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J1 UNITED STATES ATOMIC ENERGY COMMISSION
A NOTE ON THE RELATION BETWEEN ENTROPY AND ENTHALPY OF SOLUTION
0. R. Rice
C r1 T~o
This document consists of 6 pages.
Date of Manuscript: Unknown
Date Declassified: July 9, 1947
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Technical Information Division, Oak Ridge Directed Operations
Oak Ridge. Tennessee f -
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 long-range 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 energy-enthalpy 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.
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 and-free 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), together-with the subsequent use off3, and as
paqtigL-mc.iola.ljuarptI ties on-.p 538 of Frank and Evans; and-the use of equation 11(21) on p 513
NIDDC 1091 fI
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.
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) = Si-si(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) = Sio-Si(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
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 + nS0-n,R In N, -nR In N, (2)
We then obtain for the partial molal entropy of the solute
S* = ( S /On,)T,p,n, = S2-R 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)
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.
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 S-s,(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, = S10-R 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-
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
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
,olute is assumed to be composed of hard-sphere 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 long-range 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 is-7600 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 long-range 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 Landolt-Bornstein 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 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.
neighbors about a solute molecule. Each neighbor, therefore, has about 1.0 cal deg-' mole-' of sur-
face entropy, about one-fourth 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 hard-sphere 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 hard-sphere 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 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 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 logarithm-of 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 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
oIdS;(VI between these extreme types of solution. The greater part of this variation is to be referred
to the introduction of long-rangh 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'(VI-AS2.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
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)A-dS,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 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
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.
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
6. Fowler, R. H., Statistical Mechanics, 2d ed, p 844, Cambridge University Press, 1936.
7. Rice, 0. K., J. Chem. Phys. 15, 314 (1947).
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