r ,
I he writer \\ishes t o espiws his gratitude t o I)r. $1.IV. IVilliams, under wiiosc. direction this ivork \\-as carried o u t . Further. ht. tieaires t o thank Ilr. R. H. Ewart of' t>hr(kneral IJaborat.ories, I-nited States Kithbcr Clompany, Patxaic, Xew Jersey, and Dr. Gerson Iirgeles of this 1,aboratory for their unfailing interest and co6prration. lIK.:FJ.;RES< (1) I ~ O T YI>., , W A G K K RI, I . ,
SINGER, S.: J . L'hya. ( ' o l l o i t l ('hem. 51, 32 (1947). (2) FLORY, P. * J . : Chrm. R r v . 39, 137 (1946). (3) GEI.:. G . , AND TRELOAR, I.. 11. G . : Trans. Faraday S o r . 38, 147 (1942). (4) JULLANDER, 1.: rlrkiv. Kcmi, Mineral. Geol. 22, 1 (1945) ( 5 ) KEGELES, U.: J . . A m . Chem. SOC.69, 1302 (1947) (6) h N S I N G , \V. D . , ANI>K R A E h l E R , F:, 0 . : .J. Am. Cheni. SO('.54, 1369 (1935j. ( 7 ) M O S I M A N N , H . : IIclv. ('him. Acta 26, 369 (1942). (8) RINDE,H . : Ihsscrtation, I:psala, 1928. (9) SCHUI,~, ( i . V . : %. physik Chcm. B43, 25 (1939'1; for ~lisc~i~ssion wil I ~ O Y E R , R. I:.: Iiirl. Eng. C:hcni., Anal. E X . 18, 342 (1942). (10) SVEDBERG, T . : , J . Phys. ('olloiti Chem. 51, 1 (1!44i). (11) SVEDBERG^ 'r,,4N1) P m I . : K s i m , I i . 0 . : The I ' ( [ r r t c e n l r i j i ~ g c . Oxford University Press, I,o1~doI~ (1040). (12) WALES,AI., HEXI)ER,11,A I , , \ V ~ I , I , ~ , ~ ,M I . S\V.. , ANI) P:w.+R,r, I t . 1 1 . : J. Cheiii. l'hys. 14, 353 (1946). (13) \VAI,E;S. h I . , ' r H O S c i w i N , , J , 0.. \ ~ I I , I , I A M S ,. j . by.,ANI) 1';WARl. 11.: Part, 11 O i t h i s serics, i n prcparation.
I’IiOPERTIES O F COLLOIDrlL SO1,T;TIOS;S
249
oi~servetl,even when thc h i i t of mixing is nrgligil~le. Sie\.ei.al other3 (1-5, 13-1 G ) , using tliffercrit procedures, have arrived at rssential!y equivalent results. I t is now generally ;Igrre(l that this theory :ind tlic equations deduced therehy are fundamentally coriwt, though requiring some estrnsiori t o make them quantitatively applicahle to high concentrations or to actiial solutions which do not conform closely to the model chosen for the theoretical drvelopmcnt. I t is not at first apparent u-hethrr the I n i g c deviations from the ideal solution l a ~ v s ohtained , theoretically nnd experimentally, result primarily from the large sizr of the solute molecules, their shape, or their j/c.sibili/y. T o settle this qucstion? in part, ecpations for the entropy of mixing and related quantities have heen dwivetl by methods not difTering greatly from those used for solutions of cahain molecules, for solutions of large spherical arid large rodlike niole~ules.~ Tn the present paper equations are derived for the general case of rigid solute particles of any a,ssumed shape. These reduce, for the special cases previously dealt with, to thr eqiiations then ohtained. In this treatment, the hypothetical lattice assumption is a\.oitied, since it is both artificial a n d unnecessary (cf. 5). DEHIV.4TIOS O F GESEIZ.4 1. I-QI..4TIOS-i
\Ye consider a solution of -17 solvent and .IT soliiti’ molecules, occupying :L \.olume 1. 1j-e wish to calculate the dependence on concentration of the entropy of mixing, i.e., the difference between the entropy of the solution and t,he sum of the entropy of S solvent molecules in purc liqtiicl solvent and the entropy of S solute molecules in pure liquid solute. The entropy of the solution consists, in part, of the entropy associated \vith internal ra~ndom~iess of the molecules, and \vith their ~ i h i a t i o n ,rotation. aritl orientation. For the solvcnt molecules 11-c may reasonably assume, as :L closie approximation, that the rntropy (per moleciile) of thcsc types is the same Ivhether the molecules are in the (tiiliite) solution or in the pure solvent liquid. For thr soliite molc~criles,UT c m like\vise assume that the entropy (per molrciilr i of ~ I I C S P types is intlepenclcnt of t h c conchcntration in the l o \ v concentration i “ q y j r i Tvhich we arc iiiteiwtecl, differing f i m i that in thr piire liquid ssoliitc I,:L roils1 ant amount. umptions. \vc’ may limit our consitlei,ation to the distrihutioii:d crit ropy assoc*iatctl with the irregular placement of the two kinds of molecules in the soliition. For thr. pre+ent, \ve assumc thc distrit~utionto lie pe~*f’ecatly ixntlom, walizirig that 0 1 1 1 * rrwilts \vi11 ha\-e t o modifictl \vhen applying then1 to solutions in u.hic.h intcrmolccwlar force.+ tc’nd to plwlucc tlepartiires from perfect' randomnes;. ‘I‘o rompiitc the distrih~itioiialentropy of the soliit ion, \vr imagine thc voliimc, I-, to 1 ) ~at first iinoccupi(d by moleciiles ; \v(’ then ihypotheticnlly) atld t h t h
i
Thew rc,sults \ i t t tic, ( ‘ < ~ I l ( i q i i i uo~i i~ iIIigIi l’olyriiew at S t r Frariw, Sovriiii)c\r 25-30, I ! M . iThcx p r i n t e d rrpiirt 1 1 1 iiic~iuclrs11r11yt h e t watIiic’rit I J t IIP sph(~ric~a1 c a w 1 Soiiie o f t Iir (’quatioris and r,oristarits p r i ~ s r ~ i t ti hrrc d arid i n t h i s pap(’r arc equival(1iit t o ssotiiil prc~\.iou.slyclrrirrtl 1)y :I tlifTc~rriit p r r ) c ~ t I n r cIJ>~ Zirnni i l B 1 , :is v-ill ill, s h f ) \ v n .
~
250
lI.\URICE L. HCGGISS
K solute molecules, one at a time, and deduce the entropy contribution of each such addition. Assuming additivity of vOlUmeb, the volume remaining after all the solute molecules have been added is equal to the volume which N solvent molecules would occupy in the pure solvent. We may, vith negligible error, assume the entropy associated n-it11 filling thit volume of the solution with solvent molecules to be the same as that associated xith filling the same volume of pure solvent. The contribution of the solvent molecules to the entropy of mixing i; thus (practically) zero. (In effect, we assume that the distributional entropy of the pure liquid solvent within a region of given volume i b independent of the shape of that region. This assumption is certainly inaccurate for very high concentrations, i.e., when the volume filled by solvent is not large relative to the molecular volume, K i t h the foregoing assumptions and approximations we can write for the entropy of mixing of the solution: =
s - S: - SS
Y?
= ,=I
S,is the contribution of the
S,+ X: In S~I
(1)
solute molecule, added as indicated above, to the total distributional entropy of the solution, and X. is the Maxwell-Boltzmann constant. The last term takes account of the fact that all the solute molecules are alike, The partial molal entropy of mixing of the solvent, which is required for substitution into the thermodynamic equations for the equilibrium properties of the solution, is ztb
,kcording t o well-known principles of statistical mechanics,
S,= I; In (T7,iazlz)
(4)
nhere T‘, is the volume available for the center of the z t h molecule. is the volume of that (or any other solute) molecule, and a is a constant, which, for our present purpose, does not need t o be evaluated. (If the requirement is made that the molecule centers must be placed at lattice points, as in the previous derivations, a = 1.) For a = 1, obviously, ti2
1’ (5) The center of the second solute molecule (i = 2) can be placed in the solution anyyhere within the volume, I’, except \!-here overlapping of the first and second molecules would result. This limitation is ohviouily a function of the shapes and relative orientations of the tIyo molecules. In general, for any shape of molecule, 1 - 2 = 1- - k 2 1 ’ ? (6) I-*=
=
25 1
PROPERTIES O F COLLOIDAL SOLUTIOSS
where k2v2is the volume ruled out because it would produce overlapping, averaged over all relative orientations and integrated over all distances between the atomic centers. Since the volume of a spherical shell of thickness ds is 4ns2ds,
p , being the average probability of overlapping when the molecule centers are a distance s apart. The volume ruled out for the center of the third solute molecule, as a result of the presence of the first two, is approximately 2k2u2. A small correction term must be included t o take account of the chance that the first two molecules are so close together that certain regions are doubly ruled out, being too close to both the first and the second molecules. This term must be proportional t o the square of the volume of a simple solute molecule and inversely proportional to the volume available for the center of the second molecule. Hence:
The volume available for the center of the fourth solute molecule is the total volume, minus three times the volume ruled out by each of the three alreadyplaced molecules independently, plus the volume doubly ruled out (by molecule pairs 1 and 2, 1 and 3, and 2 and 3), minus the volume triply ruled out. The equation is :
-
k, 1.; (V - k2L'JV - 2h-22'2
+ k&(V
-
k2V2)-1
(9)
Similar considerations lead t o the following relationship for the volume available for the center of the i t h molecule:
I-* = 1-d,
(10)
where
and
Here hl, hz, etc., i and j are integers; the k , are constants which depend on the shape of the molecules; and W? is the volume fraction of solute, defined by the equation w 2
= V2N?/T-
(13)
From equations 3, 3 , and 1 0
o\)t:bin:
From equatior?s 11. 1 3 , ancl 1 3 : ( I 9)
(20)
F
= 1;;
- I,.
2 -
+
1,.4
2'
:md
PROPERTIES OF COLLOIDAL SOLUTIOXS
253
Performing the summation indicated in equation 21, and substituting into equation 18 yields
9,
= R (W,
n
+ K2Wi + K3w; + K * w : + KsWi +
-1 (A2), 2 1 K3 = - ( k ; - lis), 3
where IC2 =
(27)
(28)
This is the relationship n-e have been seeking. Substitution of the appropriate
k, constants for a particular molecular shape gives the desired dependence of the partial molal entropy on concentration (assuming perfect randomness of mixing). If the heat of mixing of the solution is zero, h a l = - &,/R
(32)
With this relationship and the appropriate thermodynamic equations, one may obtain theoretical equations for the osmotic pressure and other equilibrium properties of such solutions. LARGE SPHERICAL SOLUTE MOLECULES
For the case of spherical solute molecules which are largc compared with the solvent molecules, ll.2
=
8
(33)
since the center of the second solute molecule cannot be ttny\vhere within a sphere which has twice the radius, or eight times the volume, of the first solute molecule (figure 1 ) . This result can be obtained from equation 7, putting p , equal to unity for s hetween zero anti r2 (twice the molecular radius) and equal to zero otherwise. The constant k3 may be computed in the following way: The volume of overlapping of two spheres, each of radius 2r2,is
for .s between 2r2 and 3rz and zero for s greater than 4rz (figure 2). The probability that the center of molecule 2 is at a distance between s a n d s ds from the center of molecule 1 is zero for s < 2 r 2 ;for larger values of s,
dp, =
4?r.s2ds v - 8uz
(35)
254
3LkURICE L . HUGGIXS
The average volume doubly ruled out
ib
This is the last term of equation 8; hence ?is =
34
FIG.1. The center of the second sphere cannot come within a distance 2r2 from the center of the first sphere.
FIG.2. If the distance between sphere centers is less than 4r2 the two large spheres, each of radius 2r2, overlap.
An approxiniate value for k 4 , the constant concerned with the mutual overlapping of three spheres of radius 2r2, is readily computed. The probability that sphere 2 overlaps sphere 1 (to any extent) is
The value of s which Tvould give the correct ai-erage Tolume of overlapping of the first and second large spheres (equation 36) is S12
=
3.Q7r2
(39)
The probability that (with spheres 2 and 1 mutually overlapping) sphere 3 overlaps sphere 1 is
255
PROPERTIES O F COLLOID.LL SOLUTIOSS
The factor ( 1 - 0.04) takes account of the fact that the centers of spheres 2 and 3 cannot he closer together than 21.2. Its approximate magnitude is computed on the assumption that both s12and s13 have the average value given by equation 30. The probability that s23is betn-een 2r2 and 4r2, when both 1 and 2 and 1 and 3 overlap, is approsimately the area of the zone on the surface of a sphere of radius SI?, which includes all points betn-een 21'2 and 4r2 from a point on that sphere, divided by the surface area of the whole sphere. This is P?3
=
= 0.16 230'
~Iultiplyingpl?, ~ 1 3 and , 1.)?3 together gives the probability, ~ 1 2 3 ,that there is mutual oI-erlapping of all three spheres. lye may compute a rough value for the ayerage volume of mutual overlapping on the assumption that it is the same as i f , in nll cases, all three diFtances brtween sphere centers were equal to 3.07~.', as given hyequation 39. Ah exact value for this volume has not been computed. It should, hon-ever, be sonien-hat larger (perhaps 10-15 per cent) than twice the value of n cone having a lxiue equal in area to the area of mutua,l overlapping in the plane of the sphere centers and an altitude equal to the distance from that plane of either peak of the mutual 01-erlap region. In this m y the average value of mutual overlap (for those spheres shon-ing 123 overlap) has been calculated to be :
> 0.1341.:
(42) The a-erage volume of 123 overlap, for all of the third molecules added to the solution y-olume, is approximately z'i:.3
31?3
=
-I
(43)
P1?1)13p?3v1?3
Hence
Comparison with equation 9 leads to the result : k.j
> 15.3
(45)
Adding 10-15 per cent, for the reason just mentioned, we conclude that, approximately, x'4
(46)
= 18
Substituting equations 33, 37, and 46 into equations 2'7-31 gives+ 4
I n equations 54 and 55 of reference 11, the coefficient of W: should be
and -
(i- 4 2 ) ,
respectively.
' 8
- 42
256
3L-ICRICE L. HCGGISS
This may be compared with Raoult’s law, which may be put in the form
or, for large n,
The coefficients of the square and higher powers of W2 are different, but in both cases they all approach zero as n increases. Dilute solutions of spherical molecules should (if the heat of mixing is negligible) behave as nearly “ideal” solutions, obeying Raoult’s law quite closely. LARGE ROD-LIKE MOLECULES
We next consider solutions of rod-like (cylindrical) solute molecules vhich are large, both in length and in cross section, compared with the solvent molecules. Let the length and cross-sectional radius of each rod be l2 and r2, respectively. For a given angle, 6, between the axes of the first and second rods, the volume ruled out for the center of the second rod as a result of the presence of the first is that outlined by dashed lines in figure 3. Calculation of the magnitude of this volume, averaged over all values of e, leads to
The volume of the cylindrical solute molecule is
v2
2
= Rr2l2
(51)
The volume per solvent molecule, if spherical, is 3
v1 = alrl
the constant a1 depending on their closeness of packing. a1
= 2i =
1/32
(52)
If “close-packed” (53)
For “simple cubic packing”, a1
We may thus write
= 8
(54)
PROPERTIES O F COLLOIDAL SOLUTIOSS
257
and so (56)
and
(57) n
,' I
4rp 4 L
_ - - - -I, -rln- e- - - - - - -
_---
0
- - - - c -
I -*---A
I
FIG.3. T w o rods in contact, as seen in plan and elevation. The region ruled out for the center of the second rod, as a result of the presence of the first rod, is that bounded by the dashed lines. ( a s Dr. Zimm has emphasized, in discussing the subject with the writer, the coefficients in these equations are not correct when rq is not large relative to r1.1
The important thing to note here (as already pointed out by Zimm) is that, for a constant cross-sectional radius of the solute molecule, the magnitude of the coefficient of W," is independent of the length of that molecule, a result similar to that which has been obtained for randomly kinked chain molecules, but different from that (see equation 47) for large spherical solute molecules and different from Raoult's law (equations 48-49). DISCUSSION AND CORRELATION WITH PREVIOUS RESULTS
For comparison with equations 47, 48, 49, and 57, the corresponding equation for solutions of randomly kinked chain molecules, composed of segments equal
258
RIACRICE L . HCGGISS
in size to the solvent molecules (with zero heat of mixing), may be put in the form
where 1.1. is about equal to the reciprocal of the coordination number. ( p 8 is a function of the relative sizes of the chain segments and solvent molecules, hut is practically indepcndent of 71 .) Equations 57 and 58 are alike in shon-ing a dependence of the coefficient of Wg on the relative climcnaions of the solvent molecules and the solute molecule cross sections, \\-ith no dependence on the length of the solute molecule. ~ I c l I i l l a nand AIayer (12) have derived a series expansion for the osmotic pressure, Zimm (1G) witing their result in the form (59) w-hcre c is the concentration of so!ute in mass per unit volume of solution, J12 is the molecular weight of the solute, and A 2 , -43, . . . are constants. Comparison n-ith the equations previously detliicctl hy the m i t e r (e.g.,equation 9 of reference 0) shon-s that Zimm's A? and thc Tyritcr's p arc related 11y means of the equation
Here dl and d2 are the deiisitiei of the pure components. For solutions with no heat of mixing, (61)
I* = pa
Comparing equations 27, 28, and 58 of this paper, it is seen that k2 IC2 = - = n ( 3 - pLs) 2 Therefore,
(62)
For solutions of large rigid spheres, Zimm has deduced
where S o is -4vogadro's number. Substituting into equation 63, it is readily shon-n that
- x :
h2 =
-2
2
in agreement with this paper. For long rigid rods, Zimm has deduced
4
(65)
PROPERTIES O F COLLOIDAL SOLUTIONS
259
Substituting into equation 63 leads to
conforming to the result obtained here. In view of Zimm’s discussion of the significance of his results and their application to actual solutions, it seems unnecessary at this time to go into these matters further. A fern words may be added, however, concerning the effect of flexibility in the molecules. Flexibility obviously gives more randomness and so more entropy. If, though, the molecular flexibility, and so the related intramolecular entropy, do not change with concentration in the concentration range of interest (Le., in dilute solutions), this flexibility has no effect on the partial molal entropies or derived quantities, provided the distribution of shape and of relative orientations of neighboring molecules remains the same. This is likely to be the case if the heat of mixing of the solution is zero, but not otherwise. If the solute molecules attract each other (more than each attracts solvent molecules), flexibility increases the possibility of close approach of their centers and changes the magnitudes of the shape constants, kz,k 3 , etc. SUMMARY
Using a non-lattice treatment, equations have been deduced for the concentration dependence of the partial molal entropy of the solvent in a dilute solution containing large solute molecules, as a function of their size and shape. Some of the shape constants, for spherical and rod-like solutes, have been evaluated. Insofar as there is overlapping, the results are in agreement with those obtained by Zimm, extending the theoretical work of McMillan and Mayer. Flexibility of the solute molecules should have little effect on the results, unless the heat of mixing is not negligible. REFEREKCES (1) ALFREY,T., .4~1,DOTY, P . : J. Chem. Phys. 13, 77 (1945). (2) FLORY, P. J.: J. Chem. Phys. 9, 660 (1941). (3) FLORY, P . J.: J . Cheni. Phys. 10, 51 (1942). (4) GUGGENHEIM, E. A , : Proc. Roy. SOC.(London) A183, 203, 213 (1944). (5) HILDEBRAND, J. H . : J. Chem. Phys. 16, 225 (1947). (6) HUGGINS,M. L.: J . Chem. Phys. 9, 440 (1941). M . L.: J. Phys. Chem. 46, 151 (1942). (7) HCGGINS, (8) HUGGINS,M. L.: Ann. S.I-.Acad. Sci 43, 1 (1942). (9) HCGGINS, M.L.: J. Am. Chem. SOC.64, 1712 (1942). (10) HUGGISS, M .L.: Polymer Bull. 1, 25 (1945) (11) HCGGINS, M. L . : J chim. phys. 44, 9 (1947). (12) MCMILLAX, W. G., ~ S DMSYER,J. E.: J Chem. Phys. 13, 276 (1945). (13) MILLER,A. R . : Proc. Cambridge Phil. Soc. 39, 54, 131 (1943). (14) ORR, W. J. C . : Trans. Faraday SOC.43, 12 (1947). T , J. Chem. Phys. 13, 172 (1945). (15) SCOTT,R.L , A X D M ~ G A M.: (16) ZIMII, B. H . : J. Chem. Phys. 14, 164 (1946).
