THE ENTROPY CHANGES ACCOMPANYING ASSOCIATION REACTIONS IN SOLUTION1 EDWARD I. KING University of Wisconsin, Madison, Wisconsin
THE
extent to which a reaction mill proceed a t a given temperature is determined by the concentration conditions and the standard free-energy change accompanying the reaction at that temperature. The standard free-energy change is related to the changes of other thermodynamic quantities, the enthalpy and entropy, by the equation: AFn = AHo - TASo
importance of the entropy change in determining the free-energy change of a reaction. Thus, i t may be said that there are endothermic reactions which go because of favorable entropy changes, and exothermic reactions which don't go because of unfavorable entropy changes. Because the entropy change for a reaction is a measure of the change in randomness within the reaction system, it is expected that association reactions occurring in the gas phase will be accompanied by a decrease in entropy. This expectation is realized; typical data are given in Table 1.
For the purpose of gaining an insight into the factors which cause reactions to take place, it is worth while to consider the free-energy change to be a consequence of the corresponding enthalpy and entropy changes. The TABLE 1 enthalpy change is related to the difference of the bond Entropy Changes in Association Reactions Occurring in energies of the products and reactants, and in the case Gas Phase of reactions occurring in liquid media, to the difference Reaction AS(e. u.) tern^. Ref. of solvation energies of the products and reactants. The entropy change, on the other hand, is related to the change in randomness in the system which accompanies the reaction. An increase in the entropy of a system corresponds t o an increase in the randomness of that system; the greater is this increase in randomness which Mere examination of the conventional equations for accompanies a reaction, the greater is the extent to the reactions of association of ions in solution, equawhich the reaction will go. This influence which the tions which show a net decrease in the number of partientropy change of a reaction has in determining the cles accompanying reaction, might lead to the expectafree-energy change is shown by considering the two tion that the entropy change in such reactions would reactions: also be negative. This expectation is not realized. The entropy change in the reaction of hydrogen ion H,O (liquid, 25%) = H,O (solid, 25'C.) H>O(liquid, 2 5 T ) = Hdl (gaa, P = 1 cm. Hg, 25'C ) with the anion of a weak acid is always positive and an increase in entropy accompanies the formation of many The first of these reactions is not spontaneous, although it is exothermic (AH = -1650 cal.), for it has an en- complex ions from the constituent positive and negative anomalous character of associa.. tropy change of -6 cal. mol-' degree-'. Water ions. This amarentlv tion reactions in solution is more easily understood after molecules have more freedom in the liquid state than in an examination has been made of the entropies of ions the negative value of the entropy the solid state; change is just as expected. The second reaction does and molecules in solution. go in spite of its extremely endothermic character; the E ~ T ~ oF ~ DISS~LVED P I ~ ~SUBSTANCES value of AH is +10.500 cal. A laree increase in en. * A definite relationship is observed between the entropyaccompanies the reactio,n (AS = +41.5 Gal. mol-' degree-'). This large entropy increase reflects tropy of an ion in aqueous solution and its charge, the much greater randomness of the water molecules in radius, and mass. R. E. Powell and W. M. Latimer the gaseous state than in the liquid state. Correspond- (3) have shown that the standard partial mohl ening reactions for the solidification of any liquid a t a tropy, So,of a very large number of monatomic ions is tem~eratureabove the melting ~ o i n or t for the va~ori- given very closely by the equation: sation of the liquid to give vapor a t a pressure below 9 = (3/2)R In M + 37 - 270 Z/r.' (1) the saturation pressure would equally well illustrate the where M is the atomic weight, Z is the absolute value of ' Based in Part on a talk presented before a Symposium on the charge on the ion, and re is the effectiveionic radius. Equilibrium and Rate Behavior of Complex Ions held st the Uni- ~h~ value of ra was taken to be 2,00 greater than the versity of Chicago. February 21-23, 1951. This symposium ww radius and 1.00 A greater than the spon~oredby the Office of i-qaval ~~~~~~~hand the ~ $ crystal ~ ~ ifor cations ~ crystal radius for anions. These authors used the crysEnergy Commission.
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JOURNAL OF CHEMICAL EDUCATION
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0.1 ..
OJ
,
0.4
~5
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iolu.
(Figure through the courtesy of R. E. Powell and W. M. LaHmer and wlth the permiasion ofthe =&tar of the Journal of Chemical Physics.)
tal radii which are give11 by Pauling (4). The agreement between the experimentally determined values of Soand those calculated using equation (1) is shown in Figure 1 which has been reproduced from the paper by Powell and Latimer. Over a very wide variation of the quantities involved, the agreement between the observed and calculated values of Sois quite good.= The large loss of entropy which accompanies the solution of a gaseous ion, particularly an ion of high charge, is due primarily to the great loss of freedom which the water molecules suffer in becoming coordinated with an ion.' A part of this decrease in entropy is, of course, a consequence of the loss of freedom of the gaseous ion upon entering the condensed phase. I t must be pointed out that the entropies of solution of gaseous ions are more positive than might be expetted in view of the large values of the heat of solution of gaseous ions. The basis of this expectation is the Barclay and Butler empirical mle concerning the AH and AS of vaporization (6). These authors have plotted the AS ifvaporizati'o&versus the AH of vaporization at 25OC. for (a) gases from various nonassociated solvents, (b) carbon disulfide, chloroform, benzene, and chlorobenzene from acetone, ( e ) the four lower alcohols from benzene, and (d) a number of nonassociated liquids as pure substances. All of the points fall close to a straight line corresponding to the equation: AS. = +14.5 0.0011 AH, (the units are cal. mol-' for AH, and cal. mol-' degree -' for AS; the standard states are 1atmosphere for the gas, and the pure liquid for the solutes). A graph which shows the AS and AH of vaporization from solution for some ions, as well as neutral molecules, is presented as Figure 2. It is clear that the gain of entropy of the system upon trausferring an ion from the solution to the vapor phase is much smaller than would be predicted on the basis of the Barclay and Butler relationship. This conclusion that an ionic solution has "too much entropy" is explained by H. S. Frank and M. W. Evans (7) by picturing a region outside of the first coordination sphere in which the water structure is more broken down than it is in ordinary liquid water. It becomes evident that factors other than ionic charge have an important hearing on the entropy of solution of gaseous species in water when one compares the entropy changes associated with two reactions which involve isoelectronic substances:
+
-
K C (gas)
+ C1-(gas) = K + (aq.) + CI- ( s ~ . ) 2.4 (gas) = 2A (ilq.)
The values of ASo are -36 and -44 e. u. res~ectivelv.
'OO
50 100 A H (KCALS.)
150
Fi~iura2. As of sapovination versru AH of .apod.mtion of ionic end non-ionic solutes from water(and from methyl doohol i n the indicated c-a).
The solid line is the Barelay-Butler line. The values of the A S , are 8 e.u. more positive than the A S , based an the etandard states discussed here (i.e., one m o l d solution, one atm. pressure).
(Figure through the oaurtssy 01 H. S. Frank and M. w. Evans and wlth the per mission of the editor of the Journal of Chemical Phvricr.)
2 The entropy values plotted in Figure 1 are conventional ionic entropies. They are based upon S o = 0 for hydrogen ion. Since the true ionic entropy of hydrogen ion is close to zero (6), these values may be considered as true ionic entropies. (The error caused by this assumption is proportional to the charge on the ion. This follows from the fact that the directly measurable quantities are the sums of the entropies of equivalent amounts of positive and negative ions and the differencesof the entropies of equivalent amounts of ions of charge of the same sign.) 8 The values plotted as the ordinate in Figure 1 are equal to the entropies of solution of the gaseous ions plus 26.00 e. u.
FEBRUARY, 1953
73
if the standard states are one atmosphere pressure for the gas and the hypothetical molal solution for the dissolved substances. It is seeu that the presence of the charges actually decreases the amount of entropy lost in the r e a ~ t i o n . ~Solution of a nonionic solute in water is generally accompanied by a decrease in entropy. The values of ASo of solution for a number of noniouic solutes in water are plotted versus the molal volume of the solute in the pure liquid state in Figure 3. This straight line relationship was observed by Powell and Latimer (S), from whose paper this figure is reproduced. In contrast to the case of ionic substances, the ASo of solution of nonionic solutes in water is more negative than would be expected on the basis of the Barclay and Butler relationship (6) in view of the low values of the AH of solution. This is shown in Figure 2. It appears that the water molecules lose considerable freedom when a nonionic solute is dissolved in water; the entropy decrease is very much greater than could be accounted for by the loss of freedom of the solute molecules. Frank and Evans (7) picture the nonpolar solute causing the "freezing" of solvent water molecules in the vicinity of the solute molecules! As the temperature is raised, these regions of "frozen water" melt, giving rise in the case of the noble gases to enormous partial molal heat capacities, which may exceed 60 cal. mol-' degree. -' The correlation of ASo of solution with the molal volume of the solute is not inconsistent with this picture because larger molecules ~ o u l dbring about more "freezing of water" than would smaller ones. Since we wish to consider the entropy changes associated with the formation of complex species which may be ionic, it is desirable to examine the entropies of some pdyatomic ions in aqueous solution. The entropies of many such ions are more negative than would be expected from consideration of the net charge and size alone. The entropies of UOz+,UOz++, Cs+, and Ba++ are presented in Table 2. The factors of size and mass
the entropies of these uranium ions more negative than the entropies of simple ions of the same net charge. The entropies of some closely related polyatomic ions are presented in Table 3. It is seen that the entropy is TABLE 3 The Entropies of St~ucturallySimilar Polyatomic Ions in Aqueous Solution at 2SaC.
Charm
not determined exclusively by the size and net charge of the ion. There are large differences between the entropies of the ions which have the same charge, even though the sizes of these ions must be very nearly the same. The data for the three phosphate ions and the series of ions: perchlorate, sulfate, and phosphate ion indicate that the dependence of the entropy upon charge is greater than first power. This trend is important in determining the relative acid and base strengths and, presumably, the complex forming tendencies of these species.
-20
as; -2s
TABLE 2 A Comparison of the Entropy of Monatomic end Polyatomic IOM of the Same Charge +1 +2
UOzt
12 (8)
UOzC+ - l i (9)
- 30
9,d
S.*d
Charge
Ck"
ST'
PV
Cat Ba++
31.8 (5) 3 . 0 (5)
would lead to the expectation that the entropies of the polyatomic uranium ions mould be more positive than the entropies of the simple ions. The opposite is observed. In these nonspherically symmetrical ions, the solvent is not completely shielded from the higher charge on the central ion; certainly the charges on the uranium atoms of oxidation number +5 and +6 are greater than +1 and +2. If one assumes that the uranium atom and oxygen atoms are bonded by a single bond, the formal charges on the uranium atoms are +3 and t4. I t is reallv not sururisina. therefore, to find
' This compmison was pointed out by Frank and Evans (7)
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3.
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40
50
70
80
10
20
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X(.. 1 d m . ) + X(.W.. .tmd.rd .tat. 1 m o l d rolum. of th. solut. u th. . . u p
30
60
vw
th.
liquid.
mgurs through the mrteov 01 R. E. powell and W. M. &timer d w t h Wmission of the editm ol the J o u r n d of Chemical Phvricr.) The factors which determine the entropy of a substance in aqueous solution are not simple. Certain trends have been observed, however, and these will be of value in our discussion of the entropy changes which accompany association reactions in solution.
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JOURNAL OF CHEMICAL EDUCATION
TABLE 4 Reaction
H+ H+ H+ H+ H+ H+
++ NH8 = NH4+ MeNHt = MeNHst ++ MelNH = MezNH2+ MerN = MesNH+ + EtNH2 = EtNHat + I brobalticyanide ions are not more positive in view of their mide > chloride > fluoride, such as the cadmium halides charges. These ion-pairs are believed t o involve the (54) and mercury (11) halides (56), are systems in which hydrated lanthanum ion (Sf) (52); this is based pri- the bonding undoubtedly has a great deal of covalent marily on the value of "a", the distance of closest ap- character. In general it might be expected that those anions proach parameter in Bjerrum's equation for ion association (85). The magnitudes of the ASo values sub- which have large positive values of Sowould have only stantiate this view. If the lanthanum ion exists in the a slight tendency to form complex ions or ion pairs with th entropy values associated ion-pair as the hydrated ion, the ASo value metal ions. The anions, ~ ~ i known would be expected to be much less positive since the compiled by Latimer, Pitzer, and Smith ( l l ) , which
FEBRUARY. 1953
have the most positive values of So are permauganate, perchlorate, chlorate, bromate, and nitrate. These anions are certainly not noted for their complex forming tendency. One factor which makes the entropies of these ions more positive is their rotational degrees of freedom but it also seems likely that these polyatomic anions are not very heavily hydrated. Thus the contribution to the stability of the complex ion or ion-pair which arises from the entropy term would not be as important in the case of these anions. Another factor, also related to the entropy change, which may play a role in determining the stabilities of complex ions involving polyatomic anions is the loss in rotational freedom which the anion itself must undergo when it becomes coordinated to a cation. The values of the entropy change which are associated with various reactions in solution have been presented. Certain trends in these values and other generalizations have been pointed out. In some cases, these trends in the value of AS are amenable to simple rationalization; in other cases, this does not appear to be possible. The most important point, which the data illustrate, is the role which the water molecule-solute interactions play in determining the value of AS for a reaction in solution. The stability of many molecules and complex ions with respect to dissociation in solution is markedly influenced by the magnitude of the AS for the dissociation reaction.
(8) "Selected Values of Chemioal Thermodynamic Proper:tiee," Series I, Volume I, Table 81-2 (National Bureau of Standards). J. Am. COULTER,L.'v., K. S. PITZER,AND W. M. LATIMBR, Ckem. Soc., 62,2845(1940). STEPHENSON, C. C., ibid., 66, 1436 (1944). LATIMER, W. M., K. S PITZER,AND W. V. SMITE,ibid., 60, 1829 (1938). EVERETT, D. H., AND W. F. K. WYNNE-JONEB, Tram. Fapaday Soe.,35,1384(1939). Trans. Far& Soc.. . . EVANS.A. G.. AND S. D. HAMANN. 47,& (1951j. (14) Private oommunioation from J. G. X m x w o o ~to F. P. PRICEAND L. P. HAMMETT,J. Am. Chem. Soc., 63, 2392 11941) - - - - ,. (15) AWTREY, A. D., AND R. E. CONNICK, ibid., 73, 1842 (1951). (16) WINSTEIN,S., AND A. J. LUCAS,ibid., 60,836 (1938). (17) S ~ T H , 'V., ~ .0. 1, I. BROWN,AND K. S. PITZER,i b i d , 59, 1213 (1937). (18) BAXENDALE, J. H., AND P. GEORGE,T r m . Faraday Sot., 46,55(1950). (19) PITZER,K.S., J. Am. Chrm. Soc.,59, 2365 (1937). (20) MAGEE,J. L., T. RI, AND H. EYRING,J. Chem. Phys., 9 , 419 (1941). (21) HARNED, H. S., AND B. B. OWEN,"The Physical Chemistry of Electrolytic Solutions", 2nd ed., Reinhold Publishing Corporation, New York, 1950,pp. 508-516, 529-536. (22) J. Am. Chem. Soe.. . . BATES.R. G..AND G. D. PINCHING. . 71, . 1274 (1949j. (23) MCCONNELL, H., AND N. DAVIDSON. ibid., 72, 3164 (1950). (24)Kmc, E.L., ibid., 71,319(1949). (25) Rhemow~ca,E.,AND W. H. STOCKMETER, ibid., bP, 335 (1942). (26) EVANS, M. G., P. GEORGE, AND N.URI, Trans. Faraday SW., 45,230 (1949). (27) JAMES,J. C., J. Chem. Sac., 1951, 153. LITERATURE CITED (28) . . DAVIES. C. W.. AND P. A. H. WYATT. Trans. F a d a y (1) KELLEY.K.K.. "Entroniss - -~~ r - ~ of - Inoremir. Suhstanreq." .-...--... T sot., is,770(i949). - I .T. ~ u r e a ; of ~ i n e Bulletin, s 1940, 434. (29) SCHWERT,J., paper presented a t Symposium on Complex (2) BROWN,H. C., AND M. D. TAYLOR, J. Am. Ckem. Soc.,69, Ions and Polyelectrolytes held at Ithaca, New York, June 1.11'211067~ ,. 21,1951. (3) POWELL, R. E., AND W. M. LATIMER, J. Chem. P h w , 19, (30) SCHUBERT, J., E. R. RUSSELL,AND L. S. MYERS,J. Biol. 1139 (1951). Chem., 185,387 (1950). (4) PAULING, L., "The Nature of the Chemical Bond," Cornell (31) JAMES,J. C., .4ND C. B. MONK,Trans. Faraday Soc.,46, University Press, Ithaca, New York, 1939. 1041 (1950). (5) J . (32) DAvrEs, C. W., AND J. C. JAMES,Proe. Roy. Soe. (London) . . LATIMER.W. M.. K. S. PITZER.AND C. M. SLANSKY. ~ h e m . ~ h y 7,'108(1939). s., ' [A], 195,116(1948). (6) BARCUY,I. M , AND J. A. V. BUTLER,T~ana.Faraday Soc., (33) BJERRUM,N., Kg1 Danske Videmk, ' ~ e l s k a b . 7, , No. 9 119%) 34,1451 (1938). , - - -* ,. (7) FUNK, H. S., AND M. W. EVANS,J . Chem. Phys., 13, 507 (34) LEDEN,I., Z. physik Chem., 188A, 160 (1941). (1945). (35) SILLEN,L. G., A d o Ckem. Scand., 3, 539 (1949).
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