1120
J. Phys. Chem. 1982,86,1120-1125
abc # 0 , eq 6 represents the linearly forced harmonic oscillator. Pechukas and Child3 showed analytically that the (pf,c@,curve for this system was a circle displaced from the phase space origin. Now, trajectory calculations with c = 0 and abd # 0 yield an elliptical @f,gf),i curve with center at the origin and rotated axes for the quadratically forced harmonic oscillator.% Such a phase space picture is isomorphous to that obtained for the overlap integral between harmonic oscillators with identical equilibrium positions but different force constants, kl = 1 and k2 # 1. The functional form for these overlap integrals is well-known, and the phase space pictures suggest that it may be related to the less tractable expressions for the transition probabilities obtained by Lewis and Riesenfeld% for the harmonic oscillator with time-dependent force
Acknowledgment. P.M.H. is indebted to the Dow Chemical Company Foundation for research funding. The partial support of the donors of the Petroleum Research Fund, administered by the American Chemical Society, is gratefully acknowledged, as is support from the Research Corporation.
(29)H.R. Lewis, Jr., and W. B. Riesenfeld, J.Math. Phys., 10,1458 (1969).
(30)H.J. Loesch and D. R. Herschbach, J . Chem. Phys., 57, 2038 (1972).
constant. Such an oscillator is a possible model for experimentally observed30non-Landau-Teller behavior in Ar + CsI scattering. Whether used to explain results from sophisticated CID calculations or to suggest links between simple theoretical models, classical S-matrix phase space pictures appear to be an easily employed theoretical tool worthy of continued application and development.
Investigation of Association Reactions in Polar Gases Uslng the Pressure and Temperature Dependence of Thermal Conductivity L. A. Curtlss,’ D. J. Frurlp, and M. Blander Chemical Engineering Division, Argonne National Laboratory, Argonne, Iiiinois 60439 (Receivd: July 2 1, 198 1; In Final Form: August 20, 198 1)
The results of investigations into the association reactions occurring in polar gases by analysis of the pressure and temperature dependence of thermal conductivity are reviewed. The gases studied include water, methanol, ethanol, 2-propanol, 2-methyl-2-propano1,2,2,2-trifluoroethanol, acetone, acetonitrile, pyridine, acetic acid, trifluoroacetic acid, as well as the binary mixtures methanol-water, trifluoroethanol-water, and acetic acid-water. The prewure and temperature ranges of these studies are 100-2000 torr and 33&440 K, respectively. Expressions for the thermal conductivity of reacting gases derived by Butler and Brokaw, based on the work of Hirschfelder, give calculated thermal conductivities which fit the observed data very well for all the types of observed pressure dependence. The fits yield information on the associated species present (e.g., dimer, trimer, tetramer, etc.), on the concentrations of the species, and on their enthalpies and entropies of association. The thermodynamic results, in conjunction with ab initio molecular orbital calculations, provide information on the strength and structure of hydrogen bonding occurring in the cluster species.
I. Introduction Heat transfer in a pure nonreacting gas placed between a hot and a cold plate is limited primarily by molecular collisions. If a reacting gas is placed between the plates additional heat is transported as chemical enthalpy of molecules which diffuse because of concentration gradients in the gas. These concentration gradients arise since the gas composition varies with temperature. An example is a gas which absorbs heat by dissociating as the temperature is raised. Heat is transferred when the molecule dissociates in the high-temperature region and diffuses toward the low-temperature region (since there is a lower concentration of dissociated molecules at low temperatures). In the low-temperature region the gas reassociates, releasing the heat absorbed from the high-temperature dissociation. As a result of this mechanism, heat transfer in reacting gases may be considerably higher than in nonreacting mixtures. This phenomenon was probably first recognized by Nernst’ who suggested that the high thermal conductivity of nitrogen dioxide could be ascribed (1)W. Nernst, Festschr. Ludwig Boltzmann Gewidmet, 904 (1904). 0022-3654/82/2086-1120$01.25/0
to this effect. Nernst’ and later several others26 presented theoretical treatments for heat transfer in reacting gases. In 1957 Hirschfelder6presented a theoretical treatment for the thermal conductivity of a reacting gas mixture when local chemical equilibrium is assumed. Using this assumption of local chemical equilibrium, he was able to derive chemical and temperature profiles of a reacting gas between a hot and cold surface and then obtain an expression for its thermal conductivity. Hirschfelder concluded that “if reaction rates are fast in either the forward or reverse direction, the assumption of local chemical equilibrium is very good.” Subsequently, Butler and Brokaw’ presented a general expression for the thermal conductivity of chemically reacting gas mixtures with local chemical equilibrium. In this paper we will refer to the
-.-
(2) P. A. M. Dirac, h o c . Cambridge Phil. SOC.,22,132 (1924). (3)J. Meixner, Z.Naturforsch. A, 7, 553 (1952). (4)R. Hasse, Z.Naturforsch. A, 8, 729 (1953). (5)I. Prigogine and R. Buess, Acad. R. Belg., 38, Ser. 5,711,851(1952); F. Waelbroeck, S. Lafleur, and I. Prigogine, Physica, 21, 667 (1955). (6) J. 0.Hirschfelder, J . Chem. Phys., 26, 274 (1957). (7)J. N.Butler and R. S. Brokaw, J.Chem. Phys., 26,1636 (1957);R. S.Brokaw, ibid., 32, 1005 (1960).
0 1982 American Chemical Society
The Journal of Physlcal Chemistry, Vol. 86, No. 7, 1982 1121
Association Reactions in Polar Oases
theoretical treatment of the thermal conductivity of chemical reacting mixtures by Hirschfelder and Butler and Brokaw as the HBB theory. During the late 1950's and early 1960's this HBB theory was applied to the prediction and analysis of the thermal conductivity of a number of reacting systems. The most widely studied system was the decomposition reaction N2O4+ 2N02 because reliable experimental information for it was not available.'+ Other systems that were studied included the association reactions nHF H'F,, and PC15 + PClB + Cl2.l' Recently we have used the HBB theory for a systematic study of association reactions occurring in a number of polar gases. By analysis of pressure and temperature dependence of the thermal conductivities of these polar gases with the HBB theory we obtained useful information on the thermodynamics of the associated species in the vapors including enthalpies and entropies of association and equilibrium constants. In addition, ab initio molecular orbital calculations were used to provide structural information on the associated species, check on the validity of the enthalpy values derived from the thermal conductivity data, qnd provide a better understanding of the hydrogen bonding that occurs between polar molecules. The gases studied have included water, methanol, ethanol, 2-propanol, 2-methyl-2-propano1, 2,2,2-trifluoroethanol, acetone, acetonitrile, pyridine, acetic acid, trifluoroacetic acid, as well as the mixtures methanol-water, trifluoroethanol-water, and acetic acid-water. We feel that in a collection of papers honoring Joseph Hirschfelder on his seventieth birthday it is appropriate to summarize and review the results of these investigations of the interactions of polar molecules since the methods used are based in large part on theory that he developed in the 1950's. This paper will not only illustrate how remarkably well the HBB theory does in predicting the thermal conductivities of reacting gases, but it also illustrates how molecular theory combined with experiment can be very powerful for probing intermolecular interactions between polar molecules. In section I1 the method used for measuring the gaseous thermal conductivity and the data obtained are briefly reviewed. In section I11 application of the HBB theory to analyze the thermal conductivity data is described. In section IV the thermodynamic quantities of the associated species detected are summarized and compared with the ab initio molecular orbital results. Finally, in section V conclusions concerning the use of HBB theory for obtaining information on molecular interactions are given. 11. Thermal Conductivity Results The thermal conductivity, A, of the gases in this study were measured as a function of pressure and temperature by means of a thick hot wire cell immersed in a constant temperature oil bath. A relative (steady state) technique was used in which the cell was calibrated with high-purity inert gases (e.g., N2, Ar, Ne, etc.). Thermal conductivity measurements were carried out by placing a purified, degassed sample in a bulb connected to the cell, charging the cell with the sample to a pressure slightly below saturation, measuring the voltage drop across the cell as a function of pressure during a series of stepwise reductions in total cell pressure, and then determining the thermal conductivities from the reference gas calibration. The temperature and pressure ranges of these measurements were ( 8 ) B. N. Srivastava and A. K. Barua, J. Chem. Phys.,35,329(1961).
(9)K.P.Coffin, J . Chem. Phys.,31, 1290 (1959). (10)P.K.Chakraborti,J. Chem. Phys., 38, 575 (1963).
TABLE I:
Thermal Conductivity Pressure Dependence
of Some Polar Gases temp range, K
system
h vs.p
behavior"
ref
100-1000 C 14 100-1000 C 14 100-1800 B,C 15,16 100-1700 B,C 15 100-1900 B , C 15 100-1900 B,C 15 100-1300 A 11 100-1200 A 17 100-1100 A 17 200-1200 A 18 100-1000 C 12 100-500 C 19 100-1400 D 20 200-2000 D 20 100-1400 B 13 100-1200 A 17 100-600 D 21 Letters refer t o type of plot in Figure 1. Some of the
HZ0 D,0 CH,OH CH,CH,OH (CH,),CHOH (CH,),COH CF,CH,OH (CF,),CHOH CF,CF,CH(CH,)OH (CH,),CO CH,CN C,H,N CH,COOH CF,COOH CH,OH-H,0 CF,CH,OH-H,O CH,COOH-H,O a
press. range, torr
358-386 358-384 338-420 328-419 340-420 348-420 338-385 338-385 348-384 341-378 338-387 366-386 354-416 351-413 352-375 351-438 373
alcohols have two types of h vs. p plots: the linear increases (C)occur in the higher part of the temperature range and the upward curvature (B) occurs in the lower part of the temperature range.
19.0
c
5.61
0
I
I
I
200
400
600
I 800
I I 1000 1200 PRESSURE, torr
I 1400
I 1600
Figure 1. Thermal conductivity pressure dependence of some r e p resentative vapors: (A) 2,2,2-trifluoroethanoIat 370.8 K; (B) ethanol at 347.2 K; (C) water (H,O)at 386.4 K; (D)acetic acid at 397.8 K. Circles represent observed values; solid lines are from fits to the data based on HBB theory.
330-440 K and 100-2000 torr, respectively. The relative error in the measured thermal conductivity values was estimated to be less than 0.5% for each gas. For more details of the experimental apparatus and procedure for single-component vapors see ref 11 and 12 and for binary mixtures see ref 13. (11)L.A. Curtiss, I>. J. Frurip, and M. Blander, J.Am. Chem. Soc., 100,79 (1978). (12)T . A. Renner and M. Blander, J. Phys. Chem., 81,857 (1977).
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Curtiss et al.
The Journal of Hysical Chemistry, Vol. 86, No. 7, 1982
A summary of the thermal conductivity data for the 17 one- and two-component gases measured in our studies is given in Table I. Four general types of thermal conductivity pressure dependence were observed for these gases and are illustrated in Figure 1 (data points given by circles). The four types of X vs. p (pressure) plots are as follows: (1) The thermal conductivity increases linearly with pressure (plot C, Figure 1). Water, pyridine, and acetonitrile vapors exhibit this type of pressure dependence. (2) The thermal conductivity shows upward curvature with increasing pressure (plot B). The nonfluorinated alcohols fall in this class a t low temperatures. At higher temperatures their plots are more nearly linear. (3) The thermal conductivity increases to a maximum and then falls off with increasing pressure (plot D). Acetic and trifluoroacetic acid vapors exhibit thii type of pressure dependence. (4) The thermal conductivity shows a nearly linear increase with pressure, but with some bending over (plot A). The fluorinated alcohols and acetone have plots of this type. In the next section it is shown how all four of these X vs p plots can be accounted for quite adequately by the HBB theory in terms of the association reactions taking place in the vapors.
111. Application of the Hirschfelder-Bulter-Brokaw Theory to Analysis of Thermal Conductivity Data The increase in thermal conductivity with increasing pressure found for the gases in our investigations means that there are associated species present. The thermal conductivities of vapors of associating molecules can be expressed byz2 = Xf
+ + XR
(1)
where Xf is the thermal conductivity of a frozen (nonreacting) composition of all the vapor species, A, is the contribution due to an effect referred to as “collisional transfer”, and XR is the contribution to the thermal conductivity arising from the transport of association enthalpy in a thermal gradient. Generally, for an associating gas the XR term makes the major contribution to the increase in thermal conductivity with pressure and the other two terms, Xf and A,, are not very pressure dependent. As disscused in the Introduction, Butler and Brokaw7 derived an expression for XR in terms of the enthalpy changes of reactions occurring in the vapor based on the work of Hirschfe1dere6The expression for XR is given by ~~
~~
~
(13)L.A. Curtiss, D. J. Frurip, and M. Blander, J. Chem. Phys., in press. (14)(a) L. A. Curtiss, D. J. Frurip, and M. Blander, J. Chem. Phys., 71,2703(1979);(b) L.A. Curtisa, D. J. Frurip, and M. Blander, ’Water and Steam,”J. Straub and K. Scheffler, Ed., Pergamon Press, Oxford, 1980,p 521. (15)D.J. Frurip, L. A. Curtiss, and M. Blander, Int. J. Thermophys., 2, 115 (1981). (16)T.A.Renner, G. H. Kucera, and M. Blander, J.Chem. Phys., 66, 177 (1977). (17)L. A. Curtiss, D. J, Frurip, and M. Blander, unpublished work. (18)D. J. Frurip, L. A. Curtiss, and M. Blander, J. Phys. Chem., 82, 2555 (1978). (19)L. A. Curtiss, D. J. Frurip, C. Horowitz, and M. Blander, ‘Proceedings of the 16th International Conductivity Conference (1979),” in press. (20)D.J. Frurip, L. A. Curtiss, and M. Blander, J. Am. Chem. Soc., 102,2610 (1980). (21)L.A. Curtiss, D. J. Frurip, and M. Blander, “Proceedings of the 8th Thermophysical Properties Symposium”, 1981,in press. (22)D. E. Stogryn and J. 0. Hirschfelder, J. Chem. Phys., 31, 1545 (1959).
eq 2, where u is the number of independent chemical re-
Iil. actions (associations to form polymers in this case) occurring in the mixture; AH,, is the enthalpy change for the ith or jth reaction and the A , are numerical factors which depend on m, the stoichiometric coefficents; x , the mole fractions of the species in the gas; p , the total pressure; and Dkl,the binary interdiffusion cofficient6for species k and 1. The Aij’sare defined by A u, . = A11, . =
The subscripts on the coefficients, m, and mole fractions, x , refer to the kth or Zth chemical species in the ith or jth reaction. The subscripts i and j can have values between 1 and u. The parameter 1.1 is the number of distinct chemical species. The above expressions are strictly valid only if the chemical rates are sufficiently fast to guarantee that, after a short time, a steady-state condition is realized in which the pressure remains constant and the temperature and chemical composition at any point are constant.% In other words, there is local chemical equilibrium at the local temperature. For gas-phase association reactions one would expect very fast reaction rates approaching the kinetic theory collision frequency. This is due to the fact that gas-phase clustering reactions should have extremely small activation energies since no monomer bonds are broken and no extensive molecular rearrangements are taking place. Since Xf and A, are not very dependent on pressure, the pressure dependence of X as given by eq 1 will be approximately that of A., Hence, we now consider how the expression for XR simplifies in certain cases in order to provide an understanding of the dependence of thermal conductivity on pressure. If there is only one association reaction, nA(g) A,(g), occurring in the vapor then eq 2 and 3 simplify to
where p 1 is the partial pressure of the monomer, R is the gas constant, and K, is the equilibrium constant for the association reaction. The pressure-binary diffusion coefficient, pDl,,, is dependent on temperature, but not on pressure. We now consider the four types of observed thermal conductivity pressure dependences in terms of eq 4. If dimers are the only associated species present and the extent of dimerization is small (i.e., 2Kg,
