J . Phys. Chem. 1988, 92, 3623-3625
3623
Enthalpy of Dimerization of Benzene in Water D. HallCn, I. Wadso, Division of Thermochemistry, Chemical Center, University of Lund, P.O. Box 124, S-22100 Lund, Sweden
D. J. Wasserman, C. H. Robert, and S. J. Gill* Department of Chemistry, University of Colorado, Campus Box 21 5, Boulder, Colorado 80309 (Received: October 7, 1987)
We have used precise-flow and titration microcalorimetry to measure enthalpy changes associated with the dilution of aqueous benzene solutions (can,,, C 0.022 mol/L). Such direct measurements were undertaken for comparison with a thermodynamic characterization of the system reported by another group using vapor-pressuremeasurements of benzene above aqueous solutions [Tucker, E. E.; Christian, S. D. J . Phys. Chem. 1979, 83, 426. Tucker, E. E.; Lane, E. H.; Christian, S. D. J . Solution Chem. 1981,JO,1-20]. All studies so far indicate that the observed effects can be explained by a dimerization of the benzene molecules, so that the system may be a basic one for study of hydrophobic association. The extent of dimerization occurs to such a small extent that with the calorimetry experiments alone it was not possible to resolve the individual thermodynamic parameters Kdlmand from the product of these two quantities. This product is consistent between the two calorimetric techniques. However, the product obtained here is not consistent with Kdi,AHdi, from the vapor pressure studies of Tucker et al. Nevertheless, by assuming the validity of the primary quantity obtained from the vapor pressure studies-the dimerization constant-we estimate the heat of dimerization of benzene in water at 25 OC to be 4.2 f 0.2 kJ/(mol dimer).
Introduction The hydrophobic nature of apolar substances such as hydrocarbons is illustrated by their low solubility in water and their tendency to self-associate in aqueous environments. The first case, dissolution, involves hydrophobic solvation, where the properties of the isolated apolar molecule in an aqueous environment are considered. In contrast, self-association between apolar molecules already dissolved in water may be caused by several effects, but we can include all of them in the term hydrophobic association. The thermodynamic basis of hydrophobic phenomena has been the subject of many studies over a long period of time. In 1945 Frank and Evans recognized that the solution forces involved large negative entropy changes.' A decade later Kauzmann utilized the concept of hydrophobic interactions in analyzing the factors that influence the structure of proteins.2 Several general discussions of hydrophobic effects have been given r e ~ e n t l y . ~ In many studies investigators have measured the Gibbs energy changes of hydrophobic processes using solubility studies or studies of a compound's partition coefficient between a nonaqueous and aqueous phase. Entropy, enthalpy, and heat capacity changes have then been obtained by temperature derivative^.^ In general, however, direct determination of these quantities is desirable to achieve high accuracy, although measurements are made intrinsically difficult by the low solubility of these compounds. Direct calorimetric measurements have been made possible by special proceduresS for determining heats of solution as a function of temperature for pure liquids and gases into water. These studies6 provide information primarily concerning the properties of hydrophobic solvation. One notable property of a hydrophobic ~~
S.;Evans, M. W. J. Chem. Phys. 1945, 13, 507. (2) Kauzmann, W. In A Symposium on the Mechanism of Enzyme Action; McElroy, D. W., Glass, B., Eds.; Johns Hopkins University Press: Baltimore, 1954. Kauzmann, W. Adv. Protein Chem. 1959, 14, 1. (3) Tanford, C. The Hydrophobic Effect, 2nd ed.; Wiley: New York, 1980. Ben-Naim, A. Hydrophobic Interactions; Plenum: New York, 1980. (4) Wilhelm, E.; Battino, R.; Wilcock, R. J. Chem. Rev. 1977, 77, 219. (5) Gill, S.J.; Nichols, N . F.; Wadso, I. J. Chem. Thermodynam. 1975, 7, 175. Gill, S.J.; Nichols, N. F.; Wadso, I. J. Chem. Thermodynam. 1976, 8,445. Gill, S. J.; Wadso, I . J. Chem. Thermodynam. 1982, 14, 905. Dec, S. F.; Gill, S.J. Rev. Sci. Instrum. 1984, 55, 765. Nilsson, S . - 0 . ;Wadso, I. J . Chem. Thermodynam. 1984, 16, 317. (6) Dec, S. F.; Gill, S. J. J . Solution Chem. 1984, 13, 27. Olofsson, G.; Oshodi, A. A,; Qvarnstrom, E.; Wadso, I. J. Chem. Thermodnam. 1984, 16, 1041. Dec, S. F.; Gill, S. J. J. Solution Chem. 1985, 14, 417. Dec, S. F.; Gill, S.J. J. Solution Chem. 1986, 14, 827. Naghibi, H.; Dec, S. F.; Gill, S. J . J . Phys. Chem. 1986, 90, 4621. Naghibi, H.; Dec, S.; Gill, S . J . J. Phys. Chem. 1987, 91, 245. (1) Frank, H.
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solute is a large partial molar heat capacity, which depends on the number of water molecules in contact with it. A simple theoretical description of the origin of the large heat capacity of solvation has been given.' It has been much more difficult to characterize the thermodynamics of hydrophobic association processes. The systems where these effects are manifest such as proteins or lipid bilayers are themselves complicated. However, a simple situation was described by Tucker et al. in their studies of benzene in water.* From results of precise vapor-pressure measurements they concluded that benzene in water exists principally as the monomer but also forms a small amount of dimer. Their studies permitted evaluation of the equilibrium constant for dimer formation. Determinations as a function of temperature enabled an estimate of the enthalpy of dimer formation. This system consists of purely hydrophobic molecules interacting in pairs and thus is unique in its simplicity. These results have inspired theoretical investigations of this proce~s.~ An independent experimental approach is desirable to investigate the properties of benzene interactions in water. Calorimetric measurement of heats of dilution provide a direct means for determining the enthalpy of interaction and the equilibrium constant for aggregate formation.1° In this paper we describe the theoretical method for treating calorimetric dilution measurements in terms of a generalized partition function for an aggregating system. We then apply it to measurements of the enthalpy of benzene dilution in water determined using flow calorimetry (Lund) and titration calorimetry (Boulder).
Experimental Procedure Flow Calorimetry. The dilution of benzene in water was performed in a flow mixing vessel of a four-channel microcalorimeter system (LKB-2277-020). The original Teflon connection tubes were changed to stainless steel. LKB Model 21 50 HPLC pumps were used. The initial concentration of benzene in water was that of a saturated solution in a 1-L glass bottle in a 24.15 OC thermostat. The saturated solution was prepared by adding (7) Gill, S.J.; Dec, S. F.; Olofosson, G.; Wadso, I . J. Phys. Chem. 1985, 89, 3758. (8) Tucker, E. E.; Christian, S.D. J. Phys. Chem. 1979, 83, 426. Tucker, E. E.; Lane, E. H.; Christian, S. D. J. Solution Chem. 1981, 10, 1. (9) Rossky, P. J.; Friedman, H. L. J. Phys. Chem. 1980, 84, 587. (IO) Gill, S. J.; Farquhar, E. L. J. A m . Chem. SOC.1968, 90, 3039. Stoesser, P. R.; Gill, S. J. J. Phys. Chem. 1967, 71, 564. Gill, S. J.; Noll, L. J . Phys. Chem. 1972, 76, 3065.
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3624 The Journal of Physical Chemistry, Vol. 92, No. 12, 1988 excess benzene in water and stirring for 1 week. The data of Franks et al.'* were used to determine the concentration of the saturated benzene solution. Experiments were performed at 25.00 OC.
The steady-state power P in the calorimeter was measured for the process where a solution at an initial average enthalpy content per mole of benzene H I was diluted to a solution of final enthalpy content Hf.This enthalpy difference is multiplied by the moles of benzene flowing through the calorimeter per second. This latter quantity is expressed by the volume flow rate dV/dt (liters per second) times the initial molarity of benzene c, (moles per liter). Thus the operational equation for the flow system is P = ((Rf - R,)dV/dt)c, (1)
Titration Calorimetry. In the titration microcalorimeter one measures the quantity of heat produced upon addition of saturated benzene solution of concentration c, in incremental steps to a volume V of initially pure water in the calorimeter cell. The cell is totally filled with solution at all times, and addition of a small volume u of the benzene solution ejects an equal volume of the original solution. The concentration of benzene at any step n is related to the initial concentration c, by c, = c,( 1 -
(2) where D is 1 - u/V, the dilution factor. The heat q for the step is then given by
(3)
This is the difference between the enthalpy content of the final state of the system and that of the previous state. To fit the titration data, one requires an expression for H a t a given total concentration of benzene.
Theory To relate the experimentally determined enthalpy changes to a chemical process, we must describe a model that can express the molar enthalpy of the system in terms of appropriate reactions. For the dimerization of benzene we will consider the following equilibrium: M + M s D
(4)
K = [Dl/[M12
(5)
where each activity coefficient is taken as unity, indicating the assumption of ideal behavior of the uncharged component species for the dilute solutions involved in this study. The total concentration of benzene is c = [MI + 2[D] (6) and one can obtain
+ (1 + 8Kc)'I2)/(4K)
(7)
which gives the benzene monomer concentration at any total concentration of benzene. We can define a partition function Q that is the sum of the concentrations of all of the interconverting species in equilibrium. For the dimerization model these species are the benzene monomer and dimer, so that
+
(8) Q = [MI + [D] = [MI K [ M I 2 The average molar enthalpy of the system (R), relative to that of some standard state (Ho)is given byI2 (9)
where the partial derivative is taken at fixed activity of the reference species and c is the total concentration of the aggregating (11) Franks, F.; et ai. J . Chem. Sor. 1963, 2716. (12) Gill, S. J.; Richey, B.; Bishop, G.; Wyman, J. Biophys. Chem. 1985,
21, I .
cf, mol/L -(Hf - H : ) ,J/mol 0.01577 0.01 147 0.00984
CC,
12.79 21.21 24.43
mol/L - ~ H -P H ; ) . Jlmol
0.00801 0.00679
29.44 31.51
" c , = 0.02184 M; temperature for preparation of saturated solution = 291.3 K.
- -L@
,
1
_I_\ 0
.O1
.02
,
mo
/I
Figure 1. Enthalpy of dilution of benzene solutions at 25 OC determined by flow calorimetry. Each point represents a dilution from the initial concentration of 0.02184 mol/L.
species. In the present case we shall take as the reference species benzene at infinite dilution-that is, the monomer. The temperature dependence of K is a In K / a ( l / T ) = -AHD/R where AHD is the enthalpy change for the dimerization reaction. With 8 and 10 the partial molar enthalpy of the system (from 9) relative to the monomer becomes
The monomer concentration [MI can expanded about c = 0 to yield [MI = ~ ( -l KC
with the equilibrium constant
[MI = (-1
TABLE I: Enthalpy of Dilution of Benzene at 298.15 KO
F i ~ a '[ B e n z e n e ]
D")
q = HnCnV- Hwlc"-I( v - u ) - H,c,v
Hall6n et al.
+ 8 P c 2 + . . .)
(12)
and inserted into eq 11. Expansion of the resulting expression and collection of terms give the observed heat for a dilution experiment, corresponding to a given dilution experiment in the flow calorimeter to a single step in the titration calorimeter:
H f -Hi = A H D K ( ( c ~c;) - - 4 K ( ~ f -2 c,')
+ ...)
(13)
Here we see that the first-order dependence of the enthalpy change on the concentration change cf - c, is through the proportionality AH& Resolving this product into the two individual parameters depends on the concentration dependence of the heat as seen in the second-order term. Alternatively, if K is determined by an independent method, the quantity AH, can be obtained from a first-order dependence alone.
Results and Discussion In Table I we summarize the flow calorimetry of benzene dilution. In all experiments the benzene solution was diluted from its initial concentration ci to a final concentration cf, giving a heat of dilution per mole of benzene Hf- Hi.The quantity Hf - HI is plotted against cf in Figure 1, where in all experiments the initial concentration of benzene ci was constant. Such a plot is seen to be linear, and examination of eq 13 shows that as a result only the product AHDK can be resolved, with a value of 2.08 0.03 kJ L/mo12. The lack of a higher order concentration dependence is consistent with a small value of the dimerization constant. Without such concentration dependence, individual resolution of the two parameters is not possible from these data alone. The parameter AHD can, however, be estimated by using the value of K for the dimerization (0.50 f 0.01 L/mol) determined by the high-precision vapor-pressure measurements of Tucker and
*
J. Phys. Chem. 1988, 92, 3625-3629
\
I
.Bl5 , mo:/l Figure 2. Heat released in successive steps of a titration of water with a saturated benzene solution at 25 OC determined from four separate titration calorimetry experiments. Solid line: Predicted behavior using the product AHDK determined in the flow calorimeter. Dotted line: Predicted behavior using the values obtained from the vapor-pressure measurements of Christian and Tucker.8 0
.005
.O1
[Benzene]
Christian8 at 298 K. Using the product of AH and K from our calorimetric experiments, we thus obtain AH, = 4.2 kJ/mol as the heat of dimer formation. This value is much less endothermic than the value of 13 f 2 kJ/mol derived from the temperature dependence of the K values obtained from the vapor-pressure measurements.* The titration calorimetric measurements of stepwise additions of benzene solution to water yielded heats near the sensitivity limits of the apparatus. The results from four experiments are shown
3625
in Figure 2. While of lower precision than the flow calorimetric determinations, these results nevertheless show a similar result: that the heat of dimerization is less endothermic than the value reported by Christian and Tucker, which is shown in the figure by a dotted line. The solid line is drawn with the result of the flow calorimetry and can be seen to fall closer to the range of the titration data. The value of the dimerization heat obtained by the vaporpressure measurements is much different from that determined by the calorimetric techniques reported here. In general direct methods provide the best measurements of a thermodynamic quantity; thus in the face of the observed inconsistency we feel that the use of the calorimetric results, which themselves are of high precision, provides the more reliable estimate of the heat of dimerization of benzene in water. To determine this heat from the vapor-pressure studies, one must first measure the very small amount of benzene dimerization and its equilibrium constant and then determine the temperature dependence of this constant. Any systematic errors, which are often difficult to detect, will therefore become most serious with this method. Consequently we feel that currently the best characterization of the benzene self-association can be made using both the calorimetric determination of AHDK and the vapor-pressure determination of K itself. Acknowledgment. We acknowledge support by the Swedish Natural Science Research Council (D.H. and I.W.) and by N S F Grant PCM8019930 (S.J.G., C.H.R., and D.J.W.) in the performance of this work. We thank E. E. Tucker and S. D. Christian for making available their original vapor-pressure data and for numerous discussions on this problem. We also thank Suzanne Paulson and Gunilla Grantz for assistance. Registry No. Benzene, 71-43-2.
Calculation of the Free Energy of Electron Solvation in Liquid Ammonia Using a Path Integral Quantum Monte Carlo Simulation Massimo Marchi, Department of Chemistry, McMaster University, Hamilton, Ontario, Canada L8S 4M1
Michiel Sprik, and Michael L. Klein* Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 191 04-6323 (Received: November 18, 1987: In Final Form: January 13, 1988)
We have used a path integral quantum Monte Carlo simulation to evaluate the free-energy and entropy change associated with electron solvation in liquid ammonia. The entropy change at constant volume is found to be negative, essentially because the electron induces local ordering in the surrounding solvent molecules. The experimental entropy change at constant pressure is positive. We suggest that the positive value arises because of the large positive entropy associated with the work done by the electron in expanding the fluid during the solvation process. 1. Introduction
Recent developments in simulation techniques now make it possible to study quantum systems of chemical interest.' In particular, through the application of the Feynman formulation of quantum statistical mechanics one can carry out a simulation of electron solvation in polar fluids such as ammonia and water.24 Calculations of this type have established that an electron becomes (1) Berne, B. J. Thirumalai, D. Annu. Reu. Phys. Chem. 1986, 36, 401. (2) Sprik, M.; Impey, R. W.; Klein, M. L. J . Chem. Phys. 1985.83, 5802. Sprik, M.; Klein, M. L. Comput. Phys. Rep., in press. ( 3 ) Schnitker, J.; Rossky, P. J . Chem. Phys. 1987, 86, 3471. Rossky, P.; Schnitker, J. J . Phys. Chem., in press. (4) Wallqvist, A.; Thirumalai, D.; Berne, B. J. J . Chem. Phys. 1987, 86,
6404.
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self-trapped in a hole or cavity in the solvent. It now appears that either Monte Carlo (MC)2 calculations or molecular dynamics (MD)3 provide an adequate sampling of electron paths, at least in the temperature range of the liquid state. To date, interaction energies and equilibrium averages have been quite successfully calculated.' However, no attempt has been made to compute the free energy and entropy of the electron solvation process. These two quantities are well documented experimentally; for example, the entropy change for electron solvation in ammonia is AS = 18kg.5-7 Since solvent molecules are bond ordered around the (5) Krebs, P.J . Phys. Chem. 1984, 88, 3702. (6) Schindewolf, U. In Physics and Chemistry of Electrons and Ions in Condensed Matter; Acrivos, J. V., Mott, N. F., Yoffe, A D , Eds ; D. Reidel: Dordrecht, 1984. (7) De Maeyer, L. C. M . Pure Appl. Chem. 1986, 58. 1105.
0 1988 American Chemical Society
