2519
IONSOLVATION IN WATERAND PROPYLENE CARBONATE
The Thermodynamics of Ion Solvation in Water and Propylene Carbonate by Mark Salomon N A S A , Electronics Research Center, Cambridge, Massachusetts 08189
(Received January SO, 1970)
The energetics of single-ion solvation in water and propylene carbonate are evaluated on the basis of an electrostatic model. Single-ion free energies of solvation are obtained by extrapolation (to infinite ionic radius) of a plot of the differences of cation and anion conventional energies vs. l / r . The enthalpy of solvation of individual ions is a more complicated function of l / r and several methods are used in evaluating these quantities.
Introduction The free energies and enthalpies of solvation of individual ions are of fundamental importance since they constitute the basis of theories on ion-solvent interactions.’ For the most part, the Born equation
cations and anions the conventional enthalpies are defined, respectively, as
AHconv(M+) = AH~olv(fi4’) - AHsolv(H+) AHconv(X-) = AHsolJX-)
4- AHBoIv(H+)
(3) (4)
The difference in conventional enthalpies is therefore
AH,onv(M+) - AHconv(X-) =
{ AHBOIV(M+) - AHsoiv(X-)1 - 2AHsOiv(H+) (5) and the Born-Bjerrum equation
Since the term on the left-hand side of eq 5 is known from experiment, Halliwell and Nyberg proposed that AH,,,,(H+) could be evaluated by extrapolation of the plot of AHconv(M+)- AHconv(X-) for ions of equal 6)3. On this basis a value of -260.7 radii vs. l/(ri kcal/mol was obtained for AHsolv(H+). Morrise used more recent thermodynamic data along with a revised set of crystal radii’O and obtained a value of -263.7 kcal/niol for this quantity. Recently, Conway and Salomonl1J2have pointed out that the results of this method may be too high since
+
have dominated the many attempts to calculate these quantities.’ The usual approach is to empirically adjust the crystal radii, ri, of cations and anions until the plot of AH or AG vs. l/(ri 6) is linear with all the Another method introions falling on a single duced by Bernal and Fowler4 involves the assumption that ions of equal radii have equal (coulombic) energies of solvation. In order to improve on calculations based , ~ Conway and on eq 1 and 2, Laidler and P e g i ~ and Desnoyers6have considered the effect of dielectric saturation. Laidler and Pegis have also shown that for the free energy, the leading term in the Born equation is the gas phase (crystal) radii and that large corrections to this radius are unjustified. Buckingham’ showed that in addition to the Born charging enthalpy, ionmultipole and polarization energies constitute major contributions to the overall solvation enthalpy. In Bucltingham’s model, ions of equal crystal radii have equal enthalpy values from all contributing factors with the exception of the ion-quadrupole enthalpy. At this suggestion Halliwell and Nyberga proposed a significantly different method for the calculation of the absolute solvation enthalpy of the proton, AHsolv(H+), which involves the differences in conventional enthalpies between cations and anions. For monovalent
+
(1) B.E.Conway and J. O’M. Bockris, “Modern Aspects of Electrochemistry,” Vol. 1, Butterworth and Co., Ltd., London, 1954, Chapter 2. (2) W. M. Latimer, K. S. Pitzer, and C. M. Slansky, J . Chem. Phys., 7, 108 (1939). (3) E.J. W.Verwey, Rec. Trav. Chim. Pays-Bas, 61, 127 (1942). (4)J. D.Bernal and R. H. Fowler, J . Chem. Phys., 1, 515 (1933). (5) K. J. Laidler and C. Pegis, Proe. Roy. SOC.,Ser. A , 241, 80 (1967). (6) B. E. Conway and J. E. Desnoyers, Phil. Trans. Roy. Soc., London, 256A, 389 (1964). (7) A. D.Buckingham, Discuss. Faraday SOC.,24, 151 (1957). (8) H.F. Halliwell and N. C. Nyberg, Trans. Faraday SOC.,59, 1126 (1963). (9) D.F. C. Morris, Struct. Bonding, 4, 63 (1968). (10) B.S. Gourary and F. J. Adrian, Sol. State Phys., 10, 128 (1960). (11) B.E.Conway, “Modern Aspects of Electrochemistry,” J. O’M. Bockris and B. E. Conway, Ed., Vol. 3, Butterworth and Co., Ltd., London, 1964,Chapter 2. (12) B. E. Conway and M. Salomon, “Chemical Physics of Ionic Solutions,” B. E. Conway and R. G. Barradas, Ed., John Wiley and Sons, New York, N. Y.,1966. The Journal of Physical Chemistry, Vol. 74, No. 18, 1970
MARKSALOMON
2520 the assumption regarding the cancellation of the iondipole term for r+ = r- is only valid if the orientations of the solvent a t the cation and anion are mirror images. However, if the model of Verwey3is considered, then the orientation of a water molecule a t a cation is a t 52" to the radial direction while that a t anions is along this direction. For this condition the ion-dipole energy does not cancel but differs by a factor of (1 - cos 52") for anions and cations of equal radii. The plots of conventional enthalpies should therefore involve a term in l / r 2 as well as one in l/r3. In order to resolve this problem, two methods, similar to that of Halliwell and Nyberg, are employed in this paper. The method is applied t o ionic solvation in water and in anhydrous propylene carbonate (PC). Conventional Free Energies One can define a set of relations similar to (3)-(5) for the free energy, i.e. by AGconv(M+) = AGso1v(Mf) - AGso1v(H+)
(6)
AGconv(X-) = AGso~v(x-)
(7)
+ AGso1v(H+)
dAGconv = AGconv(M+) - AGconv(X-) =
{ AG~I~(RII+) - AGsoiv(X-)} - 2AGsOiv(H+) (8)
For cations and anions of the same radii, the absolute enthalpy of solvation of the proton is therefore obtained from
where dAHconv = AHconv(M+) - AHconv(X-) and Z+ - 2- = -2AHsOlv(H+). In eq 10, 6 is the radius of the solvent molecule equal to 1.38 for water and 2.62 for PC (estimated from density measurement^).'^
Results For aqueous solutions the conventional energies are referred to the solvated proton and for PC the solvated lithium ion is taken as the standard of comparison. All data correspond to the standard states at 298.15"K and 1 atm pressure. The free energies of solvation are obtained mainly from emf data. For example, consider the electrode reaction Lis
+ AgCl, = Lib+ + Clh- + Ag,
(11) where the subscripts s and h refer to the solid and hydrated species, respectively. The standard free energy of solution of LiCl is given by
+
In order to evaluate AGosolv(H+),a suitable function AGt"(AgC1,) AGf"(LiC1,) (12) in r must be obtained for AGso~v(R.I+) - AGso~v(X--). AGaoln" = AGn" I n this paper it is proposed that the function is simply where AGf" is the free energy of formation and AG11' l / r so that eq 8 can be written as is obtained from emf data. The free energy of solvation is obtained from AGaolnO by (9) AGsolv" = AGsoln" AGlat" (13)
-
+
where ri is the gas phase radius of Gourary and Adrian.'" The basis of this assumption is attributed to the compensation effect between the enthalpy and entropy of s ~ l v a t i o n . ~ JIt~ is now generally accepted that the enthalpy of hydration is a complex function (e.g. a power series) of l / r (7) and the corresponding entropy effects should also involve the same factors since those factors which lead to an increase in the enthalpy of interaction between the ions and water also tend to restrict the degree of freedom of the water molecule in the primary hydration shell, thereby decreasing the entropy.13 Since these effects oppose each other, it is reasonable to assume that the free energy contributions from terms higher than r--1 are small and that they tend to cancel for large ions when differences in energies are considered. Having obtained AGso~,"(H+)from eq 9, the corresponding enthalpy term can be evaluated by = -31.3 eu.'," taking ASso~vo(H+) Conventional Enthalpies To evaluate the enthalpy of solvation of the proton, the conventional enthalpies were fit to the least-squares relation
The Journal of Physical Chemistry, Vol. 74, N o . 18, 1970
The lattice free energy is obtained from
+
AGlat"(LiC1) = AG,"(Li,) AGt"(C1,) - AGf"(LiC1,)
+I - A
(14)
where I and A are, respectively, the ionization potential of the metal and electron affinity of the halide. Values of AGl10 in water and other combinations of metalhalide systems were obtained from deBethune's tabled5 with the exception of the Li-H2 couple.lB For the alkali halides in PC our previous results are used.l7J* I n the absence of emf data free energies of solution in P C were estimatedls from AGsO1nO = -2RT In (maatyjJ
(15)
where maatis the solubility of the salt and yf is the mean molal activity coefficient which is evaluated from the Davies equation as discussed previously. l8 (13) D. D.Eley and M. G. Evans, Trans. Faraday SOC.,34, 1093 (1938). (14) M.Salornon, J. Phys. Chem., 73, 3299 (1969). (15) A. J. deBethune, T. S. Licht, N. Swendernan, J. Electrochem. Soc., 106, 616 (1959). (16) D.It. Cogley and J. N. Butler, J . Phys. Chem., 72, 1017 (1968). (17) M.Salomon, J. Electroanalyt. Chem., March, 1970. (18) M. Salomon, ibid., submitted for publication.
2521
IONSOLVATION IN WATERAND PROPYLENE CARBONATE AGt" values were taken from several recent compilati0ns1~~~0 with the exception of R b and Cs salts which were taken from Latimer's tables.z1 Ionization potentials were obtained from a recent monographzz and the electron affinities of the halides are those measured by Berry and ReimannsZ8 Data for the OH- ion are based on the reaction
HzOli, = Hh+
+ OHb-
200
r
,
7-30
(16)
for which we can write AGoonv"(OH-) = AGi6"
+ AGfO(HzO1) AG," (Hg+) - AGf" (OH,-)
Values of AGleo and AH16" are, respectively, 19.095 and 13.526 kcal/mol. z4 Crystal radii for the alkali metals and the halogens are those based on Gourary and Adrian's data.1° For the OH- ion the crystal radius as determined by X-ray (cf. ref 21 of the paper by crystallography is 1.03 KhomutovZb). Values as high as 1.47 have been reported.* The value of 1.12 ichosen here is slightly larger than the experimental one and is justified on the basis that some increase accompanies the solvation process due to hydrogen bondinga25 The crystal radii for Ag+ and Tlf were taken from ref 22 while the remaining values are from ref 8. Lattice enthalpies a t 298.15"K when not available from the literature were calculated from = -------{1 600
+
-
-} 0.4 +
130'
I'
e3
-4
I
I
.S
.6
I .7
I
I
I
.8
.9
1.0
I
1.1
I /r
Figure 1. Conventional free energies as a function of ri-1 in water.
60
I - 190
%
0
-180
X
(17)
r+ Yr+ rThis relation is claimed to be accurate to within 1-2 kcal/mol.z6 This accuracy may, however, diminish significantly due to errors in estimating crystal radii. Hence for the alkali halide salts, eq 17 does in fact predict values close to the thermodynamic ones, but the errors in AH,,," for perchlorate, tetraphenyl arsonium (AsPh4+),and tetraphenyl borate (BPh4-) are unknown due to the unknown uncertainties in their radii. Finally, the entropies, enthalpies, and free energies for the (CH,)JV + ion were obtained from Boyd's work.z7 Table Iz8lists the values of lattice energies and solvation energies for the various 1: 1 salts. All values are based on the molal scale. Table I1 lists the conventional free energies and enthalpies of the individual ions. Figures 1 and 2 show the plots of conventional free energies 'us. 1 / ~ for the aqueous and PC systems, respectively. There are no existing data to allow one to calculate AGsolvo for the cesium ion in PC. While some data exist on the solubility of LiF in PC, they are conflicting and give anomalous values for the free energies of transfer (cf. 18). The conventional energy for F- in PC
5
-170
-
- I60 -2
.3
.4
'5
.6
.7
.8
.9
1.0
I/r
Figure 2. Conventional free energies as a function of ri-1 in PC.
AG,,,,"(F- in PC) = 207.0 kcal/mol. This value was used1*to evaluate the conventional free energies for all the alkali metal fluorides. The plots of differences in conventional energies for ions of equal radii are shown in Figures 3 and 4 for the aqueous and PC systems, re(19) D. D. Wagman, W. H. Evans, v. B. Parker, I. Halow, s. M. Bailey, and R. H. Schumm, "Selected Values of Thermodynamic Properties," N.B.S. Technical Note No. 270, 1968. (20) "JANAF Thermochemical Tables," The Dow Chemical Co., Midland, Mich., Aug, 1965, and addenda. (21) W. M. Latimer, "The Oxidation States of the Elements and Their Potentials in Aqueous Solutions," Prentice-Hall, Englewood Cliffs, N. J., 1952. (22) J. P. Jesson and E. L. Muetterties, "Basic Chemical and Physical Data," Marcel Dekker, Inc., New York, N. Y., 1969. (23) R. S. Berry and C. W. Riemann, J . Chem. Phys., 38, 1540 (1963).
The Journal of Physical Chemistry, Vol. 74, N o . 12, 1970
2522
MARKSALOMON
Table I : Lattice and Solvation Enthalpies and Solvation Free Energies (25')
HF HCI HBr HI LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KCI KBr
KI RbF RbCl RbBr RbI CsF CSCl CsBr CSI AgC1' AgBr Ad TlCl TlBr
TI1 LiBPha NaBPh4 LiC104 NaClOc (CHa)dC1 AsPhrI
...
...
... ...
...
248.6 205.9 195.8 182.1 220.7 188.1 179.8 167.9 196.4 171.4 164.7 154.5 186.6 164.4 158.1 149.2 174.7 157 6 152.0 143.8 218.7 215.9 212.3 179.0 175.2 169.0 107. 8c 103.4c 156.5" 147 .4c 132. I d 90. o c
230.0 188.5 178.7 165.5 202.1 170.6 162.6 151 1 178.4 154.1 147.6 138.2 170.2 149.0 142.9 134.4 161.4 144.9 140.2 132.3 201.5 198.7 195.4 162.2 158.4 152.3
... ...
...
I
I
...
...
...
... *
.
.
I
I
.
383.7 350.3 345.9 333.1 247.5 214.8 207.5 197.2 220.5 187.2 180.0 169.7 200.6 167.3 160.0 149.7 198.2 160.3 152.9 143.2 186.0 153.3 145.8 135.8 202.9 195.8 185.6 168.8 162.2 151.5 135.8 108.2 163.8 144.1 132.2 81.7
... ... ... .,. ...
207.6 202.6 197.4
...
182.0 177.0 171.7
... 165.8 160.8 155.6
...
159.6 154.6 144.4
...
153.0 148.0 142.8 197.7 192.7 187.5
... ... ...
139.8 114.1 166.0 150.5 129.6* 84.3
... ... ...
363.7 335.3 328.9 320.7 226.6 198.2 192.0 183.5 201.1 172.7 166.5 158.0 183.6 155.2 149.0 140.5 179.4 151 .O 144.8 136.4 175.4 147.0 140.9 132.4 188.2 182.0 173.4 157.2 150.9 142.4
207. On 183.2 179.0 173.0 183.9 (160.1) (155.8) 150.1 168.5 (144.7) (140.5) 134.9 166.5 142.7 138.5 132.5 164.7 140.9 136.7 130.7 174.2 (170.0) (164.0) 145.3 141.1 135.7
... ... ...
...
...
118.0 ,..
...
... ...
...
,..
...
* Values in parentheses are obtained from the additivity relationship for ACsolvoand/or a See text for derivation of this quantity. AH,olvO. The remainder of the data were obtained from emf work,16--1* solubilityl* and heats of solution and transfer from references AH],$' calculated from eq 17. Heats of solution in PC and HzO are 28-30. Additional details can be found in ref 18 (in press). (1) J. N. Butler, D. R. Cogley, and W. Zurosky, J . from ref 28-30. d Reference 27. e From the heats of transfer in ref 28-30. Electrochem. Soc., 115,445 (1968), free energy of solution. (2) J. N. Butler, private communication on heats of solution. spectively. From the intercepts at l / r = 0 it is found that AGAV(H+ in HzO) = -235.0 kcal/mol and AG,,~,"(Li+ in PC) = -95.0 kcal/mol. Using the value of -31.3 eu for the absolute entropy of solvation of the proton in water, the corresponding enthalpy term is -244.3 kcal/mol. The results for the various monovalent ions are listed in Table 111. Also shown in this table are the heats of transfer for the individual ions based on the present work and on the assumption that for large cations and anions (such as AsPh4+and BPh4-) the heats of transfer are e q ~ a l . ~ ~ ~ ~ ~ I/?
Figure 3. Conventional free energy differences between cations and anions of equal radii &s a function of ri-1 in water. The Jaurnal of Physical Chemistry, Vol. 74, No. I& 1970
(29) G. Choux and R. L. Benoit, J . Amer. Chem. Soc., 91, 6221 (1969). (30) C.V. Kriahsan and H. L. Friedman, J . Phys. Chem., 73, 3934 (1969).
ION
SOLVATION I N WATER
A X D PROPYLENE
treatment for solvation in PC gives the following results
Table I1 : Conventional Energies and Enthalpies at 25'" ,------AHconvo-
Ion
Li + Na+
K+ Rb+ c s+ Ag T1+ (CHz)D AsPhi + OH F+
+
c1-
Br-
IClOaBPh4-
AHconvo(l\l+) = 171.06 -
------AQs,mvO-
H20
PG
IlzO
136.7 164.1 184.1 191.1 198.2 148.3 182.3 218.1 251.4 -386.1 -384.6 -350.3 -345.9 -333.1 -300.5 -272.4
...
137.2 162.6 180.1 184,3 188.3 147.1 178.1 217.3
23.7 41.8 48.1 54.4 9.9
... 75.4 113.1
PC I
,
...
I
.
Pc
+
100
I
I
I
I
I
I
I
I
.I
.2
.3
.4
.5
.6
.7
I .8
Ilr
Figure 4. Conventional free energy differences between cations and anions of equal radii as a function of vi-1 in PC.
A second set of calculations was carried out according to eq 10 and for the aqueous system the results are AHoonv" (M +)
=
271.09 307.65 +--(Ti + 1.39)3
838.52 (Ti 1.38)2
+
(h8.6)
AHConv"(X-) = -223.47 (Ti
1945.8 1.38)2+
+
(Ti
2304.9 1.38)3
+
(10.3)
( ~ i
AHconvo(X-)= -41.40 -
+ -+
7326.0 2.62)2
( ~ i
18205.6 ( ~ i 2.62)3
+
(f1.4)
Conclusions The present results suggest a revision in the presently accepted values for the energy and enthalpy of solvation of the proton as well as for the other ions. The difference amounts to 16 kcal/mol, which is somewhat larger than our previous estimate of 8 kcall mo1."J2 In regard to other related quantities such as the proton affinity, P , no significant changes are indicated. For the process Hg+
1
0
+
The value of AHsolvo (Li+) in PC is - 106.2 kcal/mol.
... ...
a Aqueous values are referred t o the solvated proton; values are referred t,o the solvated Li ion.
240
(
5263.0 + _~ + 2.62)3
I
-207.0 -183.2 -179.0 -173.0
.
3474.1 ~ i 2.62)'
_.____
...
... .
.
...
, . .
-207.6 -206.6 -197.4 -166.0 -140.0
.
23.1 38.5 40.5 42.3 9.6 38.3
-366.1 - 363.7 -335.3 -329.2 -320.7
, . .
2523
CARBONATE
P + BzOg + H30g'
AHso~v'
HgOhf
the solvation enthalpy of H30+ can be estimated by taking its radius as 1.4 8 and from the least-squares fit it is found that AHso~vO(H~O+) is -67.4 kcallmol and that P = -176.9 kcal/mol. This is consistent with the presently accepted value of = - 18Oj3Iand values as low as -150 to -160 k c a l / m 0 1 ~ must ~ > ~ ~be ruled out. Ionic radii still appear to present some poroblems. For the OH- ion in solution a value of 1.12 A is indicated by the linear anion plot of Figure 1. A value of 1.47 8 must be considered to be in error since it is a "thermochemical" value which is based upon incorrect values for the lattice enthalpies of the alkali metal hydroxides. wadding ton'^^^ values of these lattice energies are obtained from a Mayer-Born-Madelung treatment from which he finds that ARM' for NaOH and KOH, respectively, are -197 and -175 kcal/mol. This leads to a value of -371 kcal/mol for the conventional enthalpy of OH- which is smaller than the fluoride ion, therefore requiring a larger ionic radius. Using the thermodynamic data found in references 20 and 21, values of -244.1, -212.1, and -188.5 kcal/mol are obtained for AHlato of LiOH, KaOH, and KOH, respectively. These lattice energies when combined with the heats of solution35and conventional enthalpies of
(*0.3)
The data are given in kcal/mol and the quantities in parentheses are the standard deviations of the intercepts. The value of AHsolvo(H+) found by this method is -247.3 kcal/mol. The value obtained above (-244.3) has been used in evaluating the single-ion values for AHsolvoas shown in Table 111. A similar
(31) M.DePas, J. J. Leventhal, and L. Friedman, J. Chem. Phys., 51, 3748 (1969). (32) J. L. Beauchamp and S. E. Butrill, ibid., 48, 1783 (1968). (33) D. Van Raalte and A . G. Harrison, Can. J. Chem., 41, 3118 (1963). (34) T.C.Waddington, Adv. Inorg. Chem. Radiochem., 1 , 158 (1959). (35) V. B. Parker, "Thermal Properties of Aqueous Uni-univalent
Eleotrolytea," NBS Reference Data Series NSRDS-NBS2, April 1965. The Journal of Physical Chemistry, Vol. 74, No. 1.9, 2970
MARKSALOMON
2524
Table 111: Energetics of Single-Ion Solvation at, 25''
H+ K+ Rb +
c s+ Ag+
TI + (CHB~N +
ASPhq OH -
F-
c1-
BrIc104 BPha5
235 0 97.8 72.4 54.9 50.7 46.7 87.9 56.9 17.7
+
95.0 71.9 56.6 54.5 52.7 85.4 56.7
...
... ...
... 131.1 128.7 100.3 94.2 85.7
112.0 88.2 84.0 78.0
... ...
Units are all in kcal/mol.
263.7 127.5 99.7 79.6 73.3 69.1 113.4 81.4 45.6 15.0 123.9 120.0 87.5 80.0 70.2 36.8 12.2
..,
I
Li + Na+
... ,..
b
Reference 9.
Present results.
Li+, Na+, and I
