2496
A. G. Ogston and D. J. Winzor
homolog would favor more extensive solvation by water. This may partially explain the lower observed viscosities of TMPA solutions relative to the other members of the series. TMPA is miscible with water in all proportions reflecting the strong tendency of this amide to form hydrogen-bonded networks with water. Further interpretation of the molecular behavior of these amides in aqueous solutions will require more extensive study.
References and Notes (1) P. P. DeLuca, L. Lachman. and H. Schroeder, J. Pharm. Sci., 82, 1320 (1973).
(2) P. Assarsson and F. R. Eirich, J. Phys. Chem., 72, 2710 (1966). (3) T. T. Herskovits and T. M. Kelly, J. Phys. Chem., 77, 381 (1973). (4) Corning Model LD-3 Demineralizer, Corning Glass Works, Parkersburg, W.Va. (5) Millipore Corporation, Bedford, Mass. (6) High Accuracy Products Corporation, Claremont, Calif. (7) P. G. Sears and W. 6.O'Brien. J. Chem. Eng. Data, 13, 112 (1968). (6) G. S. Kell, J. Chem. Eng. Data, 12, 66 (1967). (9) "Handbook of Chemistry and Physics", 43rd ed. Chemical Rubber Publishing Co., Cleveland, Ohio, 1961. (10) F. Franks and D. J. G. Ives, Quart Rev. Chem. Soc., 20, 1 (1966). (1 1) J. M. Corkill. J. F. Goodman, and T. Walker, Trans. Faraday SOC., 83, 768 (1967). (12) L. Benjamin, J. Phys. Chem., 70, 3790 (1966). (13) H. S. Frank and M. W. Evans, J. Chem. Phys., 13, 507 (1945).
Treatment of Thermodynamic Nonideality in Equilibrium Studies on Associating Solutes A. G. Ogston' and D. J. Winror' Department of Biochemistry, University of Oxford, Oxford OX1 3QU. United Kingdom (Received March 10, 1975)
Equations are presented for more rigorous treatment of thermodynamic nonideality in osmotic pressure studies of reversibly associating solutes. It is shown that the usual approximation concerning the interrelation of activity coefficients is strictly valid only in the case of highly charged species or of end-to-end association of highly asymmetric monomer units. However, calculations based on uncharged spherical molecules, the most unfavorable case theoretically from the viewpoint of the approximation, have shown that use of the Adams approximation still yields a reliable estimate of the thermodynamic equilibrium constant over the ranges of stoichiometry and solute concentration that are likely to be encountered in practice. The possible use of dynamic measurements to define the interaction coefficients describing the nonideality is also considered.
Use of a property such as the osmotic pressure of an associating macromolecular system to study the association equilibrium requires allowance to be made for thermodynamic nonideality. We may express this in terms of the activity coefficient of each species i by
+
In y L = aLLm, Za,m,
+ higher terms
(1)
where m,, the concentration of species i, is conveniently expressed on a molal basis; aLLand ay are constant coefficients expressing interactions of species i with itself and other species j. Except for the necessary identities (YLJ
= aJ1
(2)
there are no essential general relationships between these coefficients; their values will depend on molecular properties of the solute species such as size, shape, solvation, flexibility, and ionic charge. For simplicity, we consider the case of reversible j-merization of monomer. Neglecting higher terms in eq 1, the thermodynamic association constant must be expressed m
XI= ~ e x p ( ( a J , m+J aiJmi) - j k + aiJmJ)}
(3)
The osmotic pressure of the system (neglecting the pressure term) is then given by eq 4 in which the subscript zero is used to denote solvent The Journal of Physical Chemistry, Vol. 79, No. 23, 1975
!!!ex? = m l + mj + RT
a"
all -m12
+A , . Z + aijmimj (4) 2 l
2
Although it would be possible, in principle, to obtain values of Xj, all, all, and ajj by applying eq 3 and 4 to a sufficiently large series of measurements, this procedure would be impracticable; first, because it would demand an impossibly high standard of accuracy in the measurements; secondly, because eq 3 and 4 cannot be solved explicitly. To avoid these difficulties Adams2Y3and Roark and Yphantis4 have introduced the approximation of assuming that the exponential in eq 3 has value unity, i.e., that the exponent is zero. From this assumption it follows that
3= aiJ -jw1 ml
ja1,
-
(5)
a j j
Since all of the quantities on the right-hand side of eq 5 are considered to be constant, this could imply that mJ/ml is constant, but this could only be true for j = 1,i.e., for isomerization. Alternatively (if j # l), the ratio mJ/ml will vary; eq 5 then requires that both numerator and denominator on the right-hand side of eq 5 are zero; it then follows necessarily that the coefficients are related by jzall = jail = all
If, and only if, this relationship holds can eq 4 be written
(6)
2497
Thermodynamic Nonideality in Associating Solutes
+
+
UyIN = 4 d 1 Gill 0' - 1)r) 8rr2(6 1)11 (j - 1)rJ (32/3)rr3 ( I l b ) m = m l + jmj
(7b)
In the event that eq 6 does not hold, it is still permissible to write eq 4 as /.loo
- /.lo
RT
a
-ml+mj+-m2 2
however, then 8 becomes a quantity that varies between limits all and ajj/j2as mjlml varies between zero and infinity. Evidently, therefore, there is possibility of error in the Adams3i5 procedure, which entails fitting of experimental data with the best value of X j (assuming the exponential in eq 3 to be unity) together with the best constant value of E in eq 8. We shall therefore pose the following questions. (i) In what, if any, cases can eq 6 be expected to hold good? (ii) If eq 6 does not hold good, is the assumption likely to cause serious error in the estimation of Xj? (iii) Can independent estimates of the interaction coefficients be obtained? We shall discuss these in turn.
(i) Validity of Eq 6 In moderately dilute solutions where variation in solvation of a species with concentration is likely to be unimportant there are two principal causes of nonideality, viz., the excluded volume effect and ionic charge. The contributions of these to the excess free energy can each be expressed in the form of eq 1,and are additive; we shall discuss the volume term first. Excluded Volume. If we express concentrations on the basis of moles per volume, the coefficients in eq 1 have dimensions volume per mole, and are in fact molar covolumes, Uij (see Flory6). For molecules that are hard spheres these are given7 simply by 4rN u..= (ri + rj)3 3 where ri and rj are the molecular radii. For nonspherical rigid, or for flexible, molecules the value of Ujj must be estimated by appropriate integration of the degrees to which neighboring molecules interfere with the statistics of each other's distribution, taking into account all relevant aspects of their dimensions, mutual separation, mutual orientation, and configuration. Such calculations have been made for random-coil molecules,6 for cylindrical rods,s and for ellipsoids of revolution (see Appendix). We choose to consider excluded volume effects with two extreme models, viz., sphere-sphere and rod-rod polymerization. In the former we consider that spherical monomer of molecular radius r1 associates to spherical j-mer of radius r, = j1I3r1.Then it follows from eq 9 that 8j Ulj = (Jjj jU1'= (1+ j1/3)3
which is evidently inconsistent with eq 6. In the second case we consider end-to-end polymerization of cylindrical rods (as defined by Ogston8) such that monomer of radius r and cylindrical half-length 11 forms a j-mer of the same radius and cylindrical half-length jl, (j - 1)r. By eq 3 of Ogstons we obtain
+
U11/N = 4 d 1 2 + 16rr211+ ( 3 2 1 3 ) ~ ~ ( l l a )
UJjIN = 4 r r bll
+
+
+
+ 0' - l)r)2+ 16rr2bll + (i - 1)r) + (32/3)rr3
(llc)
where N is Avogadro's number. As 11 becomes much greater than r the proportions U1l:Ulj:UJJ approach l:j:j2, irrespective of the value of j . This case is thus consistent with eq 6. Ionic Charge. Following Scatchardg in the definition and treatment of a charged macromolecular component i of effective net molecular charge Z,, in a supporting solution of uni-univalent electrolyte of concentration mo, we may express the nonideality of the macromolecular component by =
2 2 u,,+ 2 -
(12) 2mo U,, is the effective covolume; electrostatic repulsion between neighboring molecules of like charge will of course tend to make Ut, larger than it would be for uncharged molecule of similar dimensions. T o this is added a term that reflects the ionic strength of the solvent and the charge of the interacting macromolecular species. By analogy with eq 1 2 we may write the corresponding coefficient a,,for monomer-polymer interaction "1,
In the extreme situation that the valence terms dominate the magnitudes of "11, al,, and aJl (i.e., that monomer and polymer are effectively point sources of charge), conservation of charge on polymerization (2, = j z l ) suffices for consistency with eq 6, irrespective of the shapes of the oligomers; as noted by Braswell,lo this conclusion comes directly from Debye-Huckel theory.ll In intermediate situations where covolume and valence terms each contribute significantly to the thermodynamic nonideality, a degree of disagreement with eq 6 can be expected whenever the covolume terms alone do not satisfy its requirements. However, a general conclusion is that existence of a net charge on an associating solute can only improve the degree of conformity with eq 6 and hence the validity of considering the exponent to be zero in the equilibrium expression (eq 3).
(ii) Errors in Estimating Xi Since the most unfavorable case for the validity of eq 6 is that of polymerization of uncharged spheres, we have used this case to test the Adams3s5 procedure. Values of U11, Ul,, and U,, were calculated by eq 9 for values of j likely to be encountered in practice. These were then used, with chosen values of the thermodynamic X , in eq 3, to calculate the corresponding values of ml and mJ, the range of total weight concentration again being that likely to be met with in practice. From these concentrations and the U values the theoretical osmotic pressures were obtained from eq 4. The resulting synthetic data were then analyzed by the Adams5 procedure (with use of eq 8 for the osmotic pressure and eq 3, taking the exponential as unity) to obtain estimated values of X,. Results of this series of steps for j = 2 are summarized in Table I, where, for readier comparisons with experimental situations, ml and m, have been presented as the corresponding weight concentrations w1 and w, (g/ml) of dimeric and tetrameric hemoglobin, respectively: the fairly comThe Journal of Physical Chemistry. Vol. 79, No. 23, 1975
2498
A. G. Ogston and D. J. Winzor
TABLE I: Estimation of Equilibrium Constants for Monomer-Dimer Systems from Simulated Osmotic Pressure Data ~~
~~
Est values
Concn x io3,g/ml -
I I / R T x 106,
Wi
Wj
W
m-osm
1 .ooo 0.800 0.600 0.400 0.200
22.100 13.619 7.4 52 3.251 0.804
23.100 14.419 8.052 3.651 1.004
3 .OOO 2.500 2.ooo 1.400 0.500
19.908 13.413 8.376 4.016 0.502
8 .ooo 6 .OOO 4 .OOO 2 .ooo 1 .ooo
14.195 7.682 3.315 0.811 0.201
103 W i r g/ml
103 g/ml
xj,‘ml/g
x = 2 x IO4 ml g-1 0.4149 1.042 0.2522 0.829 0.1393 0.617 0.0639 0.4 06 0.0188 0.202
22.058 13.590 7.435 3.245 0.802
2.03 1.98 1.95 1.97 1.97
1.58 1.58 1.58 1.58 1.58
22.908 15.913 10.376 5.416 1.002
x = 2 x io3 mI g” 0.4454 3.111 0.3070 2.580 0.2013 2.053 0.1084 1.420 0.0234 0.500
19.797 13.333 8.323 3.996 0.502
2.05 2.oo 1.97 1.98 2.01
1.66 1.66 1.66 1.66 1.66
22.195 13.682 7.315 2.811 1.201
x = 2 x IO2 ml g“ 0.5171 8.204 0.3248 6.145 0.1815 4.069 0.0756 2.020 0.0343 1.005
13.981 7.537 3.246 0.791 0.196
2.08 2.oo 1.96 1.94 1.94
1.94 1.94 1.94 1.94 1.94
a Value divided by the appropriate power of ten. Limiting values are 3.10 respectively.
mon practice of expressing X i on a weight-concentration scale (ml/g in the present instance) is also adopted: r; and Mj were taken as 31.3 A and 64,500, respectively. Columns 1-3 of Table I list monomeric, dimeric, and total solute concentrations for three values of X;, viz., 20,000 ml/g (the experimental value of neutral pH12), 2000 ml/g, and 200 ml/g, the calculated osmotic pressures (I’I/RT)being shown in column 4. The fifth and sixth columns present the estimated concentrations of monomer and polymer, respectively, for values of & chosen to give reasonable constancy of the estimated X j over the whole concentration range. These estimates of X j (seventh column) correspond very closely with the true (thermodynamic) values; we note also that in each case the parameter 6 (final column) lies within the required limits Ul; and Ujj/j2, Similar conclusions apply to equivalent treatment of monomer-tetramer and monomer-octamer systems (Table 11). For these calculations the values of rj and Mj used for Table I were retained, and X j was selected to give approximately 20 times as much polymer as monomer (weight basis) a t the highest total solute concentration (-0.02 g/ml) considered; in this respect the data of Table I1 are comparable with the first listing in Table I 0‘ = 2, X = 20,000 ml/g). Clearly the estimated values of X j in the final column of Table I1 are also in good agreement with the X j values used for synthesis of the “experimental” data. It should be emphasized that this agreement between estimated and input association constants results from a compensation of errors, since the Adams method5 estimates only a mean value of & over the experimental concentration range; this in turn, as Tables I and I1 show, leads to somewhat erroneous values of the concentrations w 1 and wj. However, the errors in these values largely compensate for the omission of the exponent term from eq 3, and a good estimate of X ; results. Since the case we have chosen is the most unfavorable from the viewpoint of the Adams3ps The Journal of Physical Chemistry, Vol. 79, No. 23, 1975
X
-
a b x io5
lo5 and 1.55 X lo5 as rnj/rnl approaches zero and infinity,
approximation, these results justify the experimental use of the Adams6 procedure for estimating X I , at least in the ranges of j and of concentration within which it is likely to be used. Where the monomer and polymer are not spherical (as postulated for the systems on which Tables I and I1 are based), or where ionic charge contributes to the excess free energy, eq 6 will be more nearly satisfied than in the case considered, and hence the Adams a p p r o ~ i m a t i o n ~ ~ ~ ~ ~ will be justified a fortiori. (iii) Independent Means of Estimating Interaction Coefficients Although thus well justified as a method for determining the thermodynamic association constant, the Adams2p3approximation does lead to erroneous estimates of concentrations of monomer and polymer. Better estimates would be desirable if, for example, differences in the specific enzymatic activities of monomer and polymer were of interest. To arrive at these would require better estimates of the interaction coefficients. As has been seen, these can hardly be obtained from the osmotic measurements themselves, which yield only a mean value of & applicable to the experimental range. Accessory means of estimating interaction coefficients are therefore required. There appear to be two possible sources of such estimates. (a) Electron microscopy could give information on the molecular sizes and shapes of monomer and polymer in the dry state from which, with due allowance for solvation, covolumes in solution could be estimated. (b) Measurements of dynamic properties such as diffusion (rotary or translational) or sedimentation velocity can give information about molar frictional coefficients from which deductions can be made a t least of equivalent spherical Stokes radii, or in favorable cases interpreted on the basis of an ellipsoidal model (see Ogston13). It will usually be possible to obtain the molar frictional coefficient of mo-
2499
Thermodynamic Nonideality in Associating Solutes TABLE 11: Estimation of Equilibrium Constants for Monomer-Tetramer and Monomer-Octamer Systems from Simulated Osmotic Pressure Data Est values Concn x io3, g/ml
n / R T X lo6, m-osm
ml"" g""
1.015 0.887 0.770 0.661 0.504
20.851 12.105 6.906 3.755 1.275
1 .96b 1 .96b 1 .96b 1.97b 1 .98b
j = 8, X , = 1.2 x loz2m17 g'? 1.063 0.4673 20.620 1.036 0.3800 15.998 0.947 0.2409 8.557 0.89 7 5.912 0.1917 0.704 0.0981 1.392
19.557 14.962 7.610 5.015 0.688
1 .2OC 1.13' 1.18' 1 .2OC 1.14'
U)l
W
0.950 0.850 0.750 0.650 0.500
20.916 12.142 6.926 3.766 1.279
21.866 12.992 7.676 4.416 1.779
1 .ooo 0.980 0.920 0.880 0.700
19.620 15.018 7.637 5.032 0.692
j = 4 , X , = 2 x 10'0
a
Value divided by the appropriate power of ten. With X lo5 and 0.774 X 105 as limits).
103 w t , g/ml
x,,
103 w j , g/ml
-
wt
nd3
0.4225 0.2555 0.1593 0.1007 0.0512
d =
0.394
X
g-3
lo5 (cf. 0.387 x 105 and 1.55 x 105 as limits). c With d = 0.0989
X
lo5 (cf. 0.0967
nomer by extrapolating dynamic data to zero concentration. Because of the effect of concentration on (e.g.1 sedimentation velocity, the frictional coefficient of polymer cannot be estimated reliably by similar extrapolation to high (or infinite) concentration. However, given first approximation estimates of monomer and polymer concentrations based on the ad am^^,^ approach and the sedimentation coefficient of monomer (by extrapolation to zero concentration), it should be possible to obtain that of polymer (thence its frictional coefficient) from the weight-average sedimentation coefficient, at concentrations low enough for the effects of self- or cross interactions on the sedimentation to be relatively small. Such measurements yield, of course, values for molecular dimensions based on dynamic theory. If the molecules are in fact hard spheres (and if both theoretical treatments are correct), the values of molecular radii deduced from dynamic measurements are bound to be the correct ones for calculating covolurhes. This does not necessarily hold good for equivalent spherical radii deduced for nonspherical or for solvated molecules. The only nonspherical case for which both calculations can a t present be made is that of the solid ellipsoid of revolution; for ellipsoids Perrin14 has calculated the ratio of the frictional coefficient f to the frictional coefficient f o of a sphere of the same volume as a function of ellipticity J ; and this ratio gives the ratio r,/ro of the equivalent dynamic spherical radius rs to the radius of the sphere of same volume, ro. The equivalent spherical covolume radius re can also be calculated by the method outlined in the Appendix. Table I11 gives a comparison of the ratios r,/ro and re/ro (the latter for self-interaction) for hard ellipsoids of revolution as a function of ellipticity J ; also values of re/ro for cylinders of lengthldiameter ratio J. This shows that the dynamic and covolume estimates deviate seriously only for J > 10 and that dynamic data are therefore likely to be fairly reliable for estimating covolumes for self-interaction. Similarly, the covolumes Ut, for cross interaction of ellipsoids calculated on the spherical model do not differ seriously from the true values (Table IV). Thus it would appear that where estimates of interaction coefficients from dynamic measurements can be made, they are likely to lead to better
TABLE 111: Comparison of the Hydrodynamic and Exclusion Covolume Radii of Ellipsoids of Revolution and of Cylindrical Rodsa Y$ YO
J
~J~Ll
Ellipsaids
Cylinders
1 .ooo 1 .ooo 1 A15 1.018 1.043 1.048 1 .lo9 1,112 1.234 1.229 1.330 1.347 1.457 1.489 1.775 1 .a37 2.466 2.356 3.097 2,947 a (i) The ratios r 4 / r o of equivalent spherical hydrodynamic radius (r.) to the radius of a sphere of the same volume ( r o ) for ellipsoids of revolution of varying ellipticity J; (ii) the ratios rr/rO of equivalent self-covolume radius ( r c 2 )to ro for ellipsoids; (iii) the ratios re/rg of equivalent self-covolume radius to ro for cylindrical rods of the same volume and varying length/diameter ratio J 1 1.5 2 3 5 7 10 20 50 100
1 .ooo 1.015 1.044 1.112 1.255 1.375 1.543 1.996 2.946 4.067
estimates of monomer and polymr concentrations for an associating system, because the full versions of eq 3 and 4 may then be used. Conclusion In summary, this theoretical study of the osmotic pressure of associating solutes has established (i) the conditions that are consistent with the ad am^^^^^^ assumption concerning self-cancellation of activity coefficients in the equilibrium expression; (ii) the probable reliability of the association constants obtained by +he approach even in theoretically unfavorable cases; (iii) the possible use of equilibrium studies in conjunction with dynamic measurements for obtaining the interaction constants contributing to the thermodynamic nonideality, and hence for defining more closely the true composition of a monomer-polymer system. The Journal of Physical Chemistry, Vol. 79, No.23, 1975
2500
A. G. Ogston and D. J. Winzor
TABLE IV: Comparison of Relative Values of UII, UI,,and U,, for the Interaction of Ellipsoids of Revolutiona
Y
Ratio U,,/UI1 Model
Parameter
Equiv sphere
Hydrodynamic
4 .OO
1 .oo 2.17 4 .OO
1.05 2.26 4.18 1 .ll
2.09 4 .OO
2.17 4 .oo
2.41
True 1 .oo
j = 4 J = 5
2.11 1 .oo
j = 4 J = 10
1 .oo
1 .oo
1 .oo
j = S
3.12 8 .OO
J = 10
3.38 8.00
Figure 1. Geometric model used in the derivation volume for t w o ellipsoids of revolution.
4.45 1.11 3.77 8 .a8
a The “true” values were calculated from eq 18; the “equivalent sphere” values from the equivalent spherical radii of the ellipsoids (Table 111) for self-interaction and eq 9; the “hydrodynamic” values from the equivalent hydrodynamic radii (Table 111) and eq 9. Ratios are expressed relative to the “true” value of UII.
of
e q 18 as the co-
We note that when a = b = rr the third term in eq 17 reduces to 4r(r’)2,the fourth term to 4r2(r’),and hence the whole covolume to (4ir/3)(r r‘)3. Therefore we can at once, by symmetry, write the covolume for a pair of ellipsoids of revolution whose axes are a1,bl and a2,bz (revolution taking place about the a axes).
+
Appendix. Calculation of t h e Covolume of a Pair of Ellipsoids of Revolution The first stage is to obtain the covolume of a sphere of radius r and an ellipsoid formed by rotating the ellipse x2
y2
3+2=
B1=al
about its x axis. By Figure 1, the coordinates X,Y of the center of the sphere that is tangential at x,y to the ellipse are
+ r sin8 Y = y + r cos 8
X =x
1:;”
irY2 d X
sin a = bla
The Journal of Physical Chemistry, Vol. 79, No. 23, 1975
In
+
1 cos a1 1 - cos a1
sin a1 = bllal
1 (18)
(15b)
References and Notes
(16)
cos 01 1 +cos In a ] ] (17) cosa 1 - c o s a
where
a1
(15a)
Since tan 0 = dyldx, this is easily if laboriously integrated by use of eq 14 and 15 to yield 3
sin2 a1
[2+- cos
with similar expressions for A2 and Bz. As required, eq 18 reduces to eq 17 for a2 = bz = r.
The required covolume is then
U=2
where
(1) On sabbatical leave during 1974 from the Department of Biochemistry, Unlversity of Queensland, St. Lucia, Qld., Australia. (2) E. T. Adams, Jr., and H. Fujlta. “Ultracentrifugal Analysis in Theory and Experiment”, J. W. Williams, Ed., Academic Press, New York, N.Y., 1963, p 119. (3) E. T. Adams, Jr., Biochemistry, 4, 1655 (1965). (4) D. E. Roark and D. A. Yphantis, Ann. N. Y. Acad. Sci,, 164, 245 (1969). (5) J. Vlsser, R. C. Deonier, E. T. Adams, Jr., and J. W. Williams, Biochemistry, 11, 2634 (1972). (6) P. J. Flory, “Principles of Polymer Chemistry”, Cornell University Press, Ithaca. N.Y., Chapter 12. (7) E. Edmond and A. 0. Ogston, Biochem. J., 109, 569 (1968). (8) A. G. Ogston, J. Phys. Chem., 74, 668 (1970). (9) G. Scatchard, J. Am. Chem. SOC., 68, 2315 (1948). (IO) E. Braswell, J. Phys. Chem., 72, 2477 (1966). (11) P. Debyeand E. Huckel, Phys. Z.,24, 185 (1923). (12) E. Chiancone, L. M. Gilbert, G. A. Gilbert. and G. L. Kellett, J. Biol. Chem., 243, 1212 (1968). (13) A. G. Ogston, Trans. Farada SOC., 49, 1481 (1953). (14) F. Perrin, J. Phys. Radium, 61,7, l(1936).
