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'ZYrt ',<7A PPC /0 71 MDDC 1091 J1 UNITED STATES ATOMIC ENERGY COMMISSION A NOTE ON THE RELATION BETWEEN ENTROPY AND ENTHALPY OF SOLUTION by 0. R. Rice Clinton Laboratories CVC C r1 T~o Rj hOq This document consists of 6 pages. Date of Manuscript: Unknown Date Declassified: July 9, 1947 This document is issued for official use. Its issuance does not constitute authority to declassify copies or versions of the same or similar content and title and by the same authorss. Technical Information Division, Oak Ridge Directed Operations Oak Ridge. Tennessee f  'U A NGTE ON THE RELATIONS BETWEEN ENTROPY AND ENTHA.LPY OF SOLUTION By 0. K. Rice The relation between entropy and enthalpv of solution for a serie of nonpulajr .ilute% in a given nonpolar solvent is diccu ;ed. It i. considered that solution of a gaseous salute, uithi.ut changing its concentration on going from gas phase to liquid solution phase, does not change ilt own entropy, all change of entropy being referred to the solvent. The entropy of the solvent change! bhccause 01 sur fac e effects around the solute molecule and a kind of longrange order introduced by the slute mole cules. Two extreme cases are considered: II) the case ol an ideal solution, and (2) the case of a solute of hard attractionless spheres. The difference in entropy of solution between tnhse extreme cases can be estimated. It can also be estimated by extrapolation from the t sperimental data on the entropy and enthalpv of solution, and these two estimates agree in order of magnitude. The fact that the relation between entropy and enthalpy of solution is linear is also shown to be a reasonable ex pectation, and the effect of changing solvent as well as solute is considered. The groundwork is thus laid for a qualitative understanding of this relation between entropy and entnalp\ of solution. It is well knokn that there is a relation between the entropies and enthalpiet of solution of a series of nonpolar gases or vapors in a given nonpolar liquid solvent, provided the solute molecule is not larger than that of the solvent. This was apparently first noted by Bell', who observed .that some re lation between the energy and enthalpy of solution might be expected but gave no explanation of the form of the relation nor of the order of magnitude of the quantities involved. It was further discussed by Barclay and Butler. 4 rather detailed theoretical discussion has been gihen Frank and Evans.3 In their treatment certain relations used in the discussion of pure liquids' were introduced into the theory of solutions only slightly, if at all, modified.* It is certainly of interest that this gives an equation which seems to fit pretty well the energyenthalpy relation for solutions noted above; however, the physical significance and validity of this procedure are not entirely clear. This being the case, it appears that some consideration of this problem from another point of view is still desirable. In the present note we shall consider the properties of two extreme, hypothetical types of solution, namely, (I) an ideal solution, and (21 a solution in which the solute molecules consist of hard spheres, ;llphppcC.upy space.but do not..exert attractive forces on the solvent molecules. Real solutions may usually be considered to stand somewhere between these extreme types, and it is possible to say some thing about the transition from one type to the other for any given solvent. NOTATION Since we shall be interested in entropies of gas and liquid phases for solvent and solute and changes bf entropy under various conditions, as well as a number of different volumes andfree volumes, we require a system of notation with a variety of subscripts and suffixes, and capital and small letters. In the table which follows, a subscript i is used to denote component i. In the text i takes the value I for solvent and 2 for solute. To any of the symbols we may affix an asterisk, indicating that it refers :.. 1 oar.exauple, le use l'etquation I1Il), togetherwith the subsequent use off3, and as paqtigLmc.iola.ljuarptI ties on.p 538 of Frank and Evans; andthe use of equation 11(21) on p 513 NIDDC 1091 fI MDDC 1091 to the case of an ideal solution, or we may add a subscript 0, indicating that it refers to the case in which the solute consists of hardsphere molecules. ni = total number of moles of component i Ni = mole fraction of component i in liquid phase V = volume of liquid solution containing one mole of solute ViO = molal volume of pure liquid component i vi = molal volume of gaseous component i Vf,i = molal free volume of component i in liquid solution Vf,io = molal free volume of component i in pure liquid phase (See equation 5.) S = total entropy of liquid solution Sio = molal entropy of component i in liquid component i Si = partial molal entropy of component i in liquid solution si(vi) = molal entropy of gas at volume indicated ASi(vi) = Sisi(vi) = entropy of solution of 1 mole of pure vapor component i at molal volume vi in a large amount of solution of any given concentration ASio(vi) = SioSi(vi) = molal entropy of condensation of pure com ponent i from molal volume vi in vapor phase ASm = entropy of mixing AHi = enthalpy of solution of I mole of pure vapor component i in a large amount of any given solution AHio = enthalpy of condensation of pure component I IDEAL SOLUTIONS If two liquids form an ideal solution, the entropy of mixing is given by the classical expression A Sm*= n,R In N, R In N, (1) and the total entropy of the solution is given by S* = n,Si + nS0n,R In N, nR In N, (2) We then obtain for the partial molal entropy of the solute S* = ( S /On,)T,p,n, = S2R In N, (3) Let us now consider the entropy of solution of 1 mole of solute from the vapor phase into a very large quantity of solution of any given concentration, with vi = V. This means the solute enters the solution without change in its volume concentration. We have for this process AS,*(V) = *s,(V) = So s2(V)R In N2 (4) We can write an equation defining Vf2,o as follows: S20 s(v2) = R in (V,2,o/v,) (5) MDDC 1091 The free volume thus defined includes all effects of communal entropy. An exactly ideal solution is one in which the solute and solvent are'' exactly alike "although dis tinguishable; that is, their molecules have the same size and force fields. In this case V,o = V20 and Vf,,o = Vf,2o. Since V is the volume of solution containing 1 mole of solute, we have V = V,o/N,. Set ting v, = V, introducing the expression for V into equation 5, and substituting in equation 4, we obtain S,*(V) =Rln (V,,O/V,O) = RLn (Vf,,o 'V,o (6) The ratio of molal free volume to molal volume simply represents the effect of neighboring mole cules on the space available for the motion of any molecule in the field of those neighbors. Thus the effect of solution without change of concentration is simply the effect of neighboring molecules on the free space available. NONIDEAL SOLUTIONS Equation 6 holds, of course, only for ideal solutions. We may, however, define a new quantity Vf,,' by the general equation S, (V) = Bin (Vf,, /V,o). (7) Vf,,', then, may be said to give the volume left free by 1 mole of solvent for a particular solute. It seems possible, as Frank and Evans have noted, for Vf,,' to become as large as, or even greater than, V,0. At first sight this appears strange, for it seems then that the solute is free to move around in a greater volume than that which contains it, even though this volume itself is well filled with sdlveot molecules. Frank and Evans pointed out that this could only be explained as an effect of the solute on the solvent. An explanation of the nature of this effect and an estimate of its order of magnitude is the principal aim of the present note. We now consider the process of solution from a different point of view. We suppose that we have the solute in the volume which it is going to occupy, and we pour the solvent in on it. Since this is a liquid system, dilute in the solute, the solute molecules may occupy any preassigned positions, re gardless of whether the solvent is present or not. We may thus say that the partition function of the solute is unaltered by the presence of the solvent, and we may, somewhat artificially perhaps, refer all entropy effects to the solvent.0 In considering the entropy of the solvent, we may assume that all solute molecules are held in fixed positions, since in the partition function for the solute all possible positions are included. From this point of view equation 7 gives the effect of the solute on the entropy of the solvent. However, a slight correction is required, for it is evident that the process we have just considered gives the total entropy of solution of I mole of solute rather than the partial molal entropy. The total entropy change will be Ss,(V)n,S,0, instead of A S2(V) = ,s, (V) .Since S = n,, + n, = n,, + 12 (since n, = 1), we see that the entropy change for the process considered is equal to AS2(V) + InS,n,S,0. For a dilute solution this may be shown u to be equal to A S2(V) + R. For a solution that is ideal as well as dilute, the total entropy of solution A S,*(V) _t R of 1 mole of solute molecules will be negative. This follows from equation 6 because Vf,,o is always very consid erably smaller than V,0. But in an ideal solution the solvent molecules in the immediate neighborhood. *Somewhat similar considerations have been carried out by Fowler and Guggenheim,' but they did not attempt to consider the solvent effects fully in setting up the chemical potentials. See also Barclay and Butler, reference 2, p 1454. tWe have 9, = S10R In N,, since Raoult's law holds for a dilute solution. N, = n,/(n, + 1) with n, = 1. With n, )> 1, this becomes N, 1 n,, and In N, n"I, whence the relation follows imme diately. MDDC 1091 of a solute molecule are in the same environment as any other solvent molecules, since the force fields are the same. The lowering of entropy of the solvent implied by equation 6 cannot, therefore, be r(le rred to any change in the range of motion of the individaul solvent molecules in the neighborhood of solute molecules. It must, on the contrary, be attributed to the introduction of a certain degree of longrange order, produced by having the solute molecules held in fixed position. This restriction in position is transmitted through the neighboring molecules to the solvent, even through the range of motion of these molecules about their equilibrium positions is not altered. It is rather the equilibrium positions themselves which are affected. To understand better the situation in nonideal solutions. let us consider the case in which the ,olute is assumed to be composed of hardsphere molecules, which exert no force on the solvent mole cule.;. IThis means that the enthalpy of solution .:; Is actually positive, because of the energy nec esiarv to produce the hole in the solvent into ;ihich the solvent molecule is going to go. Roughly, as suming that the hole is the same size as a solvent molecule, we may say that A H, is equal to dH,o.)* If the solute is like a hard sphere, we may expect that a fixed arrangement of solute molecules will be much less effective, if effective at all, in inducing longrange order in the liquid. Furthermore, since the energy will not be lowered by pruxinmitv of solvent to solute molecules, the solvent molecules around a solute molecule xill be reasonably free to arrange themselves in such a way as to allow a maximum of freedom of motion. There will thus be a gain in entropy, similar to the gain in entropy when the free surface of a liquid is increased. This gain in entropy is to be equated to A S),,o(V) R. APPLICATION TO THE EXPERIMENTAL DATA Let us now apply these ideas to solutions in a typical solvent, acetone. We use the data collected by Frank* and by Frank and Fvans. From the entropy of vaporization of acetone (using equation 5 applied, however, to the solvent instead of the solute) and from its density, we calculate that at 25'C the value of Vf, o.'V,0 = 0.0030, whence R In (V1,3o'Vo) = 11.5 cal mole' deg'. The heat of con densation,AHo0, of pure acetone is7600 cal mole"'. Hence for a solute forming a perfect solution 4S,*(V) = 11.5 andA H2* = 7600. The table of Frank and Evans shows howS3,(V) and dH vary from solute to solute. (Actually they list the energy and enthalpy for evaporation from solution to form a gas at I atmosphere at 25'C, so their values differ by a sign and a constant additive amount from ours.) By extrapolation we find that, for a hard sphere solute for whichAH., = dH,, = + 7600, we can set S.,,(V) equal to about 8. Had there been no surface effect,dIS,,,(V) R would have been zero, assuming that there is no longrange order under these conditions. It is, therefore, natural to com pare the 8 + R = 10 cal mole' deg' with the surface entropy to be expected. The surface entropy of a liquid is closely related to the Eotvos constant,d(V,'/1ldT, where 7 is the surface tension. If we divide this by N'3, where N is Avogadro's number, the expression may be interpreted as the surface entropy per molecule. (This follows because Y is the surface free energy per unit surface, and (V, NPi'' is very close to the surface occupied by a molecule at the sur face.) The Eotvis constant for acetone is, from data in the LandoltBornstein Tables, 1.8 erg mole"2/3 deg', which makes N^.'1d(V,' 3y)/dT equal to 2.5 x 10`' erg deg'. Comparing this to the Boltzmann constant k = 1.37 x 10'6 erg deg', we see that the surface entropy is a little less than 2 R or 4 cal deg' per mole of molecules at the surface. The surface area about a solute molecule is of the order of four times the area occupied by a molecule at the surface. One would not expect the entropy connected with unit area to be the same as for a flat surface, and the 10 cal mole' for the surface entropy in the solution of a hardsphere gas is certainly of the correct order of magnitude ... ' We may look at this from a slightly different point of view. There will probably be about ten nearest *H. S. Frank, reference 4; see especially p 499. t H. S. Frank, and M. W. Evans, reference 3, see pp 514515. MDDC 1091 neighbors about a solute molecule. Each neighbor, therefore, has about 1.0 cal deg' mole' of sur face entropy, about onefourth as much as a free surface molecule. In a recent paper' we estimated the surface entropy per molecule of a liquid by a rough statistical calculation and found a value about half as great as that given by the E6tvds constant. In our calculation we neglected the decrease in density at the surface laver. which undoubtedly results in an increase in the surface entropy. On the other hand, this decrease in density is probably not appreciable in the neighborhood of a solute mole cule. Further, a molecule at the surface of a small spherical hole inside a liquid in not nearly as free as a molecule at a flat surface. Therefore, the value of 1.0 cal deg' mole' for the solvent molecules about the hardsphere solute molecule seems entirely reasonable. We may now be in a position to understand the linear relation between A S2(V) and A H,, simply as ,,,i the start of a series expansion. It has, as we have noted, been pointed out by Bell that there should be some sort of relation betweendS.(V) and A H, for a given solvent, and this is also obvious from the dis ,, cussion just given.: We expand about the values for a hardsphere solute, writing S1SJVVI =A Sa,o(V) *a, (AH, AH,,l a. (AH, AHa,12 .. (8) The question then reduces to a decision as to whether the first term in this expansion is the dominant one.A S2(V) can be divided into two parts, as previously discussed, the surface entropy and the entropy Al negative in sign, of course) associated with longrange order; and each one of these can be expanded in a series like equation 8. Let us consider the surface entropy first. As we have noted, this part is contributed by all the .nearest neighbors of a solute molecule, and in the range from ideal solution to solution of hard sphere it goes from n to 1.0 cal per mole per deg for each nearest neighbor, in the case of acetone. This cor responds to a change in the effective free volume of each nearest neighbor molecule by a factor 1.65. Over so great a range the free volume change might be expected to deviate somewhat from being a linear function of the. force exerted by the solute molecule on its neighbors, and hence onArH,; and the logarithmof the free volume, which determinesdS,(V), would also be expected to deviate to some extent from being a linear function of the free volume itself. However, these deviations would not be expected to be exceedingly great, even with a 65 per cent change in free volume, and the experimental data do not actually cover more than about twothirds of the range between the ideal solution and the hardsphere solute. Also, the change in surface entropy:contributes only a small part of the variation oIdS;(VI between these extreme types of solution. The greater part of this variation is to be referred to the introduction of longrangh order. But lhis is an'effect' which is actually shared among many molecules of solvent, so that the change in free volume for any one molecule will be so small that one need have no surprise if the experimental range does not extend beyond that in which the first term of the series expansion of equation 8 suffices. This, of course, is not a rigorous explanation of the linearity betweenAHl., anddS,(V), but does make it seem plausible. Franl: obtained a linear relation, Sb'dit thisFiwas done by carrying an empirical linear relation between theAH s and S's of pure liquids into the equation for the solutions. The slope of ihedS,(V) vsAH line, which may be expressed as a =AS'(VIAS2.lSo(V (JH,'.<,l0), (9) is almost the same for a considerable number of solvents. This may be understood in terms of the variation of A S.(VI withhH,* for a series of different solvents. This, of course, is the same as the variation of A SofVl withAil,.0 'This depends upon the existence of a fairly simple relation between the force exerted by neighbor molecules and the mutual energy, which means that the potential energy curves must in all cases be of similar shape. cI MDDC 1091 Let us writedAS,* V) and6dH,* for the differences between the respective indicated quantities for tao solvents. Then sinceAH.,, = 3H we will have : iH d H,,ol = 2AdH,. This means that if 6 is to be the same for the two different solvents, we should, from equation 8, have S. A[ S*fV) AS.,,(Vl] 2a, 6 H;*. 6AdS2,(V) may be expected to be close to zero, because AdS,,0(V) is contributed entirely by the surface entropy, and the Eotvos constant has roughly the same value for most common solvents. We might thus expect to find 6 AS '(V) 23a, dH,'. Actually, empirically, it is found that 6dd,*(V) "a,6 AH.,*, since the slope of the AS,o(fV vs 3 Ho line for different solvents is approximately the same as that of the AS,(V) vsm H, line for solutions with a common solvent. But, for the usual solvents,d6AS,(V) is such a small fraction ofAS2*(V)AdS,o(V) that the empirical relation between AS,O(V) and H,o does not require that theAS2(V) vsAH, lines for the different solvents be appreciably different. It thus appears that the similar value of a for the different solvents merely reflects the similarity of the solvents used. On the other hand, the fact that theAS,o(V) vsAH,o line has about the same slope as most of the A4S(V) vs A.H, lines for the' various solvents is a remarkable fact, noted by Barclay and Butler but still not fully explained. In going from one solvent to another, for whichdH,o is greater, there is no change in surface entropy, since this is zero with either pure solvent or ideal solution; the decrease in entropy arising from longrange order, however, is caused by two factors, (1) the increased force exerted by the particular atom which is condensed into the solution, and (2) the increased force which all neighbor atoms exert on each other. On the other hand, in going from one solute to another which has a greater JH. in the same solvent, there is a decrease in entropy on account of the surface effect, but there is no decrease in entropy on account of factor (2) of the preceding sentence. The surface effect in the case of changing solutes must, therefore, approximately balance factor (2) in the case of changing from one pure solvent to another. REFERENCES 1. Bell, R. P., Trans. Faraday Soc. 33, 496 (1937). 2. Barclay. I. M., and J. A. V. Butler, Trans. Faraday Soc. 34, 1445 (1938). 3. Frank. H. S., and MN. W. Evans, J. Chem. Phys. 13, 507 (1945). 4. Frank, H. S., J. Chem Phys. 13, 493 (1945). 5. Fowler, R. H., and E. A. Guggenheim, Statistical Thermodynamics, pp 372 ff, Cambridge University Press, 1939. 6. Fowler, R. H., Statistical Mechanics, 2d ed, p 844, Cambridge University Press, 1936. 7. Rice, 0. K., J. Chem. Phys. 15, 314 (1947). '.(rty'nn i(d ~'' x ,.: d" *A.. ** Z.* Ea^. ,*: . ~ ~ ~ fj ..... ^ ': .. : l *. :** '. *> * ~ ~ ~ ~ ~ ~ i. .*:*i ''^ v ^ ^ sy '.a. ... L O 'iiiv4&Mv. V0 ,4.i . I' ~ ~ ~ ~ :! :*t *1*n.., 'Y ' '` i :' I. '.i. i: 'ir..ii:::;  ,, ~i~lrc 5. u i. :. , 3b~rI:1 ?E~. .:1;E ~' ~i :~' L 151M1Ii llIIHII~l 3 1262 08907 9817 4f .4 I 
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