration, the index w indicating the maximum radius of the solution, and a indicating the radius at the meniscus. If IC is constant, values of n* may be substituted for c in equation 2. The equation follows from the fact that the total amount of solute in the cell at equilibrium is the same as at the start of centrifugation. 2. Charged Solutes.-Sedimentation of ionized solutes depends on the charges as well as on molecular weight. The effect of charge varies with concentration of the solute and of supporting electrolyte, and by varying their concentration ratio, that molecular weight and charge can be selected which best satisfies all results. I n principle, the ( 1 ) This doaument is based on work performed for the U. 8. Atomic Energy Commission a t the Oak Ridge National Laboratory, Oak variation of concentrations in a single centrifugaRidge, Tennessee, operated by Union Carbide Corporation. tion is sufficient for simultaneous charge and ( 1 4 Department of Chemistry, Massachusetts Institute of Techmolecular weight determination, but in practice nology, Cambridge, Massachusetts: Consultant, Chemistry Division Oak Ridge National Laboratory. more experiments frequently are required. (2) J. 9. Johnson, K. A. Kraus and G. Scatchard, THISJOURNAL, Equations for computing molecular weights from 68, 1034 (1954). centrifugation results as a function of assumed (3) J. S. Johnson, K. A. Kraus and R. 1%’. Holmberg, J . Am. Chem. Soe., 18, 26 (1956).
(4) See e.&, J. W. Beams, N. Snidow, A. Robesqn and H. M. Dixon, LII. Rev. SCi,Ifl8&,16,295 (1954).
co being
(5) T. Svedberg and IC. 0. Pedersen, “The Ultracentrifuge,” The Clarendon Press, Oxford, England, 1940, p. 312. (6) 0. Lamm, ATkiv. Kemi. Mineral. Qsol., ITA, No. 25 (1944).
JAMESS. JOHNSON,GEORGESCATCHARD AND KURT A. KRAUS
788
charge were developed for an idealized system12 which will also be used for the present discussion: a polymeric component PX,, which is ionized in solution as P + z xX-; a supporting electrolyte BX; and solvent. I n the assumed system, the partial specific volumes and solution density are constant. Charge per monomer unit 2’ (primes refer to quantities expressed in terms of monomer) is also constant. The discussion will be limited to a monodisperse polymeric solute. I n such a charged system the supporting electrolyte does not sediment independently of the polymeric solute; Le., BX is distributed differently in the background and solution compartments. To minimize this difficulty and to make derivation of the equations more convenient, the components are redefined in a manner used by Scatchard’ for interpretation of osmotic pressure measurements on similar systems, and also employed in light scattering work.8 The polymer component (2) is redefined as (PX, - (d2)BX) or (PX,/2B-,/J: the concentration c2 of this component &the same as that of PX,, but its activity is u2 = CpCX”/2CB-2/2qpgXJ2gB-2/2 = C P C X ~ / ~ C B where - ~ / ~ Gg ~ indicates , activity coefficients of ions and G the appropriate activity coefficient products. The equilibrium condition for component (2) is given by
+
d In uz = d In
+ d In
CPCX*/*CB-~/~
d In cz
+
G2
=
1 fv ( z / 2 ) d In 1 v d In Gt = A2 d(z2) (3)
+
where The subscript i indicates the component in question. Contrary to our earlier procedure12activity coefficient terms will be left in the expressions. Sufficient information to evaluate them usually is not available, and they are assumed independent of radius (ie., d In G,/d(x2) = 0). The activity of the supporting electrolyte, component (3), is given by the product u3 = CBCXgBgX = CBCXG~, but its concentration in presence of component 2isc3 = CB (z/2)c2 = cx - ( 4 2 ) ~ With ~. this definition of components, it was shown2 that, a t centrifugation equilibrium
+
d In
= d In abg = 2 d In
d In (1
c3
+
- q 2 ) + d In G3 d In
Gbe.
+
= 2 d In Cbg = A s d(zz) = A b g d(z2) (4)
where the subscript bg indicates BX in the background compartment. The quantity 9 usually is small enough compared to unity to allow neglect of the term d ln(1 - v2) in equation 4. Thus, with components defined in this way, the distribution of BX in the background compartment approximates the distribution of component (3) in the solution compartment, Le., ratios of concentration of component (3) a t two radii will be nearly the same as the ratios of cbg. The refractive index difference between solution and background is given by the equation (7) G. Scatohard, J . Am. Chsm. Soc., 68, 2315 (1946). R. Lontie and P. R. Morrison, ibid.,
( 8 ) J. T. Edsall, H. Edelhoch, 73, 4641 (1960).
?&*
k2‘C.%‘
+
Vol. 63
ks(c8
- cbg)
(5)
in which k f 2 = bn/dc‘2 = dn/bctpxz - (2‘/2) (an/ ~ C B X )and k) = bn/dcsx. Usually the initial concentrations of BX in the two compartments are approximately the same. Then the difference in the initial concentrations of component (3) (cap) and of BX in the background compartment (Cbgo) is given by the equation c30
- Cbgo
=
(6)
(zf/2)C20’
Further, a t equilibrium the difference between c3 and Cbg is approximately equal to (z’/2)ct20 at all radii. Values of c2‘ necessary for computation of molecular weights would be given to good accuracy by n*/lcf2only if solutions are made up so that c30 = Cbgo. However, usually z’ is not known beforehand and iterative experiments would be required. The error incurred by neglect of the difference between c8 and Cbg is usually small if interpretation is based on refractive index gradients, but must be considered with interference optics, and a computational procedure to eliminate it is desirable. The greater precision attainable with interference optics makes it desirable to eliminate some other approximations made in the interpretation of data obtained with a schlieren optical system. To do so makes the arithmetic rather burdensome, and a procedure adaptable to machine computation was developed. To reiterate, the problem is the computation of the degree of polymerization N implied by the results of a centrifugation for a series of values of X’ covering the expected range. After comparison, those values of X I and N best satisfying all results can be selected. It is assumed, for the present, that the menisci in the solution and solvent compartments are a t the same radius. 3. Computational Procedure. (a) Input Information.-The centrifugation results are recorded as photographs of fringes stemming from recombination of light passing through the solution and solvent compartments. On the same photograph are reference fringes, resulting from slots in the counterbalance, which enable alignment of the plate. The horizontal (radial) positions of maxima (or minima) of darkness are determined with a comparator and corrected for cell distortion. With knowledge of the radial magnification of the optical system, and of the radius at one point on the photograph, computation yields a list of radii for which there is a known and constant difference in n* between adjacent points. Additional input information required are the partial specific volumes and refractive index increments of polymeric solute and of supporting electrolyte, molecular weights of the supporting electrolyte and of the monomer unit of the polymeric solute, density of the solution, ,speed of rotation and temperature, and the difference in n* corresponding to one fringe interval (X/h). (b) Integration Procedure.-If refractive index increments k, are constant ‘i7,n*(ro2
(kz’cro‘ + - Xa2) = =J:n* d(r2) = n a * ( r J - ~ 2+)
- 5a2)
E
kaC3c
JI
kaCbgo)(Zw*
(n* -
Itcr*)
d(z2) (7)
1
L.
x
EQUILIBRIUM ULTRACENTRIFUGATION OF CHARGED SYSTEMS
June, 1959
where no*is the initial value of n* and na* is the equilibrium value of n* a t the meniscus. The value of (n* - n,*) for any radius may be obtained from the number of fringe intervals between xa and z. The integral (n* - na*)d(x2) can therefore be evaluated by standard methods, for example by the trapezoidal rule. Since the change in n* per fringe interval is constant, Simpson's rule may also be used, with (xo2 - z2) as the varying quantity. In the latter case, an estimate of the contribution to the integral by elements a t the limiting radii may be made by a trapezoidal computation. With the integral evaluated, na* and therefore n* at any fringe position may be computed. (c) Computation of cbg.-Values of Cbg at various radii are needed in subsequent steps of the computation. From equation 4
so
dCbR
=
d(z2) -
(As/2)cbg
2d 1n
(8)
Gbg
and
(2/A3) Cbga
=
[
(Cbgw
= 2[1
+
Cbga)
cbgw(Gbgw/Gbga)'/2
(cbd2)
d In
exp((A3/2)(za2 -
- Xa') - (1/Aa)
Aa[Cbgo (20'
Cbgo
-
- (Gbao/Gbga)'/z
sao
cbg
Gbg]
(9)
2~'))
(10)
d In Gbg]
exp((A3/2)(aaZ- zw2)) (11)
Once Cbgo is known, Cbg a t other values of z may be computed. (d) Computation of czJ (2' = O).-For z' assumed equal to 0, c3 = CBX in the solution compartment, and c2' may be computed by a modified form of equation 5 c2f =
n*
- kaCbg{ [ ( c ~ / ~ b ~ ~ ) G ' /-z l 11 (h')
(12)
where G =
(Gar~Gbg)/(GbgoG3)
Equation 12 follows from the fact that d In (1 q2)/d(z2) = 0, and when z' = 0 (see also footnote 10). (e) Computation of cz'(zJ # O).-For computations of c2' with assumed polymer charge other than zero, it is convenient t o make use of a quadratic equation in q. After substitution of 2qc3/z' for cz' (13)
With the approximation that the initial value of the activity of component (3) and of the background solute occur at the same radius a t equilibrium, we obtain from equation 4
Combination of these equations yields (1
- q 2 ) ( 1 - d Y n * + k v ~ , =) ~ (1 C'bo
Cbgo/C30
1 - 70
(15)
PO2)[(2k2'W/zf)
+ kdaG
(16)
This equation must be modified to allow for some complications resulting from details of construction of the equipment, as well as departures from assumed experimental conditions which are inconvenient to control. (i) The initial concentration of CBX in solution and background compartment may not be precisely the same. The assumption that they are was not made in the equations involving refractive index (5 and progeny) but is made in equation 15. Correction may be made by adding a term A = (Cbgo CBXo)/C30, to the right side of this equation.g (ii) I n equations 7 and 14, use of the initial background concentration (Cbgo) involves the assumption that the menisci in solution and solvent compartments occur a t the same radius. In practice, this is seldom precisely the case; indeed, it is well to fill the background compartment more completely than the other, so that the fringe pattern will cover as much of the polymer solution as possible. The value needed for Cbgo in the equations is that concentration which would have to be introduced into the compartment to give the actual distribution of Cbg, if the menisci in the solution and background compartments were located at the same radius. Once Cbgo has been computed for the actual limits of the background solution, the required value of Cbgo may be computed by substituting xa2 and zo2for the solution compartment into equation 11. (iii) I n the mask at the upper collimating lens of the Spinco optical system, the slits are frequently located unsymmetrically, ie., the slit for the solution compartment is centered along the optical axis, while that for the background compartment is off center. As a result, light going through a given radius of the solution compartment interferes with light from a slightly different radius of the background. The difference E will vary with radius, but for present purposes it is sufficient to use an average value; thus light from the solution compartment, radius x, will be assumed to recombine with light passing through the background compartment a t x* = z E, where e = 0.0123 cm. for centrifugations reported here. Three modifications of the procedure are incurred by this. (a) The light will be cut off in the background compartment between the solution compartment radii zo - E and xu, and to cover the full range of the polymer solution in the integration (equation 7), a short extrapolation is necessary. (b) The concentration Cbgo in equation 7 must be replaced by C*bgo computed for the limits za* andz,* (the limits of the polymer solution E) in the manner described in (ii). (c) The value of Cbg in the refractive index equation 5 and progeny must be replaced by C*bg, the
+
+
(9) If A is defined a little differently, ((cbg)xro
-
csxo)/cao, where is the value of the background concentration at the radius for which a3 = c a 2 ( l +)UP = aw = caoa(l voZ)G30,the approximation that the initial activity of both component (3) and the background solute occur at the same radius (equation 14) would be eliminated. The difference between the two definitions of A usually may be neglected for practical purposes. (cbg)m
From equation 6
789
-
-
JAMESS. JOHNSON, GEORGE SCATCHARD AND KURTA. KRAUS
7 90
concentration of background solute at a radius greater by E than the corresponding solution radius. Note that cbg in equation 14 is not changedvalues computed for the same radius as that in question for the solution are used. (iv) With the multicell (Analytical G ) rotor now available from Spinco, simultaneous centrifugation of five solutions is possible. The fringe patterns for the five cells are separated by wedging of the centerpieces, and consequently there is a slightly different length of light path for the two compartments. Although the difference in thickness of the column of solution (at most about 0.007 cm. out of 1.2 cm.) is the same at all radii, the difference in light path varies somewhat with radius, since the total concentration of solute varies. Correction is made by multiplying C*bg by a factor h&, where hl is the thickness of the solution compartment, and h2 the thickness of the background compartment. (The value for An*/fringe interval is computed for the solution compartment, ie., X/hl). Note that only the terms C*bg which arise from equations involving refractive index are multiplied by h&; however, the correction also must be made in computing the value of the integral, equation 7, Le., (hZ/hl)C*bgo is substituted for Cbgo. With these modifications equation 16 becomes (1 - S Z ) ( l - 70 A)'(?%* f k3(h~/hl)C*bg)~
+
C'bg
(1
- 57o2)[(%'9/zf) + kalzG
Vol. 63
machine, but where this course is not feasible an equation derived by making certain approximations may be useful. It has given values of N within 3% of those obtained by the more complete procedure in the few cases so far tried, in spite of the fact that allowance is not made for the factors discussed in Sec. 3e. Taking the logarithm of both sides of 5 , followed by differentiation, yields, for constant activity coefficients d In n* d(xe)
d In c2' d(x2)
-
(20)
If the approximations are made that the original concentrations of cJ2, c3 and Cbg occur at the same radius, that v2 is negligible in comparison with unity, and that d In c3/d(x2)is small compared with d In cf*/d(x2),the equation d In n*/d(z*)
- d In cz'/d(z2)
=
is obtained. After a table of n* as a function of x has been computed from the fringe positions, one may compute N for assumed values of z' by iteration with equation 21 and an approximate form of 19.a
(16a)
or rearranged as a quadratic in q [(I
- 90 f
+
A)'(n* k3(h2/hl)C*bg)2 f C2bg(l - ?a2) (4k'22/2~'2)G]~2 Jr [C'bg (1 - 9o')G 4knfka/Z']9 f C2bg(l - vo2)ka2G - (1 - 70 A)' (n* ka(ha/hl)c*t,g)' = 0 (161s)
+
+
When 3 is evaluated, c3and cJ2are given by the equations c*f = 2carl/z'
(18)
With the values of c2' computed at the radii of fringe positions, the value of N implied for the x' in questionlomay be computed with the equation
which is obtained by substituting Nz' for z and N A t 2 for A2 in equation 3. The above procedure has been coded for the Oak Ridge National Laboratory digital computer, the OBACLE, with activity Coefficients assumed constant, and tests are presented for centrifugations of known systems. Computations are most satisfactorily effected by (10) I n the computation of e'z for z' = 0, equation 12 is modified somewhat b y allowance for the complications discussed above.
A small approximation is involved since
Experimental Centrifugations were carried out with a Spinco Model E Ultracentrifuge, equipped with a Rayleigh interference optical system, and with an Analytical G (five-cell) rotor. Most of the centrlfugation technique is standard or has been described earlier3*11and for the most part only changes resulting from the use of interference optics will be discussed. It has already been mentioned that the data are recorded as photographs of fringe patterns, resultingfromrefractiveindex differences between solution and solvent compartments. Reference fringes from gaps in the counterbalance permit alignment of the plates. Progress of centrifugation is followed, and attainment of equilibrium established, by examination of the plates with a comparator capable of measurements in both vertical and horizontal directions. For an arbitrary vertical setting relative to the reference fringes, the horizontal osition of the fringes is determined. When there is no signii!cant change in position from day to day, equilibrium is considered established. Cell distortion is evidenced in centrifugations with water in both compartments by departures of the fringes from the horizontal. Normally water centrifugations are carried out both before and after centrifugation of solutions. The distortion indicated by these runs is reproducible to about f O . l fringe interval. Distortion of as much as 1.5 fringe intervals has been found, but the extent for a given cell does not seem to change rapidly with time, nor does it seem very sensitive to speeds of rotation in our usual range of operation, from 14,000 to 30,000 r.p.m. In reading plates for interpretation, correction is effected by altering the vertical setting of the comparator to compensate for departures from the horizontal with water. An average of the distortion found in the water centrifugations before and after the run is used. Because of the acidity of some of the solutions studied, cell centerpieces of pure e oxy resin were used, rather than the usual aluminum fillef type. Normall t h e cells were not disassembled between centrifugations, %ut were rinsed, shaken overnight filled with water, rinsed again and dried. Values of the thickness It of the column of solution were obtained by subtracting the thickness of the quartz windows (11) J. 8. Johnson, K. A. Kraus and T. F. Young, J . A m . Chem.
In equation 12% the exponential term is taken as unity.
Soc., 76, 1436 (1954).
EQUILIBRIUM ULTRACENTRIFUGATION OF CHARGED SI-STE~IS
June, 1959
(assumed not to compress) from the distances between their outer faces in the assembled cells. A Baird interference filter assembly was used for isolation of the 546 mp Hg line, and photographs were made on Eastman Spectroscopic IID plates. Temperature, controlled by equicment now standard with the machine, was 25.0 i ca. 0.1 . Refractive index increments were determined with a Brice-Phoenix Differential Refractometer. Densities were measured pycnometrically. Sodium erchlorate solutions were prepared by neutralization of 6c1o4with NaOH; concentrations were established by density measurements.12 Concentrated bismuth stock solutions were prepared by dissolving Bin08 in HClO4 solutions, and characterized by bismuth and perchlorate (precipitation of tetraphenylarsonium perchlorate) analyses. All chemicals used in these and other preparations were C.P. or reagent grade. The composition of the N&MoO,. 2H20 used in pre aration of solutions was checked by precipitation of PbhdOd.
Results and Discussion In order to test the performance of the equipment and to see how far some real systems depart from the assumptions made in the derivation of equations, we have carried out studies of three solutes having known molecular weights: Na2Mo04in 1M NaC104, BiOClO4 in 1 M NaC104, and BaClz in 1 M HCI. The centrifugation results, corrected for cell distortion, are presented in Fig. 1 as deviation plots of log n* - (S/2.303)z2. The points are computed from the value of n* at individual fringe positions, and S is an average slope, d In n*/d(z2), computed for all the points. Since these solutes are monodisperse, graphs of log ct2 0s. ~2 should be linear, except for charge and activity coefficiellt effects- The graphs of log n* US. 9'2 are close enough to linear for the deviation plot in Fig. 1 to be a useTABLE I CENTRIFUGATION CONDITIONS, REFRACTIVEINDEX VOLUME DATA Molarity component 2
0.1029 .lo03 .0558
.0535 .0497 .0255 .0249
AND
kz' for
Apparent Centrifspecific ugation (546 m p ) vol., cc. conditions' NamMoO4-1 M NaCIOd z' = 0
0.0334 .0336 .0336
0.168 .177 .166
.0336 .0335 .0336 ,0336 .0335b
S
A1 A2 A3 A4 A5
0.00607 .00608 ,00603 ,00597 ,00631
B3 Bl B5 B5
0.02598 ,02488 .03012 .03508
.170 .167 .156 ,159 16gb
.
BiOClOrl M NaCIO.
0.1004 .0602 .0200 .0080
0.0300 .0299 .0314
0.155 -153 .151
.0300b
.154b
BaCh-1-M HCI
0.1009 ,0575 .0350
0.0295 0.143 A4 0.00794 .0295 .I43 A5 .00885 .0297 .138 A1 .00832 .0295* 143b a Letters: speed of rotation: A, ca. 27,690 r.p.m.; B, ca. 14,290 r.p.m. Numbers: cell number. bused in computations.
.
(12) H. E. wirth and F. N, Collier, iW., 72, 5292 (1950).
~
$
.
g -
5 ,
'791
627 629
- 3 853 - 3 655
- 5 334
...,
e-*-*
7.m. -3
-3 346 35
40
45
50
54
x2.
Fig. 1.-Deviation plot of ultracentrifugation results, interference optical system (values of s listed in Table I). (Each fringe position is represented by a point.) One point at z* = 39.3, not shown, from centrifugation of 0.008 M BiOC104 was about 5% in n* below the average line for the other points.
in principle d log n*/ ful test of precision, d(z2) is only approximately constant. Most of the points scatter less than 3=0.5% in n*. The precision is much better than with schlieren optics. (See e.g., the comparable graph in ref. 3, Fig. 1). Values of S and the exDerimenta1 conditions are summarized in Table I. 1. Volumes and Refractive Index Increments.The results of refractive index and apparent specific volume measurements of sodium molybdate, barium chloride and bismuth oxyperchlorate in the supporting .electrolytes are summarized in Table I. In the case of NazMo04,values are given for several solutions for which centrifugations are not reported. Within the accuracy of our measurements, no dependence of specific volumes or refractive index increments on concentration was observed, and average values were used in the centrifuge computations. Since the precision of these measurements depends on the concentration of solute, the averages were weighted with factors proportional t o concentration. Further , (averaged) apparent specific volumes were taken as the partial specific volumes. For BiOC104 in 1 M NaC104, the present value of the volume agrees with that reported earlier.'a * Literature values were used for partial specific volumes and refractive index increments of the supporting electrolytes. For NaC104,ka = 0.007414 (13) R. W. Holmberg, K. A. Kraus and J. 8. Johnson, ibid., 78,5506 (1950). (14) H. Kohner, Z . physik. Chem., B1, 427 (1928).
These measure-
792
JAMES S. JOHNSON, GEORGE SCATCHARD AND KURTA. KRAUS
Vol. 63
4% in N for such a light solute indicates a substantial improvement in precision over schlieren optics, with which a coherent result would be diEcult to obtain, with cells of the same thickness. About 2% of the variation of the mean from N = 1 0.20$. may be attributed to neglect of the contribution to n* of pressure differences between the two compartments.’l The rest lies easily within the range either of possible experimental errors or of activity coefficient effects (see below). 0.15 t3. BiOC104 in 1 M NaC104.-A system more of the type we have ordinarily studied is hydrolyzed Bi(II1) in NaC104. Earlier inve~tigationsl~ with the schlieren optical system indicated a degree of polymerization of 5-6 and some complexing of 0.10 io perchlorate ions by the polymer. Since then, X0 0.5 , f.0 2 . ray scattering measurements on concentrated Fig. 2.-Degree of polymerization N of BiOCIOI in 1 M BiOC104solutions, carried out in this Laboratory,l’ NaClOd as a function of charge 2 ’ . have strongly suggested that the Bi atoms are ar-_ --- - ~ 0 6 ranged octahedrally, which would indicate a hexamer. Olinll* on the basis of e.m.f. measurements l 0 0,101 M BaClz 1 carried out in SillCn’s laboratory, . . also favors this 0056 M BaCIp A 0035 M BoCI? 1 species. -0 7 We have centrifuged BiOC104 solutions of (initial) concentrations 6,008 to 0.100 M in 1M NaC1o4 I supporting electrolyte. To prevent precipitation N the solutions contained also 0.01 to 0.02 M HC104 1 2 0 r +. in excess of the composition BiOC104. Under these conditions the hydroxyl number is two, and-exi.00 c cept for perchlorate complexing-the predominant species is ( B ~ O ) N + ~ . ’ ~ The results of the centrifugations are summarized 0 00 L as plots of 1/N vs. z’ (Fig. 2). Previously, similar I I - -1 0 0.5 1.0 1.5 2.0 2.5 results have been presented as N us. z’ graphsa; =. ’. the reciprocal 1/N is used here, since the graphs Fig. 3.-Degree of polymerization N of BaClz in 1M H C are more linear. The curves for all concentrations computed as a function of charge 2’. approach one another most closely for N = 6 and 8 = 0.373 cc.12 were used and for HC1, Jc3 = (within a spread of about 2% in N ) . The corresponding value of 2’ indicates that about two per0.008316 and fj = 0.533 cc.lB chlorates are complexed per polymeric aggregate TABLE I1 and that hence the average formula of the species REPRODUCIBILITY OF MEASUREMENTS +4. is BisOa (C104)~ 4. BaClz in HC1.-Three solutions of BaClz (5-cell rotor, wedged cells, 0.056 M NazMo04-l M NaC104) (initial concentrations 0.035 to 0.100 M ) were cenCell h1(cmJa hz(cm.)b N&2 trifuged in 1 M HC1 supporting electrolyte. The 1 1,202 1.195 1.07 results are presented in Fig. 3 as graphs of 1/N 2 1.201 1.197 1.06 computed as a function of assumed charge x’. The 3 1.198 ’ 1.198 1.05 curves should, of course, cross at N = 1 and x’ = 4 1.193 1.197 1.03 2. The values of N computed for x’ = 2 are 5 1.196 1.203 1.06 about 5% high. The discrepancy is not far outside Theoretical degree of polymerization N+2 = 1.00. a Thickness of “solution” compartment. Thickness of of probable experimental uncertainties for such a “background” compartment. low molecular weight solute. However, we shall examine how far it might result from activity coef2. Na2Mo04in 1 M NaC104.-Reproducibility effects since in this case data on the pertiof the experiments and adequacy of the procedure ficient used to diminish effect of cell distortion were tested nent three-component systems are available.lg The by centrifugation of identical solutions of 0.056 M activity coefficients enter principally in two ways, as d In Gz/d(x2)in the sedimentation equation of the NazMoO4-l M NaC104 in the five cells of the “Ana- component of interest and as a correction for the lytical G” rotor. The values of N computed for differences in the sedimentation of the supporting the known charge, z’ = 2, are given in Table 11. electrolyte in the solution and background comThe values fall within k0.02 of N = 1.05. (If com- partments (d In y+Hci/dz2). puted for z’ = 0, N is about 0.5.) A spread of only Estimates of activity coefficients were made with menta were for Na-D light, but there is little change of an/& for this the equations salt with wave length. 0.251
I
1
I ~
,4
>I:
,
-
--0
L
(15) 0. R. Howell, J . Chem. Soc., 2039 (1927). The value given is anlac for 1 M HCl, interpolated for 546 mp for measurements a t other wave lengths. (16) H. E. Wirth, J. Am. Cham. Soc., 62,1128 (1940).
(17) H.Levy, M.D. Danford, P. Agron and M. A. Bredig, unpublished results. (18) A. Olin, Acta Chem. Scand., 11, 1445 (1957). 16, 5924 (1954). (19) H.8. Harned and R. Gary, J . Am. Cham. SOC.,
m
June, 1959
hABmA
EQUILIBRIUM ULTRACENTRIFUGATION OF CHARGED SYSTEMS
f bBBmB f dAABmA' f
2dABBmAmB
f dBBBm2B (23)
where b and d are constants or interaction coefficients20; A and B indicate HCI or BaClZ,respectively; p is the ionic strength ( Z m i x i z / 2 ) and m the moldit~y. The coefficients i? these equations were I osmotic coefevaluated from tablesz1of Y ~ H C and ficients, 4, of BaC12,and the constant aI2for HCI in BaCl2.I9 The values of Y*HC~ and 4BaCI2 in water were represented by equations having a DebyeHuckel term, in addition to terms linear and quadratic in molalityz2; a12 was assumed to vary linearly with p . Correction for d In Gz/d(xz) was found to lower N (computed for x' = 2) by ea. 3% for 0.035 24 BaClz; by ea. 4y0for 0.058 M BaClz; and by ea. 6% for 0.1 M BaCL Corrections for the effect of digerences in d In -y&/d(x2) between solution and background compartments were smaller. The N for 0.1 M BaCls was raised by about 1%; corrections for the other two solutions could not be estimated accurately enough for application but appeared to be less. I n addition, corrections for the effect of differences in pressure on the two compartments'l were estimated to decrease N by about 2% in all cases. The net effect of these corrections lowers the computed values of N from ca. 1.05 to 0.97 for 0.1 M BaCI2, to 0.99 for .058 M BaCIz, and to 1.00 for 0.035 M BaC12. It appears that in this case, differences between known and observed values of N are indeed of approximately the magnitude expected from activity coefficient variations. 5. General Discussion.-We may conclude from the coherence of the results in the three examples that considerably more accurate estimates of the degree of aggregation of charged solutes may be obtained with interference optics than with schlieren optics. Further, with proper corrections, use of wedged centerpieces allows simultaneous centrifugation of five solutions with no significant loss in accuracy. In equilibrium ultracentrifugations with deep cells,z3this permits a substantial (20) The values of the coefficients of equations 22 and 28 are: = 0.210, dAAA 0.012 (from y * H C l in water); bBB = 0.2241, dBBB= 0.0468 (from r$BaClz in Water); bAB = 0.2478, dAAB = 0.0327; dABB = 0.0882. (21) R. A. Robinson and R. H. Stokes, "Electrolyte Solutions," Butterworths Scientific Publications, London, 1955, p. 476, 471. Original source: HC1, H. S. Harned and R . W. Ehlers, J. A m . Chem. A'oc., 56, 2179 (1933); BaC12, R . A. Robinson, Trans. Faradav Soc., 36, 735 (1940). (22) G . Scatchard and L. F. Epstein, Chem. REUS., SO, 211 (1942). (23) Recently, techniques have been described for equilibrium ultracentrifugations with columns of solution shallow in the radial direction (K. E. Van Holde and R . L. Baldwin, THISJOURNAL, 62, 734 (1958)). B y this device, much less time is required to attain equilibrium. bAA
793
reduction in the time required for a single set of results, particularly since with digital computers the time needed for computations is greatly decreased. It has been stated earlier and reiterated in the examples that determination of the degree of aggregation implies simultaneous estimation of the charge of the ions. I n many cases this implies that through equilibrium ultracentrifugation some information regarding complexing reactions of the solutes with ions of the supporting electrolyte may be obtained. Thus, we concluded that the Bi(II1)hexamer (BiaOe+6)is to some extent complexed by perchlorate ions, Further, equilibrium centrifugations gave the expected result that Ba(I1) is not complexed by chloride ions in the medium studied and that the molybdate ions are not complexed by sodium ions. In an earlier studyz4of considerably less precision, since it was carried out with schlieren optics, we pointed out that In(II1) in bromide solutions appeared to be considerably less complexed than some have stated. One method of estimating x' involves determination of an intersection of the curves of N us. assumed charge z' (see Figs. 2 and 3). With right solutes in light supporting electrolytes these curves are too nearly parallel for this to be a sensitive procedure. In such cases it is sometimes better to determine 2' a t the point where such curves cross the proper value of N . For this to be effective the activity coefficient derivatives, particularly the derivative d In Gz/d(z2),should be small. Often it is more effective to centrifuge in a relatively high molecular weight supporting electrolyte since then the curves of N us. z' will be steep and more sensitive to the concentration ratio C'Z/C~. If one deals with a monodisperse solute, the charge z obtained should equal the average charge of all species ( z = Z x i F i where x i is the charge of the species i and Fi = m i / 2 m i is the fraction of the solute as species i). In principle, knowledge of the average charge as a function of ligand concentration permits computation of complex constants; however, a t present it appears that such an objective is not practical. Experimental difficulties, as well as expected variations in activity coefficient terms, make estimates of x' too uncertain for evaluation of complex constants. I n t.he BaClz and NazMo04 experiments, without activity coefficient corrections, z' would be estimated with an accuracy of approximately 0.2 in charge units. These experiments were carried out a t moderate ionic strength, in the region where most activity coefficient us. concentration curves have a minimum. It is difficult to predict a t present if results useful in the study of complex reactions may be obtained a t substantially higher ionic strength, where activity coefficient derivatives should be larger and more difficult to estimate. Acknowledgment.-We are indebted to Miss Neva E. Harrison for technical assistance. (24) J. 8. Johnson and K. A. Kraus, J. A m . Chem. ~ o c . ,79, 2034 (1967).
