Thermodynamic Properties of
2479
a Hard-Sphere Solute
The hydrolysis schemes discussed above permit calculation of the concentrations of individual ions as a function of temperature. Thus, for a given total metal concentration of M , a t temperature T, and acidity pH(T), the concentration of species AlX(OH),'3x - > ) + is given by eq 14 where m, the free metal in the system, is equal to the positive root obtained by solution of eq 15. For the present purpose, it is convenient to define of parameter Pxy equal to the percentage of total metal tied up as the complex Alx(OH),(3X--5)+.An analytical expression for this parameter is easily derived from eq 14 and results in the following equation for Px>
P,,
=
(lOOxQx,mxml-~)/M
(19
In examining the variation of Pxy with temperature we have chosen to use pH(T)-temperature data for a 0.038 mol/kg Al3+ solution in 1 mol/kg KC1 with a starting pH(25) of 2.59 (Figure 2). Also, the two species (A12(OH)24+,A114(OH)34*+) hydrolysis scheme proposed by Mesmer and B a e ~ 5was ~ selected since it reproduces the experimental hydrolysis data to a higher temperature (149.8") than do the three species schemes also listed (good to 124.8"). Plots of Pxy us. temperature for A13+ ( i e . , free metal ion), A12(OH)24t, and A114(OH)34~+are shown in Figure 10. These curves demonstrate that a t temperatures less than 125" the principal species involved in the hydrolysis of Al3+ are the free metal ion, Al3+, and the dimer, Alz(OH)24+. At higher temperatures, however, the large polymer, Ah4(OH)348+, becomes important. This species is probably the precursor to the formation of boehmite since precipitation was only observed to occur a t
temperatures greater than 150". Also, the Z ( T ) us. temperature plots shown in Figure 7 indicate that the formation of boehmite is irreversible with respect to changing temperature while a t lower temperatures almost no hysteresis in the Z ( T ) us. temperature plot was observed. Therefore, the following scheme is proposed to provide a quantitative description of the hydrothermal hydrolysis of Al3+ in aqueous KC1 solution a t elevated temperatures
+ 2H20 = Al,(OH),'+ + 2H' 14A13+ + 34H20 = Al14(OH)348++ 34H+ 2A13+
(20)
(21) followed by the precipitation of boehmite according to 6H20 8H+ ( 2 2 ) A1l,(OH),,e+ + 14y-AlOOH
+
+
References and Notes (a) Present address, Department of Chemistry, Victoria University of Wellington, Wellington, New Zeaiand. ( b ) Summer student MaySept 1972. (a) H. C. Helgeson, "Complexing and Hydrothermal Ore Deposition," Macmillan, New York, N . Y . , 1964; (b) G. Kortum, "Treatise on Electrochemistry," 2nd ed, Elsevier, Amsterdam, 1965. D. D. Macdonaid, P. Butier, and D. Owen, to be submitted for publication. R. E. Mesmer, C. F. Baes, Jr., and F. H . Sweeton, d. Phys. Chem.. 7 4 , 1937 (1970). (a) R. E. Mesrner and C. F . Baes, Jr., inorg. Chem., 10, 2290 (1971). ( b ) Mesmer and Baes have assigned boehmite the formula a-AIOOH which in fact corresponds to diaspore (ASTM 5-0355) D. D. Macdonald and P. Butler, Corros. Sci.. 13, 259 (1973). A. S. Quist and W. L. Marshall, J. Phys. Chem.. 69, 2984 (1965) R. A. Robinson and R. H. Stokes, Electrolyte Solutions," Butterworths, London, 1959. "Handbook of Chemistry and Physics," 50th ed, The Chemical Rubber Publishing Co., Cleveland, Ohio, 1969. G . C. Akerlof and H . I. Oshry, d . Amer. Chem. SOC., 72, 2846 (1950).
Thermodynamic Properties of a Hard-Sphere Solute in Aqueous Solution at Various Temperatures in Relation with Hydrophobic Hydration M. Lucas Departement de Genie Radioactif, Commissariat a I'Energie Atomique, 92260 Fontenay-aux-Roses, France (Received Aprii 2. 7973) Pubiication costs assisted by Commissariat a I'Energie Atomique
The limiting variation with the solute molality m of some partial molal thermodynamic properties of a nonpolar molecular solute has been calculated by means of the scaled-particle theory. The variation of calculated dVp/drn with temperature appears to be closely related to the unusual variation of the pressure derivative of the compressibility of pure water. The calculated heat of dilution L Z is also related to the temperature derivative of the water compressibility. Although the fit between calculated and experimental Vz/dm and Lz is only qualitative, it suggests that the variations of these quantities with the temperature are not directly related to the solute influence on the water structure.
Introduction Considerable importance has been ascribed to the fact that the partial molal volume Vp of a hydrophobic solute decreases with increasing molality and that the relative partial molal heat content Lz of such a solute is positive.
On this basis conclusions have been drawn regarding the ability of the solute to increase the hydrogen bond strength between water molecules.1 However, the relative partial molal heat content Lp of a solute such as tert-butyl alcohol increases markedly when the solution temperature The Journai of Physical Chemistry, Voi. 77, No. 20, 1973
M. Lucas
2480
is r a i ~ e d , suggesting ~-~ that it might be negative at temperatures lower than -lo", although nmr measurements suggest that this alcohol is a stronger water structure promoter a t low temperature than a t high temperatures.5 A straightforward extension of the scaled-particle theorye-8 to aqueous solutions of nonpolar molecular solutes makes it possible to calculate some thermodynamic properties o f the solute and their variation with the solute molality. These calculations suggest interpretations of these properties which are different from the current interpretation in terms of water structure promotion. This paper presents the scaled-particle calculation, the comparison between calculated properties and experimental ones for the tert-butyl alcohol-water system, and an interpretation of some solute thermodynamic properties. Theory According to eq 10 of ref 6, the partial molal value Vz o f a molecular solute interacting with water through dispersion forces is
where T is the absolute temperature, ,B is the solution isothermal compressibility, G, is the molar free energy for cavity formation, and G, is the molar free energy of interaction. G, is given by the second term in the right of eq 9 o f ref 9. (The third term in the equation which includes the contribution of the solute polarizability is neglected since it is small compared to the second.) Then
since b y l b p isequal to .Py, dXlbp =PX, anddylbp E
PY. Then
bGC bP + RTP
+ + 3 d X ) ( 1 - y)' + aNd3 (9X2d*+ 3 d 2 Y y + 3 d X y ) ( l - y ) + 9X2d2yl + 7 = RTP(1-~)-~{9dzX2+ 3(1-y)(dX + d 2 Y ) +(l-y)'l + = R T P ( 1 - y r 3 { ( l 3d2Y
since
Then
bG, __ bP =
32,N
+
G,
-In(I-
=
y)
TPN + __ RT d 3 +
(3X
-6yX + 34?X)
which is the hard-sphere part of eq 2-7 in ref 8 (the sign before In (1 - y ) is misprinted in ref 8). In this equation P is the pressure and N is Avogadro's number. In addition we have
+
r b r ( 5 5 a 0f rnd3)(6V0~,o)-1(55,5mu)-'
.Y =
X
Y
=
T N (55a' f rnd2)(6v0~,o)-'(55.5 4- mu)-'
=
+
TlV(55a + md)(6V0~,o)-'(55.5 mu>-'
When m is vanishingly small, since then X = y / a and Y = y / a 2 , VQz is given by the equation
G,
= -ln(I-y)
i r N p d > 3d'y 9X2d2 +m +-1 - y +-+2(1-y)*
we have The Journalof Physical Chemistry. Vol. 77, No. 20, 7973
3dX 1-y
+ c2(3y+6y2) +
where d / a = c. The last expression is also equal to -RTpo 2.67 y ( c l z l k r ) (1 + c ) ~ since , then y = ~ N a 3 / 6 P ~ , ~ . Equation 3 may be derived from the equations of ref 6 except that in ref 6 dispersion terms are omitted. From eq 1 and 2 it may be derived that 1 bV2 = -1 BP Vo2- ?rNa3c3/6 bm(m -+ 0) Po m(m -00)
1 3Y -55.5 1 -
( c 3- v)
+
1 +{ ( c- u)(3yc2 - 3y2c9 -t 55.5f(y)
- ~ ) ( 1 8 y ' c+~ 3 y c - 3 y ' c ) - (c3- u ) Y ( -~ 2y + 3cy t 3c2y) + ( 1 - y)*y22.67[3y(c3- U ) ( t , p / k T ) ( l +c ) + ~ ( 1 - y ) ~ ' ( t ~ ~ / h T X l+ (1 c )-~ ) ( t 2 2 / k T ) S c ' I \ ( 4 ) where f ( y ) = (1- Y ) +~ cZ(3y + 6 y 2 ) + c(3y -3y2) - ( 1 (c'
-y)%?.67y(l
with u = V~/VQH,Owhere Vz and v O H , ~ are respectively the solute apparent molal volume and the pure water molal volume, d and a are respectively the solute and water hard-sphere diameter, and m is the solute molability in the solution. From the equation
-__I_
then V , = ( 1 ) t (2).
Vo2 RTPo(1- yjW3[(l- y)' where p , is the particle j number density, €2, is the depth of the potential minimum in the Lennard-Jones (6-12) potential, and uZ1 = dz + d,/2 where d~ is the solute hardsphere diameter and d, is the diameter of the particle j . G , is given by the equation
+
55.5 tI2 ( a d)3 55.5 m u k T 8
-,-PRT( 9v HgO
+~)~(t,*/kT)."
In addition, we have
Since iff = ghi we have
2481
Thermodynamic Properties of a Hard-Sphere Solute
where PO2
3
1 bVOZT -vo2 bP
-6
by the same procedure the equation PO2
=
(Voz-xNa3c3/6)Po VO'
is derived from eq 3. Equations 3, 4, 5, and 6 make it possible to compute bV2/i3m ( m 0) for a nonpolar solute (with or without dispersion forces) in water. When no dispersion forces are present L2 may be calculated as follows. The variation of the solution expansion coefficient is given by
-
J 0
50
t'
100
Figure 1. Plots of experimental and calculated and 1/ T -taPa/PobT vs, temperature.
a 80/aP
with cYoz
3
1 bVoz Vo2- n N a 3 c 3 / 6 V o z bT = V02
-
for the solute, t l l / k for water, the pure water compressibility Po, and the water expansion coefficient a@a t various temperatures. The last two quantities are taken respectively from Tables IV and I1 from ref 11. The experimental values of (1/(/30)2(&3/aP)( have been obtained from eq 33 and Table 10 of Kell's paper on the PVT properties of water.lz Figure 1 shows the plots of this quantity a t various temperatures and of the quantity [(l/Po)(dBo/dT)+ ( l / T ) ] p compared to the values these quantities would have if the hard-sphere equation of state applied to water, in which case PO would be equal
l/n
where 010 is the pure water expansion coefficient. From eq 14 of ref 7
U
=
(aT/p)(V2-nNa3c3/6)
(9)
Relating the molal internal energy charge for the solute transfer from the gas phase to its aqueous solution of molality m and setting the heat of dilution L2 equal to U U O = m bU/hm ( m 0 ), we obtain
-
1 -
L2
Uo m(m-*O)
- 1 bol cy0 bm
1 bp
+
Po b m 1
V o z- nNa3c3/ 6 d m (10) with
which leads to 1 bP@ (pO)z a p -
- 1 - 4y(l -y)-'
- 2y(l+2y)-'
(13)
and
Uo
= (aoT / /30)(Voz - lINa3c3/ 6 )
(11)
This calculation is interesting in that it makes apparent the importance of the term (l/Po) (bpo/bT)+ 1/T which appears in eq 8. When dispersion forces are present, H i is given by
+
( d +a)3 32RTnN [55.5 ( e I 2 / k T ) md3t2,/kT 9VoH,,(55.5 mu)
+
1
H is calculated a t different molabilities assuming V2 = v D 2 and V D H ~ Oconstant a t all molabilities and that the 0 ) is equal to L2/m ( m 0). limit of H - H o / m ( m The assumption that Vz = VO2 a t all molalities implies that the parameter b in the equation H - Ho = a m + bm2 is not correct but does not change the value of the limit of H - Ho/m ( m 0 ) . For the numerical calculations the necessary parameters are the water and solute hard-sphere diameters, t 2 2 / k
-
-
-
1 T
+ --Po1
dP@ - = c~o[l+4y(l-y)-' bT
+ 2y(l+2y)-'I
(14)
Calculations are carried out with a = 2.90 A. Discussion Calculations are carried out with a solute size equal to water (2.90 A) and twice this size (5.80 A) and with tzz/k = 0 (hard-sphere solute without dispersion forces) or tzz/k of the same magnitude as for a nonpolar solute. For rare gases it is found experimentally that tzz/k = 146(d - 2.58 A) K,14 which gives tzz/k = 46 K for d = 2.9 A and 465 K for d = 5.8 A. For water t l l / k is taken equal to 96 K . t1z/k is equal to (€11 e z z ) 1 : 2 / k . 1. Hard-Sphere Solutes l t 2 2 / k = 0). Figure 2 shows and L z / m for plots of VO2, bVz/bm, 4'K = -[&7@~/dP]T, the two solute sizes considered, against the temperature. Consideration of the plots shows that dVz/bm and Lz are negative a t low temperature and positive a t high temperaThe Journal of Physical Chemistry, Voi. 77, No. 20, 7973
V O
2
cm3 mole-'
3
C
k-l
a
Fl
-1 K p I mole 1 -
100
100
-
50 50.
1 -
___---
t- BuOH
-6
"L
___..---
OIL 0
5 0 t'100
0
5 0 t.100
-8
o
0
50 t'lO0
0
50 t'lOO
Figure 2. Plots of the solute partial molal thermodynamic -quam tities vs. the solution temperature: dotted lines. d = 2.9 A , t / k = 0 K ; full lines, d = 5.8 A, c,'k = 0 K.
ture and that d°K increases with the temperature. The shape of the curve for dVz/dm is especially interesting in that it shows a minimum. The shape of the curve for apo/dP (Figure 1) is similar and calculations show that its variation with the temperature is the cause for the corresponding variation of dVz/dm. In the same vein the important variation of Lz/m with the temperature and especially its negative value under ca. 40" is related to the similar variation of 1/T + ~ B O / P OT for pure water. The calculations show also that if this quantity is given by eq 14 (that is if water behaved as a hard-sphere fluid) then the variation of Lz/m with the temperature should be much smaller. It has been suggested that a hard-sphere solute enhances the H bonds between water molecules a t temperatures lower than 4" and weakens these bonds at higher temperatures.15 The calculated values of dVz/dm and L z / m for such a solute are apparently related to the peculiar variation of the derivative ape/@ and @Io/bT for pure water rather than to the solute structural influence on water. In other words, the sign and magnitude of bVz/dm are probably not evidence for or against the water structure promotion by a solute ( a table for the calculated thermodynamic quantities is given in the microfilm edition; see ref 10). 2. Nonpolar Molecular Solutes u t h Dispersion Forces (tzzlk # 0). The introduction of dispersion forces with small values ( c z z / k = 46 K ) for a solute with the same diameter as water does not significantly change the properties (plotted in Figure 3). However, the properties of the solute with a diameter twice that of water are significantly modified by the existence of dispersion forces with tzZ/k = 465 K. VOz is significantly smaller and the minimum of VOz is shifted to lower temperatures. bVz/bm is less negative. The modifications of L z / m are more difficult to rationalize. In the same figure, we have plotted experimental values of the corresponding quantities for tert-butyl alcohol as solute. It would have seemed more appropriate to have used a nonpolar solute. However, due to their poor solubility in water, data as bV2/dm or Lz/m are comThe Journai of Physicai Chemistry, Voi. 77. No. 20, 1973
50 to100
o
so
toloo
o
so
t-ioo
o
so r i o o
Figure 3. Plots of the solute partial molal thermodynamic -quantities vs. the solution tem erature: dotted lines, d = 2.9 A , c / k = 46 K, full lines, d = 5.8 t / k = 467 K.
1,
pletely absent from the literature. On the other hand, they are available for a solute as tert-butyl alcohol and have been ascribed to the nonpolar part of the molecule. The experimental bV/bm has been computed from data of Franks and Smith.lG From their very accurate measurements it may be deduced that bVz/dm is distinctly less negative a t 0.5" than a t 5". Although the calculated and experimental values are different, the existence of a minimum in both cases is significant. It is difficult to explain this minimum if it is ascribed to the enhancement of H bonds by the solute since one should then have to hypothetize that it is a maximum at 5". Nmr data show that the water structure promotion by the tert-butyl alcohol seems to increase monotonously when the temperature is decreased.5 The calculated $OK is very small a t 0". Experimental determinations of the adiabatic 4°K by Franks, et al., show that it is generally negative a t 5" for a variety of solutes and positive at 25".17 The isothermal 4'K is derived from the adiabatic one by addition of a .18 Then the isotherterm nearly equal t o 2/30(bvOz/a0dT) mal 4'K is certainly negative a t 5" since dvOz/aT is negative, and probably positive above 25". The experimental variation with the temperature is certainly larger than the calculated one. Finally, we have plotted in Figure 3 Ln/m for tert-butyl alcohol. The experimental limiting values of Lz/m have been calculated from data in ref 2, 3, and 4. The variations of experimental and calculated Lz/m with temperature are qualitatively similar except that the calculated one is more negative a t all temperatures. In view of this fact it is difficult to accept the usual interpretation that a positive Lz is caused by water structure promotion by the so1ute.l The discrepancy between calculated and experimental quantities may be due to the polar character of t-BuOH not accounted for by scaled-particle theory and also some shortcomings of the scaled-particle approach. First the use of a constant-temperature independent hard-sphere radius may be questioned.9 The equations giving the influence of dispersion forces also incorporate crude assumptions, for example a random distribution of molecules in the solution. Somewhat different equations have been used by
Free S o l v a t e d E l e c t r o n in HMPA
Tiepel and Gubbins.9 The main difference is that the pressure term in their eq 11, which in Pierotti's and this paper yield the term LVa3c3/6 in the equation for VZ, yields a more complex term. The fit between calculated and experimental dVz/brn is much better but the fit for L z / m is much worse when Tiepel and Gubbins equations are used, so that we cannot decide between the two sets of equations. However, the main conclusion of this paper is that scaled-particle theory makes it possible to ascribe the variation of the solute partial molal properties to the anomalous behavior of a(3"ldP and @"/d?' for pure water rather than the solute structural effects. Supplementary Material Available. The derivation of eq 4 will appear following these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementary material from this paper only or microfiche (105 X 148 mm, 20x reduction, negatives) containing all of the supplementary material for the papers in this issue may be obtained from the Journals Department, American Chemical Society, 1155 16th St., N.W., Washington, D. C., 20036. Remit check or money order for $3.00 for
2483
photocopy or $2.00 for microfiche, referring to code number JPC-73-2479. References and Notes (1) J. E. Desnoyers, M. Arel, G. Perron, and C. Jolicoeur, J. Phys. Chem., 73,3346 (1969). (2) J. J. Kozak. W. S. Knight, and W. Kauzmann. J , Chem. Phys., 48, 675 (1968). (3) W. S.Knight, doctoral dissertation, Princeton University, 1962. (4) J . Kenttamaa, E. Tommila, and M. Martti, Ann. Acad. Sci. fennicae A I/93. 1 11959). M. M.'Marciacq-Rousselot and M. Lucas, J , Phys. Chem.. 77, 1056 (1973). R. A. Pierotti, J. Phys. Chem., 67 1840 (1963) R. A. Pierotti, J. Phys. Chem., 69, 281 (1965). J. L. Lebovitzand J . S. Rowlinson, J. Chem. Phys.. 41, 133 (1964). E. W. Tiepel and K. E. Gubbins, J. Phys. Chem., 76, 3044 (1972) See paragraph at end of paper regarding supplementary material. F. Franks, Ed., "Water, a comprehensive Treatise," Plenum Press, New York, N. Y., 1972, Vol. 1. Chapter IO. G . S. Kell and E. Whalley, Phii. Trans. Roy. Soc., Ser. A , 258, 565 (1965). F. H.Stillinger, J. Chem. Phys., 35, 1581 (1961). H . D. Nelson, Ph.D. Dissertation, Utrecht, 1967. M. LucasJ. Phys. Chem., 76,4030 (1972). F.Franks and H. T. Smith, Trans. Faraday Soc., 64, 2962 (1968). F. Franks, J. R . Ravenhill, and D. S. Reid, J. Sol. Chem., 1, 1 (1972) J . E. Desnoyers and P. R. Phiiip. Can. J. Chem.. 50, 1094 (1972)
The Free Solvated Electron in Hexamethylphosphoric Triamidel G. Dodin and J. E. Dubois" Laboratoire de Chirnie Organique Physique de I'Universitb Paris VI/, associe au C. N.R.S., Paris, France (Received April 16, 1973) Publication costs assisted by the Laboratoire de Chimie Organique Physique de i'Universite Paris V i /
The width of the esr narrow line from the solvated electron in hexamethylphosphoric triamide (HMPA)sodium solutions arises from exchange narrowing and the activation energy for the hyperfine coupling modulation as well as the volume of the solvated electron increase with increasing solvent viscosity. When the viscosity is extrapolated to zero, the line width has a nonzero value resulting presumably from electron-nuclear spin dipolar coupling. The resonance line in the ideal nonviscous state might he attributed to a species we shall refer to as a "free" solvated electron which would consist of an electron trapped in a cavity formed by a layer of solvent molecules. The contributions to the experimental line width variation from both viscosity effects and technological artifacts arising from cavity shifts and 100kHz modulation are easily differentiated provided certain experimental conditions are satisfied.
Introduction Esr spectroscopy has been widely used to study the paramagnetic species formed in the solutions of alkali metals in organic solvents.2 For solvents with low dielectric constants (ethers and amines), a hyperfine resonance signal arising from the interaction of the electron spin with the nuclear spin of the metal is observed: the spectrum is assigned to the monomer M.2-* In solvents with higher dielectric constants such as liquid ammonia and hexamethylphosphoric triamide (HMPA) the resonance spectrum consists of a single line and is attributed to the solvated electron. In metal-ammonia solutions, the width of the resonance line is due to exchange narr0wing;s.G the process of hyperfine coupling
modulation is, however, controversial. According to the Kaplan and Kittel theory,7 this modulation is due to the rotation of the solvent layer around the electron involving an interaction whose correlation time is the Deb ye dipolar relaxation time. On the other hand, Dewald and Lepoutre,8 and more recently Lambert.9 have suggested that the modulation of the hyperfine coupling could be achieved by the passage of the electron from one cavity to another uia a tunneling mechanism resulting in a correlation time for the interaction corresponding t o the lifetime of the electron in a given cavity. Whatever the modulation process, the exchange rate should follow the Arrhenius law; when log AN (where A H is the line width) is plotted against 1/T a straight line is indeed observed.9 The Journal of Physicai Chemistry, Vol. 77, No. 20, 1973
