Acknowledgment.-We should like to express our appreciation to the United States Atomic Energy
Commission for financial support under Contract AT-(40-1)-2733.
THE DETERMINATIOh- OF DESSITY DISTRIBUTIOSS ASD DEXSITY GRADIESTS I N BISARY SOLUTIOSS *4T EQUILIBRIUM IS THE ULTRACESTRIFUGEl BYJAMES B. IFFI',~ DOXALD H. V O E TAND ~ JEROUE VIXOGRAD Gates and C'rellzn Laboratorzes of C h e r n i ~ t r yCalifornia ,~ Inslzfzcfe of Technology, Pasadenu, C'alifoi niu R e c e a e d December 1 4 %1960
The calculatioii of density gradients and density distributions ill binary solutions a t sedimentation equilibrium in an ultracentrifuge is presented. The differential equat'ion describing sedinient'ation equilibriuni is p( p ) d p = d r d r where p is a function of the t,emperature and t,he activity, molecular weight, and partial specific volume of the solute at a density p. Values of p have been calculated from published dat,a for densit,y and activity over the entire concentration range for CsCI, KBr, RlsBr, RIKY, LiBr and sucrose a t 25.0". For each binary system p ( p ) has been expressed as a cubic polynomial in p. The integral form of the above equat,ion was then solved for the integration constants using a digital computer for the first' four salts ment.ioned above. The integrat,ion constant is the radial distance a t which the initial density of the solution occurs. This is i,he isoconcent,rationposition. Experimental confirmation of the density gradient and of a density distribution for CsCl are presented.
Introduction The aiialysis of results obtained with the recently developed procedures for sedimentation equilibrium of macromolecules in a density gradient in the ultracentrifiige6 requires a knowledge of the density and the gradient of density in the liquid column. For a single dilute species, the apparent molecular weight1'of the macromolecule is given by the relation
where dn/dr is the measured refractire index gradient in the schlieren optical system and dpldn is obtained experimentally by refractometry of solutions of known composition. In the physical chemical method, the density gradient is calculated from the relation for sedimentation equilibrium in a two-component system
where po is the buoyant' densit'y of the solvated species of molecular weight, AI,,,, u is the standard deviation of the Gaussian concentration distribution of the nnacromolecule, (dp/dr)o is the density gradient) a t the position ro, t'he cent'er of the Gaussian band, artd w is t'he angular velocit'y. In previous work, the buoyant density has been evaluated by asserting that the isoconcentration dist,ance, re, and the corresponding original density, pe, are to be :found a t the radial ceiit'er of tmhe liquid column. For bands near the center of the liquid column, to a good approximatioii
where 111, a and fi are the molecular might, activity a i d partial specific volume of the solute species. In this paper a method is presented for evaluating re. The method is based upon integrating a form of equation 5, using tlie conservation condition to obtain the integration constant. A. a part of the procedure, it has been necessary t o tabulate the values of the dpld? as a function of p and to express these as polynomials. The results for a series of solutes in water are hrst presented. The results of the integrations are given in the latter portion with some experimental coiifirmation of the results.
PO = pe
+ (dp/dr),Ar
(2)
where Ar is the displacement of the mode froin the center of the liquid column. The density gradient is evaluated by oiie of tx-o alternat,ive n ~ e t h o d s . ~The optical method makes use of the relation (1) (a) This investigation was supported in part by Researcli Grant H-3394 from tlie National Institutes of Health, U. S. Public Health Service; (b) this work was presented in part before the 33rd National Colloid Symposiu:n. Division of Colloid Chemistry, American Chemical Society, Minneapolis, hlinnesota, J u n e 18 t o 20, 1959. (2) U. S. Pubiic Health Service Research Fellow of the National Cancer Iiistitiite, 19,59-1960; Division of General Medical Sciences, lQG&lSFI. (3) Sulrpurted :'or 0ne siiinmer b y t h e Kational Science Foundation Cnderpra:luate Iiesearcli Partiripation Program, N.S.T.-C7954. 14) Contribiiticmn No. 2li.5. ( 5 ) >I. Rlescison. 1:. JV. hl and .J. Vinoarail, P i m . .Val/. ..Icad. Sci., 4 3 , ?.XI (195:7). ((3) R . .I_. Bxldrr-in, ? b i d . , 45, 93'3 ( 1 9 X ) . ( 7 ) hf. Rieselsun, 1'11.1). I)issertation, California Institute of Teciinology, lt57.
(3)
Results and Discussion The Calculation of the Density Gradient.---For niiy binary solution, equation 4 may be TT ritten
where P(P)
=
d 111 a -dp-
IZ1'
m j L f r
The quantity /3 was evaluated by graphical methods from published dat,a of densitiesg9 and activity coefficients.l o The partial specific volume, e, was (8) "International Critical Tables," Vol. :i,lIcGra\~--lIill Book C o . , S e w York, h-.I-., 1938. (9) F. Rates a n d Assoviaten, " l ' ( J h I ' i I ~ l F l t l J ' , Saci:liarirrirtry m i l tilit S u g a r s . " Circular 440, Sational Rurcau [jf Standards, JVasliinptun 11. C . , 1942, p. 628. (10) R . .i. Robinson anti R . 1-1. Stokes, "Elc.ctrolj te Suiritiuns," Butter\\-nrilm. London England, 195.5.
,July, 1961
DESSITYDISTRIBUTIOSS A N D DENSITY GRADIENTS IS AN CLTRACESTRIFDGE 1130
evaluated by the chord method on a large scale plot of 1/p 1's. Z2, the weight fraction of the solute. Thus a graph of D 1's. p was obtained. Activities were calculated from data relating mean ion activity coefficient and molality. Corresponding density data were derived from the almost linear weight fraction us. density plot. The variation of dpld In u. with p was evaluated graphically by determining tangents to the 61 cs. ln a curve. Thc values of dp/d In a and D a t corresponding densities were tabulated and p ( p ) calculated. The results for C S C I , ' RbBr, ~~ RbC1, KBr and sucrose are given in Table I. The data for LiBr are given separately in Fig. 1. These results may be used directly to evaluate the density gradient in a phase of density p a t equilibriuin in the ultracentrifuge. T ~ B LI E VARIATIOA O F P
1.02 1.03 1.04 1.05 1.OB 1.075 1 .08 1.10 1.12 1.125 1.14 1.15 1.175 1.20 1.225 1.250 1.275 1.30 1,325 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85
Sucrose
a WITH SOLUTION
KRr
8.091 6.780 5.605 4.643 4,019
DENSITY
RbBr
RbCl
6.729
9.817
3.643
5.532
2.536
4.109
2.491
2.122
3.445
1.984
1.772
3.172
1.715
1,635
3.083
1,546
1.528 1.434 1.372
2.777 2,334
1.430 1.346 1,286 1.245 1.216 1.197 1.190 1.190 1.199 1.215 1.236
CsCl
7.496 3,449 3,237 3.121
6.121 5,229
3,091 4.594 4.151 3.848 3.637 3.469 3,330 3.213 3.112
I n this treatment, the effects of pressure havc beeii iieglected. Seglecting these effects is equivalent to stating that the solution is incompressible and that p ( p ) is the same a t atmospheric pressure and a t the pressures in the rotating liquid column. The solution, however, is not incompressible. The compression gradient a t density p is K p W r , where K is the compressibility coefficient. ilt the center of the sectorial cell for CsCl solutions, this compression gradient is of the order of 10% of the density gradient due to salt redistribution. The effects of pressure in density gradient systems are to be presented in a future publication. ( l o a ) T h e reciprocala of p ( p ) f o r CsCl h a l e been calctilated b y X. T r a u t m a n . Arch. Bzochtm. Baophys , 8'7, 289 (19601, by a sonienhat different procedure T h e d a t a for CsCl in Table 1 h a r e heen cornpared with Traiitman's d a t a a n d differ by less t h a n 1 %
.
10 I 1.o
I
1
I
I
I
1.2
I
I
1.4
1.6 1.8 g. ~ r n . - ~ . of p with solution density, LiBr p,
Fig. l.-I-aristion
The latter Ttatement regarding the effect of pressure 011 p ( p ) appears to be valid, as is demoiistrated in the experimental section, a t the center of a 1.287 g. em.? CsCl solution ( 2 57 molal) in a sector cell a t 56,100 r.p.m. The density gradient calculated from equation 3 agrees n ith the gradient determined by schlieren optical methods to withiii 0.7Cj,. In this experiment, the density gradient due to the redistribution of CsCl and H20 at equilibrium is evaluated by subtracting the schlieren elevation photographed just after reaching full speed from the elevation obtaiiied a t equilibrium. This procedure substantially eliminates the effects of pressure on the liquid, the cell windon. and the centerpiece. The deiisitv di$tributions in CsCl and RbBr solutions and in KBr and RbCl solutioiih should be similar in the density region, 1 . I d t o 1.30. In adclition, the former two salts produce higher density gradients than the latter tiyo in their coinmoil deiisity regions. The Construction of the Polynomials p(p).I n order to use the tabulated values of p ( p ) most k
effectively! a polynomial ave
(yi)
=
J=o
ajtj, was
fitted to each set' of data by the method of least squares. The polynomials were constructed by orthogonalizing the linearly independent functions 1,x,x2,.. . ,xk by the Gram-Schmidt orthogoiialization procedure. This method of curvc fitting is described in Bennett and Franklin. Tables of the iiumericul values of the f ' s are given by Fisher aiid l-atcs.'? Thus, polynomials of the form B(pj =
Po
+ +P d + .. . +i 3 d PIp
(6)
were obtaiiied for the solutes previously inentioned. The values of the first' four cocfficieiits are giveii in Table 11. The accuracy with which the polyiioniial B ( p ) gives the curve represented by the original p z's. p data depends upon IC, the degree of the polynomial and upon a, the number of points used to calculate the polynomial. In this work, third degree polynomials were used. In the case of CsC1, a fifth degree polyrioinixl reduced the maximum deviation of the p values calculatcd from the polynomial equatioii from the /? values originally derived from (11) C. A . Brnnett a n d N. I,. Franklin. "Statistical .Inalysis in Chemistry a n d the Chemical Industrg-," John TTilcy a n d Sons, 3 - c ~ Y . , 1954. PI,. 255-264. (12) R . A . Fisher a n d F. l a t e s , "Statistical Tables," Oli:-er and Boyd, London, 1957, pp. 90-100.
,J. B. IFFT,D. €I. VOETAND JEROXE VISOGRAI)
1140
Salt
a
Density range
BO
TABLE I1 POLYNOMIAL COEFFICIEXTS~ el x 10-9 x 10-9
et x
-85 3724 -1,416 8024 -2,878.2226 - 2,280 4.3& -2,320 1077
47.9726 620.9619 1,222.1450 '366.7593 860.6168
CCl 1.15 -1.85 1.05 -1.45 RbRr IlbCl 1 . 0 5 -1.40 KBr 1,075-1 325 Hucrosv 1 02 -1.14 Sce tcxt for polynomial equation.
1-01.6.5
es x 10-9 - 10 4312
10-9
51 7764 1,079 2080 2,264 0729 1,802 8082 2,085 6993
-273 -593 -475 -625
7280 3408 9704 0000
where c is the concentration a t u(r), ce the initial concentration, and B is the total liquid volume in the cell. Since these concentrations are in units of masslvolume
2.0
2.1
c =
( 9)
zzp
Examination of the relation between ZZand l / p shows that these quantities are linearly related over the maximum density increments of 0.1 t o 0.2 g. cm.-, encountered in a gradient column. Combiniiig this linear relation
2.2
2.0 m.
= ki(l/p)
2 2
I
2
+ kz
'
(10)
n-here ki and k z are coiistants, with equation 9 and with the conservation relation 8, we obtain
x 1.8
sov
91
P
1.6
Pe
dv
(11)
= __-
V
For sector and cylindrical cells, equation 11 takes the forms
1.4
1.2
rrb
arid 1.(I 1.2
14 p, g. cm.-3.
1.8
lc,
PI!
activity and density data from 3% in p to l?. The third degree polynomial was adequate to give the values of the ieoconceiitration points, re, to withiii 0.17 exccpt for RbBr. The value of n used depended on the shape of the p ( p ) curve and varied from 1.5 to 21. The maximum deviation of the p values calruluted from the third degree polynomial from the corresponding original p values was rf3 1% for KBr, *3y0 for CsCl and sucrose, .tO%l, for IibCI, and =k12y0 for RbBr. The agreement between the derived relations mid the original data is illustrated in E'ig. 2 for CsCI. Integration of the Polynomial Expansion of the Sedimentation Equilibrium Equation.-Equation 5 represented as a third degree polynomial may be integrated to give a fourth degree polynomial. ~
-
fc4)
3-
aJf3
+
- P,3) + CuZ(P2 - P B 2 ) - p e ) + w2(re2 - r2)/2
ml(p
=
0 (7)
where the a ' s are constants. The integration ronstaiit, re, WLS evaluated using the outside condition of coriservation of mass of the solute
sovv
c du
'C
=
(8)
dr
1's
(lib)
- rd
(Tb
Fig L'.--Coinp:mson of polynomial p( p ) curve and oriqirinl p( p ) curve, CsC1. Solid line, original p( p ) curve; dashvtl line, pols nomid p( p ) curve.
W(P4
.
= ___
respectively, where r, and ?-b are the radii a t the top and bottom of the cell. The solution of equation 7 incorporating the conservation conditions l l a or l l b was performed with a Burroughs 205 digital computer. Input data for the computer are the constants ai, CYZ, cy3, cy4, pe, o, r,, rb, n, the number of evenly spaced intervals in the liquid column, either 4 or 6, slid R,(1) and Re(2),two bracketing estimates for re. Equation 7 rewritten for a specific p1 and ri is a4pi4
+ a m 3 -t
+
a?pI2
alp,
a4pe4
=
+
a2 (TI2
anpea
- r e 2 )+
+ mzp,* +
a l p , = -a0
The constant, cyo, is romputed using Re(l) and one of the values of rl. Then the quartic polynomial i q solved for the corresponding pl. Each of the n 1 points (Ti, pI) is computed similarly and Simpson's rule used to compute the integral Il = l p r dr, in the case of the sector. The procedure is repeated for Re(2). A linear interpolation is used to find R e @ ) . I , is calculated and compared with Io = Pe(rb2 - raZ))/2. The procedure is continued with parabolic interpolations until satisfactory agreement is reached between I , and lo,whereupon the last set of values ofp, and r, and the final value of T , are printed out. The program was operated for aqueous solutions
+
July, 1961
1)ER'SITY
DISTRIBTTTIONS .4ND DENSITY GR.4DIENTS
of CsC1, RbC1, RbBr and KRr for both sector cells and cylindrical cells containing 2, 3 and 5 ml. of solution over a range of solution densities and for several angular velocities. The limiting radii were chosen to correspond to a 0.70-ml. liquid volume in a 12-mm. 4' sector in the analytical ultracentrifuge or to the above volumes in 5 ml. plastic tubes in the SW-39 swinging bucket rotor in the preparative ultracentrifuge. Representative density distribution data are given as follows: Fig. 3 presents the data for CsCl in cylinders of vaxying length and initial density. The distributions are as expected in that they become more linear as the cell lengths decrease and they are steeper for the higher pe's reflecting the higher gradients. The maximum density difference which can be obtained for CsCl is seen to be about 0.4 g. ~ m . - ~ . Figure 4 gives the distributions for the four salts in 3-ml. cylinders a t varying angular velocities. The slope of the curves increases with increasing velocity as required by the density gradient expression. The isoconcentration point is not invariant with changes in o as is suggested by the small scale plots, Fig. 4, but varies as shown in Figs. 6 and 7. The numerical values for the CsCl density distribution in sectors are given in Table 111. TABLF: 111 DmsIm DISTRIBUTIOXS CsCl Sector
-W,
p .
1.2
1.3 1.45 1.6 1.7 1.2 1.45 1.7 1.7
r.p.m.
