Feb., 1953
INTERACTION OF ALKALI IONSWITH LINEARPOLYELECTROLYTES
THE INTERACTION OF THE ALKALI IONS WITH SOME LINEAR POLYELECTROLYTES BYARTHURVEIS
189
.
Research Division, ATWUTand Company, Chicago, Illinois Receiued April 1, 1961
Small ions in solutions of charged high polymers are immobilized to some extent by the'electrical fields surrounding the large molecule. This immobilization, "binding," is readily shown by conductance studies and accurately measured by equilibrium dialysis experiments. Conductimetric titration curves are particularly revealing. Studies were made of the interaction of sodium and potassium ions with arabic acid, agar acid and nucleic acid. In unbuffered arabic acid systems at Oo, it was found that binding was a function of the free cation concentration. Binding increased rapidly for both cations in the range from 5 X lo-' to 3 X 1 0 - 3 N sa!t and then leveled off to a constant value. Within experimental error sodium and potassium ion binding were identical at high salt concentrations. The maximum amount bound in each case, expressed as equivalents of cation bound per mole of arabic acid was 78 & 8. In terms of equivalents of polymer acid the binding quotient was 0.20 rtr 0.02. Qualitatively similar results were obtained with agar and nucleic acids. Alkali ion, nucleic acid binding increased with increasing pH. Donnan effect calculations for arabic acid systems showed that binding could be qualitatively explained in terms of that theory. Osmotic pressure measurements, however, indicated that simple Donnan calculations were not adequate. The interaction is discussed in terms of the polyelectrolyte configuration.
Introduction The electrostatic and statistical forces governing the configuration of linear polyelectrolytes has been the subject of much recent study. The principal avenue of approach to a quantitative description of these forces has been based upon measurements of the physical properties of sqlutions of the polymers and the effects of added salts upon these The high charge density on the polymer coil has a detectable .effect on the distribution of small ions also in solution.*-" It seemed reasonable to study the polyelectrolyte charge and configuration from the point of view of direct measurements of polymer-small ion interaction.
even if there is no electrostatic interaction between the charged groups. The equilibria involved and their general equilibrium constants are
The concentration of any ion species is then
I. The Ionization Constants of an Acidic One may assume that each similar functional group Polyelectrolyte on the polymer has the same intrinsic dissociation The polyelectrolytes studied all contained weakly constant, k. However, we are not interested in acidic functional groups. It was, therefore, essen- individual functional groups but in polymer ion tial to have some idea of the dissociation constants, species. For the first dissociation constant, Kn, Kj, of these groups as a function of the total polymer there are n individual ways in which the n acid charge. groups can dissociate, each group having the same Overbeek4 and Katchalsky, Kunzle and Kuhn,B "probability" k of dissociating or have suggested similar equations for K i , based k:, = nlc (2) upon an intrinsic dissociation constant, k , and a For the second dissociating group there are correction term for the electrostatic free energy of the charged coil.' These treatments take no acCi" = n(n - 1)/2! (3) count of the fact that the apparent dissociation ways in which species (Hn-lA) can dissociate to constants of successive ionizing groups will depend upon the number of previously ionized groups give the species (Hn-2A). Suppose that each acid group is labeled, then for the dissociation of the (1) Parts of this paper presented at the 7th Annual Southwest particular molecule of species (Hn- 1A) with funcRegional A. C. S. meeting, Austin, Texas, Dec. 1951. Abstrabtsd tional group S already ionized, S ( H ~ - ~ A to ) , give from a thesis submitted to the Graduate College of Northwestern University in partial fulfillment of the requirements for the Degree the particular molecule of species (Hn-2A) with of Doctor of Philosophy, 1951. we) ,have groups S and T ionized, S T ( H ~ - ~ A
(2) F. T. Wall and E. H. deButts, Jr., J . Chem. Phys., 11, 1330 (1949). (3) J. H.Hermans and J. Th. G. Overbeek, Rec. trau. chim., 67, 761 (1948). (4) J. Th.G. Overbeek, Bull. aoc. chim. Belg., 61, 252 (1948). (5) A. Katchalsky and J. Gillis, Rec. trav. chim., 68, 879 (1949). (6) A. Katchalsky, 0. Kunzle and W. Kuhn, J . Polymer Sci., V, No. 3, 263 (1950). (7) See Symposium on Polyelectrolytes and Complex Ions, TEIS JOURNAL, 66, 1-140 (1952). ( 8 ) E. Hammarsten, Biochem. Z.,144, 383 (1923). (9) 0.Wilander, Skand. Arch. Physiol., 89, Supplement XV (1938). (10) H. Svensson, Arkis Kemi, Minerol. Qeol., 22A, No. 10 (1946). (11) W. Kern, Macromol. Chem., 2, 279 (1948).
(4)
Thus K*l
E
(ways of making pairs ST) (ways of making S) STK- 1
= -n - 1
2
IC
ARTHURVEIS
190
Vol. 57
Generalizing, we have
tributed uniformly throughout the coil. The functional groups may be considered as the centers of small sub-spheres, all packed into the polymer sphere. Assuming hexagonal or cubic close packing Wall and deButts2 have developed an ionization constant function for a polymeric acid which in the of these sub-spheres, each group would have twelve case of no electrostatic interaction between charged nearest neighbor groups. A rough calculation groups reduces toequation (7) for the fraction ionized for a typical polyelectrolyte, arabic acid, shows that the distance between any functional group of a monobasic acid of dissociation constant K . and those in its nearest neighbor layer is on the f = K/[Hl K (7) order of 3 X lo-' em. The electrostatic free If one defines f as the fraction of acid groups which energy of ionization of a particular carboxyl group are ionized, then for the polymer acid one can show in arabic acid could then, as a first approximation, that be evaluated as the work required to place a single negative char&, - E , o n the group-in 2KnKn- 1 [ H A ] .+. , + i G1K,[&E&! + j=o' [H]i "' the presence of a charge - Z E on the [HI =nearest neighbor layer of carboxyl (8) i-1 n Kn[HnAl +. . . r zr K,I ; ; :[ . .1 groum. This calculation. similar to [[H.AI +[HI i=O --I chat bf the Guntelberg charging procwhere both bracketed terms are closed series ending ess often incorporated in the derivation of the a t the term i = n. From equation ( 6 ) it follows Debye-Huckel equation, renults simply in equation that (13). . ,
+
,
+.
$
K,-i = n(n
j=O
-
1). . . (n
(i
+ l)!
- i),
]
+
which reduces to equation (7). This end result is analogous to that given by Wall and deButts for the situation of no electrostatic interaction but it allows one to estimate the intermediate values of Kn- i without recourse to an equation containing three arbitrary parameters. Since one may write4 the complete ionization constant as
P
+
AF'eiect
(9)
When values from equation (9) are substituted in equation (8) it can be seen that a term by term differentiation of the denominator of (8) with respect to [HI is equal to -1/[H] times the numerator and that denominator is the expansion of the binomial (1 Ic/[H])n. Therefore
7
=
2Dd Z€2
(13)
where Z = charge on the nearest neighbor layer of carboxyl groups; d = distance between carboxyl groups; D = dielgctric constant of the medium; E = charge on the electron. If free ions are also present the amount of work required for ionization will be diminished by the simultaneous formation of an ionic atmosphere about the newly formed ion. The effect of this ionic atmosphere will be identical with that given by the Debye-Huckel treatment, if we make the same assumptions as in that theory. Thus AF"
eleat, ionic, atm. due t o other than polymerbar-
_- -
.baxyla
€*K
2 0 (1
+ Ea)
(14)
where K and a have their usual significance. The use of the Debye-Huckel approximations severely restricts the validity of such an equation in polyelectrolyte systems. Nevertheless, as in other treatments, it is used for lack of a better approach. The total change in the electrostatic free energy on the ionization of a single carboxyl group is then
electrost&tIc interaction
the total free energy for any ionization step becomes AF'n-i
= -RT In k
- RT In
+ (:)
+
AFn-i,
eieot.
( 12)
Overbeek4 has deduced an equation 'for AFelect which evaluates this quantity in terms of the total electrostatic free energy .of the polymer coil. This gives results which appear to be too high.12p13 It seems reasonable to suppose that the electrostatic forces affecting the ionization of any particular group on such an open coil depends on relatively shorter range forces. If one assumes that the functional groups are distributed evenly along the polymer chain then, as a result of the random configuration of the polymer, the functional groups are probably dis(12) Andre 0 t h and Paul Doty, T H IJOURNAL, ~ 66, 43 (1962). (13) G. E.Kimball, M. Cutler and H.Samelson. ibid., 66, 50 (19533.
(15)
The complete expression for the free energy of ionization of the i'th carboxyl group becomes AFn-i
= -M'lnlc-ETlnn~+
z+
e2
1
[(F) + Ra) (1
2Dd (1
Kd]
(16)
+ Ka)
One may use equation (16) to calculate values of pK,i as a function of i. Values are given in Table I for arabic acid. For comparison, values of pK,-i have also been estimated by combining Overbeek's electrostatic free energy equation, equation (17), with equation (12). AFeiect
= (0.434)
ie2 1 -G5 DkT R [ 1 + 0.6KR + 0.4R2R2j
(17)
where R = average radius of over-all polymer coil.
.
INTERACTION O F ALKALI IONS WITH
Feb., 1953
Potentiometric titration curves calculated from equations (16) and (17) will be compared with experimentally determined curves to indicate the magnitude of the electrostatic free energy of the polymer coil which probably regulates the polyelectrolyte-cation interaction.
LINEARY O LYELECTROLYTES
191
ter was used but several modifications were required in order to achieve the desired accuracy. Calibration curves were established for each analysis run. The analyses were accurate to +=0.025 X 10-3 equivalents/liter in the ran$e to 10-2 N salt and f 0.025 X equivalents/liter in the range 10-4 to 10-3 N salt, that is, to about I l . O % . A blank set was run along with each acid dialysis set. All data are reported in terms of the. difference in the acid tube and blank tube salt concentrations per equivalent of acid present.
TABLE I ELECTROSTATIC FREEENERGY OF IONIZATION A N D TOTAL 111. Results and Discussion p K , - 1 FOR ARABICACID A. Titrimetric Data.-It has been recognized 17,18 T = 250, p = 10-2, n = 245, R = 0.8X10-6 cm, d = 3 X 10-7, pK = 2.5, a = 4x10-8 cm." that a high molecular weight polyelectrolyte molei
eq. 15
AFeieot
eq. 17
P&-i
eq. 16
PKm-i eq. 17
2.35 2.02 2.59 3.21 3.69 2.90 3.20 4.17 4.69 3.56 3.69 4.89 5.35 4.05 4.36 5.68 6.27 4.84 Estimated from mean ionic radii of sodium ions and the 50' 100 130 160 190 200 220 230 240
5
AFeleot
141 340 461 580 700 739 819 859 898
591 1182 1536 1891 2246 2364 2600 2719 2837
carboxyl of acetate ions, data of Kielland.14
11. Experimental Procedures A. Preparation of the Po1yacids.-Each of the acids was purified from the commercially available salt by electrodialysis following the method of Thomas and Murray.'S Arabic acid, a polycarboxylic acid, was obtained from gum arabic; agar agar acid, a polysulfuric acid mono ester, was obtained fromagar;and nucleic acid, fromaSchwarelaboratoriespreparation of yeast ribonucleic acid. The acid solutions were lyophilized and the resulting solids could be stored under refrigeration for months without bacterial action. Subsequent solutions were made up on a weight basis. B. Conductimetric and Potentiometric Titrations.-A titration cell was constructed in which the conductimetric and p H data could be obtained simultaneously. The measurements were made with a standard bridge arrangement and a Beckman Model G glass electrode p H meter Base was added from a 0.2-ml. Pyrex micro-buret whose tip was immersed to a constant level in the solution of the titration cell. Additions could be measured and controlled to f O . O O 1 mt. Resistances were measured within &0.05%, p H to f 0.05 unit. All conductances are expressed in terms of reciprocals of the observed resistances. The cell was arranged so that it could be completely immersed in the constant temperature bath. The titration mixture was kept under a nitrogen atmosphere and nitrogen gas was used for stirring. The bath temperature was 25 i 0.02".
C. Equilibrium Measurements: Differential Dialysis. dialysis distribution Alkali Ion Distribution.-Differential measurements were made following the general procedure of Klotz and Curme.18 The dialysis membranes were Nojax sausage casing, " / a ~ " extended diameter. Two ml. of the polymer acid solution was used inside of each bag and 5 ml. of salt solution outside. All experiments were run a t 0" except for a few with nucleic acid at 36.7". Pure arabic acid and agar acid solutions were dialyzed against a pure sodium or potassium chloride solution. Since base was required to put the nucleic acid into solution, these solutions were brought to p H 4.5 with the appropriate base. All of the nucleic acid was soluble at this p H . Equilibrium could be reached in 6 hours in these experiments but all dialysis runs were allowed to shake overnight, for a total period of 12 to 15 hours. The dialysis bags were removed from the outside solutions at the close of this period and the outside solutions were then analyzed for sodium and potassium ion concentration. A Beckman flame photome(14) J. Kielland, J . Am. Chem. S o c , 59, 1675 (1937). JOURNAL,32, 676 (15) A. W. Thomas and H. A. Mnrray, Jr., THIS (1928). (16) I. AI. Klotz and H. G. Curme, J . A m . Chem. Soc., 70, 939 (1948).
cule itself contributes only negligibly to the conductance of its solutions except at high concentrations, all measured conductance being due to ambient counter ions. With such a background, one may compare the conductance of polymer solutions with no added salt and with salt added. If, as is true here, the experimental conditions are such that the conductance of a salt is essentially a linear function of its concentration, one may translate conductimetric data into binding data as follows. Measure the resistance, po, of the polymer acid solution as a function of degree of neutralization, i.e., construct a conductimetric titration curve with the appropriate base, then measure the resistance, R, of a polymer acid-salt solution in the same way. The conductance of the acid counter ions in the salt solution is then given by
where Bois the measured resistance of a pure salt solution. The binding of counter ions to the polyeledtrolyte a t any particular degree of neutralization, i, will then be proportional to [ ( l / p o ) i ( l / p ) i J . From either equation (17) or equation (15) one should expect that ( l / p ) i > ( l / p o ) i jf there were no counter ion binding, since the acid would appear stronger in solutions of higher ionic stsength. Typical data are shown in Fig. 1for the particular system of agar acid-sodium chloride-sodium hydroxide. The fact that the values of ( l / p ) are everywhere lower than those of (l/pa)i indicates the immobilization 'of counter ions. These readings are in fact, minimum binding values since from these experiments, no information is available on the binding of the counter ions present in saltfree polymer solutions. Representative conductimetric binding data for sodium and potassium ions evaluated in the manner outlined above are given in Table 11. That such data are not completely descriptive of the situation is shown by a comparison of conductimetric titrations of the same acid with both sodium and potassium hydroxides of the same concentration, Fig. 2. In the titration of a simple acid, such as hydrochloric, all values of ( l / p K O H ) i would be greater than ( l / p i q a o ~ ) i due to the higher specific conductance of the potassium ion. Despite the differences in the conductimetric titration curves, the pH titration curves are only slightly different for the two bases. These data me not in accord (17) H. Saverborn, "A Contribution t o the Knowledge of the Polyuronides," Upsalle, 1945. (18) D. Edelson and R.M. Fuoss, J . Am. Chem. Soc., 73,306 (1950).
ARTHURVEIB
192
VOl.
,
57
nesium and, sodium hydroxide titrations. On the other hand, Thomas and MurrayL5 titrated arabic acid to the same endpoints with barium and sodium hydroxides. 3 9 From the similarity of pH behavior, one might conclude that I 8 the alkali ions are those being E" 2 & bound and that nearly one equiva2 7. lent of counter ion is immobilized x 2 per equivalent of polymer acid 3 1 P 6 present. Any more quantitative statement does not appear to be justifiable from the conductance data alone. An arabic acid titration with 1 sodium hydroxide in the presence of 9.566 X N sodium chlo3 ride corresponds to the conditions 0 0.05 0.10 0.15 0.20 listed in Table I. In Fig. 3, pH Base, ml. titration curves calculated from Fig, 1.-Agar wid titrations, 2.5") base, 0.06220 N NaOH. Concentration of both sets of pK values of Table I added salt: (1. 0 ) zero; (2) 4.736 x N iX,CI; (3, 0 ) 9.566 X lods N are comDared with actual titration NaCI. data. fi is obvious that the agreewith those of Mukherjee and Ghoshlg who found ment between the experimental and calculated that for arabic acid the potassium hydroxide curves is best for equation (16), corresponding to titration required 5.6% more base than the sodium rather low values of the electrostatic free energy. hydroxide titrations. They report a deviation in Even in this case, however, the electrostatic free end-points amounting to 11.4% in comparing mag- energy of ionization term increases too rapidly as the total charge on the coil increases. 4
11
TABLE I1
hfINIMtJM CONDUCTIMETRIC B I N D I N G VALUER,
25"
I, 4.736 X 10-3 N NaC1; IT, 9.566 X IO-* N N.aCI; 111, 2.051 X 10-8 N KCl; I V , 5.902 X 10-3 N KC1
System
(!- - ') W
P
x
Equiv. cation bound Equiv. of 105 -AC X IO' polymer acid
Arabic acid, pl3 3 . 5 +
I
I1 I11
4.00 13.2 1.80 4.80
IV
2.05 6.79 0.815 2.17
0.31 1.02 0.12 0.33
.4gar acid, pH 3 . 5 +
I I1
111 IV
Nucleic acid, pH 4 . 0 + I I1 111
IV Nucleic acid, pH 5 . 5 + I I1 I11 IV
Nucleic acid, pH 8 . 5 + I
I1 I11 IV
5.60 14.8 2.3 3.8
2.88 7.61 1.04 1.72
+ appropriate base
0.043 1.13 0.16 0.26
1.8 12.1 1.9 4.5
0.926 6.22 0.860 2.36
0.13
2.2 8.7 1.2 4.2
1.13 4.47 0.544 1.90
0.16 .61
1.29
0.18 .54 .16 .39
0
0.05
0.10
0.15
0.20
Base, ml. Fig. 2.-Agar acid titrations, 25"; no added salts: (1) 0.06220 N NaOH; (2) 0.06220 N KOH.
.85 .12 .32
+ appropriate base
+ appropriate base 2.5 7.6 2.5 6.3
3.91 1.13 2.85
.08 .26
(19) S. N. Mukherjee and K. B. Ghosh, J . Indian Chem. Soc., 9 6 , 277 (1949).
0.2
0.6 0.8 1.0 Fraction neutralized. Fig. 3.-Arabic acid titrations with NaOH, 25': (1) ex erimental titration in 9.566 X 10-8 N NaC1; (2) c a h a t e d titration in 0.01 N NaCI, p k = 2.5, eq. (16); (3) calculated titration in 0.01 N NaC1, plc = 2.5, eq. (17) and (12).
0.4
Feb., 1953
INTERACTION OF ALKALIIONS WITH LINEARPOLYELECTROLYTES
0.00
193
1
2 4 6 8 Free cation concentration, N X lo8. Fig. 5.-Comparison of calculated and experimental binding curves for arabic acid: (1) calculated, C1 = 1.4 X equivalents per liter; (2) experimental; (3) calculated, C1 = 4 X 10-4 equivalents per liter. 0
in the ratio of positively charged groups to negatively charged groups on the nucleic acid molecule. St,enhagen and Teorel120have shown for a thymus nucleic acid (not strictly comparable) that the electrophoretic mobility increases from pH 2 to pH 6 and then rapidly levels off to a constant value above pH 6.5 or so indicating that the maximum charge has been reached a t this pH. Gulland and Jordan21 have concluded that from pH 2.5 to 6.3 one is titrating the amino groups of guanine, adenine and cytosine while from pH 8.0 to 12.0 one titrates the enolic hydroxyl groups of guanine and thymine. There appeared to be no significant differences in the binding of sodium ions and potassium ions by this method of analysis. A Donnan equilibrium analysis of this physiologically important system is not feasible with the limited data available.
m'
E 2
8 %
D "
s 3e 0.16 E
2;
.*d ' j
2 w" 0.08
W
0.6 0.4
.s 0.3
$5
."
s' s' 0.2 0.1
(C*")*= 0 (18)
under discussion. Cl is the equilibrium concentration of ionized acid groups, Cp the initial outside cation concentration and CZ"the equilibrium outside concentration. Donnan binding values have been estimated for C1 = 1.4 X and 4 X equivalents/liter, limiting values of C1 in the arabic acid binding experiments. Figure 5 shows a comparison between the calculated and observed sodium ion binding curves. One might use such comparisons to calculate CI, if it were not that binding of the same approximate magnitude has been demonstrated in systems without artificial membranes (conductimetric data). The data on the nucleic acid systems are summarized in Fig. 6. The marked dependence of binding on the pH probably reflects the decrease
0.0
0
2 4 6 8 10 12 14 16 Free cation concentration, N X lo3. Fig. 6.-Binding of nucleic acid: (1) binding with NaCI, 0", pH 3.9; (2) binding with NaCl, O", pH 5.0; (3) binding with KCI, O", pH 5,4; . (4)binding with KCI, O", pH 6.1, 36.7", pH 6.4; (5) binding with IICI, ODJpH 8.1.
IV. Conclusions The data can best be summarized in terms of the properties of the polymer if we consider the behavior in three pH ranges, say for arabic acid. The general conclusions will then apply to other electrically and structurally similar polyelectrolytes. (20) E. Stenhagen and L. Teorell. Trans. Faraday Soc., 31, 743 (1939). (21) J. Gulland and D. 0. Jordan, "Nucleio Acid," Symposia of the Soc. for Experiments1 Biology, Cambridge, 1947, p. 57.
194
RICHARD J. GOLDBERG
There are of course no abrupt transitions in properties from one p H range to the next. A. pH 2 to 3.3.-1n this range the potentiometric titrations show that the electrostatic free energy (Feiect) of the polymer coil is low. Equilibrium dialysis binding data may be explained on the basis of Donnan calculations but some ion inactivation of the same order of magnitude is shown by conductance measurements. The polyelectrolyte molecule probably has nearly the dimensions of the statistical coil. B. pH 3.3 to 5.5a-FeleGt becomes larger due to the increased charge but apparently the coil does not expand proportionately. The calculated and observed potentiometric titration curves agree fairly well in this range. The nucleic acid binding data demonstrate that counter ion binding increases with increasing charge and charge density. This may also be seen by comparing the conductimetric titration curves for arabic acid in solutions of different cation concentration but at points of equivalent degrees of polymer neutralization. Thomas and Murrayl6 determined the osmotic pressure ( T ) of arabic acid as a function of pH. When sodium hydroxide and hydrochloric acid were used to regulate the pH, the following behavior was observed. At pH 2.5 and below, T was a constant near 15 mm. of solution, at pH 2.5, R increased abruptly and rose to a maximum of 390 nim. of solution a t pH 4.5. Then, just as abruptly, R fell with increasing pH to a value of 240 mm. solution a t pH 5, subsequently decreasing more slowly to near 180 mm. a t pH 10. In other words, the osmotic pressure of arabic acid solutions goes through a decided maximum in this pH range indicating considerable ion-polyelectrolyte interaction. The increased interaction is equivalent to a lowering of the activity of the ambient ions in the polyelectrolyte solutions.
VOl. 57
C. pH 5.5 to 8.0.-A comparison of calculated and experimental potentiometric titration data shows that Felect does not increase as rapidly as predicted by any previous equation, but the charge increases. This may be explained in either of two ways. (1) Binding increases, reducing the potentials inside the polymer coil or (2) the coil expands, reducing Felect and consequently, alkali ion binding would. level off or decrease rather than increase. The second conclusion is supported by the marked increase in conductance in this region corresponding to a relative decrease in cation binding and by the decrease in osmotic pressure in this range, indicative of decreased ion-polyelectrolyte interaction. Contrary to these conclusions, 0 t h and Doty12 have recently concluded that the coil must not expand too greatly as the charge increases above 80% neutralization. The conductimetric data show some differences in the binding affinity of the same polymer for sodium and potassium ions. These results should not be too surprising because of the known differences in the physiological behavior of these ions and the different binding capacities of soils for sodium and potassium. However, the equilibrium data do not show significant differences. The most important conclusion to be drawn from these data is in regard to future investigations. Ambient ion distribution experiments may be entirely as fruitful in elucidating the configuration and electrical nature of polyelectrolytes as are direct measurements of these properties. A great wealth of informative data may be obtained by studying polyelectrolytes which are weak acids and bases other than as completely ionized salts or in their completely unionized form. Acknowledgement.-The author is indebted to Professor I. M. Klotz, who suggested this problem, for his encouragement, advice and critical comment.
SEDIMENTATION I N THE ULTRACENTRIFUGE1 BY RICHARD J. GOLDBERG Department of Chemistry, Universitv of Wisconsin, Madison, Wisconsin Received April 7, 1968
A general theory of sedimentation is developed with the aid of thermodynamics. It reduces to the usual expression for systems of two components, but contains previously neglected terms arising from thermodynamic interactions’between solutes in systems of more than two components. The theory is used to compare osmotic pressure and light scattering with sedimentation e uilibrium. Solvation and aggregation as well as the calculation of interaction constants are discussed in the light of this Leory. The theory also provides for a description of the variations of the moments of mass with the depth in the ultracentrifuge cell. The application of the statistical method for the determination of mass distribution is described. An expression for sedimentation velocity is developed which includes a new frame of reference term. It is shown, as has been surmised, that the appropriate density is that of the solution. A theory is developed which describes the position of a moving boundary in the sector-shaped cell as the square-root of the second moment of the concentration gradient curve.
I. Introduction Although sedimentation investigations of macromolecular systems may be carried out under the conditions of thermodynamic equilibrium or the conditions of transport, they furnish more interpretable information under the former conditions than the latter. Such a situation arises as a result of the complexity of the theory of transport in (1) This investigation was supported by the Office of Naval Research, Contract NSonr76300
general and the friction coefficient in particular. The system in transport involves interactions which do not exist in the system in equilibrium. For example, when the hard sphere approximation for solute molecules is made, the excess chemical potentials of the solutes vanish, whereas the interactions described by the friction coefficients of the solutes are concentration dependent. Concentration dependent work is required to “clear a path” for a molecule. “Drag forces” arising from fluid
