MDDC - 1091 UNITED STATES ATOMIC ENERGY COMMISSION A NOTE ON THE RELATION BETWEEN ENTROPY AND ENTHALPY OF SOLUTION by O. R. Rice Clinton Laboratories •SAr, 2*2&** 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 author(s). Technical Information Division, Oak Ridge Directed Operations Oak Ridge, Tennessee A NOTE ON THE RELATION BETWEEN ENTROPY AND ENTHALPY OF SOLUTION By O. K. Rice The relation between entropy and enthalpy of solution for a series of nonpolar solutes in a fcifen ncnpolar solvent is discussed. It is considered that solution of a gaseous solute, wiuU^augin con entrauon on gomg from gas phase to liquid solution phase, does not change ts own en rZ all change of entropy being referred to the solvent. The entropy of the solvent changes because of sur- u s TwoeT S ° 1Ute m ° leCU!e a0d a ^^ ° f lOTg - range 0rd « •**&&£ by theso ut "1- e t S ' ^ W ° e f Z^ CaS6S are consldered: (1) the case of an ideal solution, and (2) the case of a solute of hard attractions spheres. The difference in entropy of solution between thes extreme cases can be estimated. It can also be estimated by extrapolation from the experiment! dlta on the TZ\ IT'"" ° f SOlUti0n ' and th6Se tW ° eSUmateS a ^ ree in «d« <* magn^de The act t a the re ahon between entropy and enthalpy of solution is linear is also shown to be a reasonable ex pecta ion, and the effect of changing solvent as well as solute is considered. The gro^S ^ s thus latd for a qualitative understanding of this relation between entropy and enthalpy of solution of nnnnn, **" ^^ ^ th6re % * TelMon betWeen the Copies and enthalpies of solution of a series iLZ T f^TZ VaP ° rS " " glVe " n ° nPOlar liqUid SOlvent > ? rovided the solute molecule £ not larger than that of the solvent. This was apparently first noted by Bell' who observed tha some l lation between the energy and enthalpy of solution might'be expected but gave no ^explanation of t he o"rm appears that some consideration of this problem from another point of view is still desiraWe In the present note we shall consider the properties of two extreme, hypothetical types of solution namely, (1 an ldeal solution> and (2) a ^^ ^ which ^ ^^ ^ cons st of hard sph res usuanv be""" T - *° "* "*" '?*** ^^ 0n the S ° lvent ™ lecul <- Real solutions may ' usually be considered to stand somewhere between these extreme types, and it is possible to sav some thing about the transition from one type to the other for any given solvent. V NOTATION Since we shall be interested in entropies of gas and liquid phases for solvent and solute and changes of entropy under various conditions, as well as a number of different volumes and free volumes in th table a wh yS h7n "^i™^ * "*&* ° f SUbSCriptS and Suffixes - a " d "P^ and smal letTs ' or solvent a! f Ti " ? "^ ' ** "*** * ^^ C ° mP ° nent L In the text * **** the value 1 for solvent and 2 for solute. To any of the symbols we may affix an asterisk, indicating that it refers *For example, the use of equation 11(10), together with the subsequent use of and as partial molal quantities on p 508 of Frank and Evans; and the use of equation n(21) on p 513 MDDC - 1091 r 1 2] 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 hard-sphere molecules. nj = 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 Vj° = molal volume of pure liquid component i Vi = molal volume of gaseous component i V f j = molal free volume of component i in liquid solution Vf,i° = molal free volume of component i in pure liquid phase (See equation 5.) S = total entropy of liquid solution Sj° = molal entropy of component i in liquid component i Sj = partial molal entropy of component i in liquid solution Sj(Vj) = molal entropy of gas at volume indicated ASj(vj) = Sj-s^Vj) = entropy of solution of 1 mole of pure vapor component i at molal volume v- in a large amount of solution of any given concentration AS i (v^ = S^-Sjtvj) = molal entropy of condensation of pure com- ponent i from molal volume Vj in vapor phase AS m = entropy of mixing AHj = enthalpy of solution of 1 mole of pure vapor component i in a large amount of any given solution AHj = 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 AS m *=-n,R InNj-njRInNj (1) and the total entropy of the solution is given by S* = n.S, + n-jS^-n,!* In N,-njR In N 2 (2) We then obtain for the partial molal entropy of the solute §,* = OS /BnjT.p.n, = S 2° _R m N * ftj 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 vj = V. This means the solute enters the solution without change in its volume concentration. We have for this process 4S 2 *(V) = S 2 *-s 2 (V) = S 2 °-s 2 (V)-R lnN 2 (4) We can write an equation defining Vf 2 ° as follows: S 2 °-s 2 (v 2 ) = Rln(V f>2 °/v 2 ) (5) MDDC - 1091 [ 3 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° = V 2 ° and Vf i° = Vf j°. Since V is the volume of solution containing 1 mole of solute, we have V = V 2 °/N 2 . Set- ting v 2 = V, introducing the expression for V into equation 5, and substituting in equation 4, we obtain SftV) = Rln (V f )2 °/V 2 ° ) = Rln (Vf^o/V, (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 efi'ect 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) = Rln (V fil /V°). (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 l °. 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 solvent 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.* 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 1 mole of solute rather than the partial molal entropy. The total entropy change will be S-s^VHnjS! , instead of A SjV) = Sj-s^V) . Since S = n^ + n^ = n,§, + S^ (since n, = 1), we see that the entropy change for the process considered is equal to A SjjOV) + n^-n^S, . For a dilute solution this may be shown t to be equal to A S^iV) + R. For a solution that is ideal as well as dilute, the total entropy of solution A SfiV) + R of 1 mole of solute molecules will be negative. This follows from equation 6 because Vf x <> is always very consid- erably smaller than V®. But in an ideal solution the solvent molecules in the immediate neighborhood. * Somewhat similar considerations have been carried out by Fowler and Guggenheim, 5 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 §, = S,°-R In N 1( since Raoult's law holds for a dilute solution. N, = n^nj + 1) with a, = 1. With ^ >> 1, this becomes N, — 1-n,, and In N x -*• -n," 1 , whence the relation follows imme- diately. 4 ] 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 referred 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 long-range 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 solute is assumed to be composed of hard-sphere molecules, which exert no force on the solvent mole- cules. (This means that the enthalpy of solution 4 :L is actually positive, because of the energy nec- essary to produce the hole in the solvent into which 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 -AH, .)* 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 long-range order in the liquid. Furthermore, since the energy will not be lowered by proximity of solvent to solute molecules, the solvent molecules around a solute molecule will 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,, (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 Evans. t 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,, /^ = 0.0030, whence R In (V fil ° V,°) = -11.5 cal mole" 1 deg" 1 . The Jieat of con- densation,/! Hj°, of pure acetone is-7600 cal mole"'. Hence for a solute forming a perfect solution zlS^CV) = -11.5 and4 H 2 * = -7600. The table of Frank and Evans shows how4 3 2 (V) and4H 2 vary from solute to solute. (Actually they list the energy and enthalpy for evaporation from solution to form a gas at 1 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 which^H, - AH 2 , = + 7600, we can set ASj.oCV) equal to about 8. Had there been no surface effect, AS 2 , {V) + R would have been zero, assuming that there is no long-range order under these conditions. It is, therefore, natural to com- pare the 8 + R = 10 cal mole" 1 deg" 1 with the surface entropy to be expected. The surface entropy of a liquid is closely related to the Eotvos constant, -dtV^/^dT, where 7 is the surface tension. If we divide this by N 2 ' 3 , where N is Avogadro's number, the expression may be interpreted as the surface entropy per molecule. (This follows because 7 is the surface free energv per unit surface, and (V^/N) 2 / 3 is very close to the surface occupied by a molecule at the sur- face.) The Eotvos constant for acetone is, from data in the Landolt-Bornstein Tables, 1.8 erg mole" 2 / 3 deg" 1 , which makes -N' 2 / 3 d(V, 2 / 3 y)/dT equal to 2.5 x 10" 1B erg deg" 1 . Comparing this to the Boltzmann constant k = 1.37 x 10" 16 erg deg"', we see that the surface entropy is a little less than 2 R or 4 cal deg -1 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" 1 for the surface entropy in the solution of a hard-sphere 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 514-515. MDDC - 1091 ( 5 neighbors about a solute molecule. Each neighbor, therefore, has about 1.0 cal deg" 1 mole" 1 of sur- face entropy, about one-fourth as much as a free surface molecule. In a recent paper 7 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 Eotvos constant. In our calculation we neglected the decrease in density at the surface layer, 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" 1 mole' 1 for the solvent molecules about the hard-sphere solute molecule seems entirely reasonable. We may now be in a position to understand the linear relation between A S 2 (V) and A H^ simply as 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 between4S 2 (V) and A H 2 for a given solvent, and this is also obvious from the dis- cussion just given. We expand about the values for a hard-sphere solute, writing AS 2 (V) =AS 2<0 (V) + a, (AH 2 -AH, )0 ) + a, (AH 2 -AH 2)0 1 2 + (8) The question then reduces to a decision as to whether the first term in this expansion is the dominant one. A S 2 (V) can be divided into two parts, as previously discussed, the surface entropy and the entropy (negative in sign, of course) associated with long-range 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 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* on4H 2 ; and the logarithm of the free volume, which determines43 2 (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 two-thirds of the range between the ideal solution and the hard-sphere solute. Also, the change in surface entropy contributes only a small part of the variation of4S,(V) between these extreme types of solution. The greater part of this variation is to be referred to the introduction of long-range order. But this 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 between4H 2 and4S,(V), but does make it seem plausible. Frank obtained a linear relation, but this was done by carrying an empirical linear relation between the4H s and A 3 s of pure liquids into the equation for the solutions. The slope of the4S 2 (V) vs4H line, which may be expressed as a =[4S 2 *(V)-4S 2j0 (V)]/(4H.,*-H 2(0 ) ) (9) is almost the same for a considerable number of solvents. This may be understood in terms of the variation of A S 2 *(V) with4H,* for a series of different solvents. This, of course, is the same as the variation of A S_°(V) with4H.,°. *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. 6 ] MDDC - 1091 Let us write 6AS*(V) and<54H 2 * for the differences between the respective indicated quantities for two solvents. Then since4H 2 , = -4H,* we will have d(4H*-4H 2 , ) = 26AH 2 *. This means that if 6 is to be the same for the two different solvents, we should, from equation 8, have 5 [4S 2 *(V)-4S 2 , (V)1 = 2a, 6 H 2 *. <54S,, (V) may be expected to be close to zero, because4S,, (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 A S 2 *(V) ~ 2a 1 54H 2 *. Actually, empirically, it is found that 6ASh*(V) "= a,6 4H 2 *, since the slope of the4Sj°(V) vs A H,° line for different solvents is approximately the same as that of the 4S 2 (V) vs4H 2 line for solutions with a common solvent. But, for the usual solvents, i54S 1 °(V) is such a small fraction of43 2 *(V)-4S,, (V) that the empirical relation between /4S,°(V) and4H,° does not require that the4S,(V) vs4H, 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 the4S,°(V) vs4H, line has about the same slope as most of the 4S,(V) vszlH, 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 whichzlH, 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 long-range 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 4H, 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 M. 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, O. K., J. Chem. Phys. 15, 314 (1947). MINIMI 3 1262 08907 9817