Wen Yaung Lee and L. J. S&tsky
2602
The probability for trap destruction due to molecular rearrangement should be lower in a more relaxed matrix containing fewer frozen configurations which are unstable a t the given temperature. The matrix irradiated a t a temperature higher than that of observation is more relaxed, and the matrix irradiated a t a temperature lower than that of observation is less relaxed than that irradiated a t the temperature of ob~ervation.~ Because of this rate of trapped hydrogen atom decay a t the given temperature, e.g., 77 K, cf. Figure 2, regardless the difference in initial yields displayed in the relative units, is greater in the matrix irradiated at 63 K, cf. curve 63/77, and lower in the matrix irradiated a t 90 K, cf. curve 90177, than in the sample irradiated a t 77 K, cf. curve 77/77. Comparing the effect of temperature decrease on the rates of trapped hydrogen atom decay in matrices irradiated a t 77 K, cf. curves 77/77 and 77/67 in Figure 2, and 90 K, cf. curves 90/90 and 90177 in Figure 2, we have estimated nearly equal numerical values of the apparent activation
energy for trapped hydrogen atom decay, 2.3 and 2.2 kcall mol, re~pectively.~ This is consistent with our assumption of trap destruction due to molecular rearrangement which, despite the difference in probability, should proceed with the same apparent activation energy at both temperatures. References a n d Notes (1) E. D. Spraque and D. Schuite-Frohiinde, J. Phys. Chem., 77, 1222 (1973). (2) J. Krohand A. Pionka, Int. J. Radiat. Phys. Chem., 8, 211 (1074). (3) J. Kroh and A. Pionka, Chem. Phys. Lett., 28, 186 (1974). (4) It is worthy to note the importance of irradiation temperature and relative insignificance of preirradiation temperature. It was conclusively shown in ref 1 that, at least with respect to the traps available for hydrogen atoms produced by y-irradiation at 77 K, no changes in th! 8 M H2S04 matrix occurs on warming to 87 K before irradiation. (5) Much greater numerical values, 4.1 and 5.79 kcal/mol, of the apparent activation energy are reported in ref 1 for hydrogen atom decay immediately and 200 hr after the irradiation, respectively. Because of experimental differences it is difficult to compare the numerical data. if, however, there was no great deviation of the decay curve at 87 K from the first-order decay, a much smaller apparent activation energy can be lnferred from the results presented in Figure 3, ref 1.
Heat of Vaporization, Infrared Spectrum, and Lattice Energy of Adamantane Wen Yaung Lee and L. J. Slutsky" Department of Chemistry, University of Washington, Seattle, Washington 98 195 (Received May 27, 1975)
The vapor pressure of adamantane can be expressed as In P,, = 50.27 - (8416/T) - 4.211 In T (2' in OK) between 278 and 443 K. The heat of vaporization at 300 K is 14.210 kcal/mol, the cohesive energy a t 300 K is -15.445 and -15.848 kcallmol at 0 K. Frequency shifts of the infrared-active internal modes of adamantane with temperature and on vaporization are reported.
I. Introduction Adamantane, by virtue of the high (Td) symmetry of the molecule and the simplicity of the crystal structure1 of the high-temperature, cubic close-packed phase, constitutes a simple system in which approaches to the computation of intermolecular forces,2 lattice-dynamical frequency distrib u t i o n ~ ,intramolecular ~,~ potential function^,^ thermodynamic properties,6 infrared spectra, and a priori deduction of the crystal structures7 of organic solids can be tested. In all such studies the static lattice energy as deduced from the experimental heat of vaporization and related thermodynamic data enters, either as one of the quantities which determine the parameters of the intermolecular or interatomic potential function or as a critical test of the adequacy of the assumed potential. We wish here to report new results on the heat of vaporization and infrared spectrum of adamantane and to briefly discuss the calculation of the static lattice energy and the parameter of the intermolecular potential function. 11. Results and Calculations The initial purity of the adamantane as specified by the supplier14 was 99+%. Repeated fractional vacuum sublimation did at length give a sample with an equilibrium vapor The Journal of Physical Chemistry, Vol. 79,No. 24, 1975
pressure invariant upon partial sublimation. The residual pressure in the vacuum system was 2 X lov9 Torr, and the pressure with the adamantane a t liquid nitrogen temperature was 2 X indicating effective removal of any occluded atmospheric gases. The vapor pressure was measured to f0.06% with an MKS capacitance manometer. The temperature regulator was an adaptation of that described by Larsen,15 and the temperature was measured by an NBS calibrated Leeds and Northrup platinum resistance thermometer. The vapor pressure of adamantane as a function of temperature as determined in this work and by Bratton and Szilard? Boyd: and Wu, Hsu, and Dows7 are given in Figure 1,where the solid curve represents a least-squares fit of the results of Boyd, Bratton, and Szilard and the present work to the form In P,, = 50.27 - (8416/T) - 4.211 In T , where T is in O K . The infrared frequencies in the cubic solid at 225 and 298 K and in the vapor are listed in Table I. The heat of vaporization a t 300 K is 14.210 kcal/mol. Within the framework of the quasi-harmonic approximation, the energy of the solid phase a t a temperature T when the lattice parameter is a may be expressed as U ( a ) E o ( a ) + E(a,T),where U ( a )is the potential energy, Eo(a) the zero-point energy, and E(a,T) the thermal vibrational
+
Heat of Vaporization of Adamantane
2683
TABLE I: The Fundamental Infrared-Active Frequencies (in c m - l ) of Adamantane in the Cubic Solid and Vapor Phases Solid, 225 Ka Solid, 298 K Vapor, 448 K a See ref 7 .
968.8 967 968.8
799.9 796 798.6
1103.8 1102.8 1099.3
1353.1 1351 1355.6
1450.6 1447.5 1455.6
2848 2853.5 2858.4
2923.5 2917.5
2904
6t
where a is the coefficient of thermal expansion, P the isothermal compressibility, and PO the compressibility of the static lattice. Reference to Table I will indicate that the fractional changes in the frequencies of the internal modes on vaporization are small and thus that the fractional changes in the internal frequencies associated with thermal expansion may be neglected and the Gruneisen parameters of the internal modes are negligible so Evih in eq l a is the vibrational energy associated with acoustic and torsional lattice vibrations. Breitling, Jones, and Boyd2 have determined the room-temperature compressibility (@= 2.37 X cm2/dyn) and coefficient of volume expansion ( a = 4.7 X K-l) and deduced C, = 9.3 cal/K mol from Westrum’sll determination of C,. The molar volume at 300 K is 127.33 cm3/mol,l hence y = 6.43. If the potential energy of interaction between a pair of molecules separated by distance rij is Uij = -Arij-6 Br,-12, then the cohesive energy per mole in terms of the nearest-neighbor distance r may be written
+
2.2
3.0
2.5 I
x
103
3.5 3.6 (OK)-’
Flgure 1. The natural logarithm of the vapor pressure of solid ada-
mantane as a function of reciprocal temperature. The open triangles are the present results; solid triangles, ref 8; solid circles, ref 9; open circles, ref 7 . The curve represents In P,, = 50.27 - 84167-1 4.211 In T. energy. The zero-point energy and the thermal energy associated with the acoustic modes and torsional lattice modes of adamantane may be estimated from the frequency distribution spectrum (g(v)) deduced by L ~ t ythe , ~ pertinent relations being, Eo(a) = (h/2)Jvg(v)dv and E(a,T) = Jhv[exp(hv/kT) - l]-lg(v)dv. The zero-point energy so calculated is 0.380 kcal/mol and the thermal energy at 300 K is 3.240 kcal/mol; the total energy associated with torsional and translational lattice modes is 3.620 kcal/mol. Over the range of vapor pressure employed in the deduction of the heat of vaporization, gas imperfections may be neglected so AHvap= 4RT - 3.620 - U ( a ) ,and at 300 K U(a = 9.426 A) = -15.445 kcal/mol. In the quasi-harmonic approximation U(ao), the cohesive energy at the static equilibrium value of the lattice parameter, may be estimated from the experimental value of a and the Gruneisen constant y = d In v/d In V if a form for the variation of U with a is assumed. The relevant relations arelo
where N is Avogadro’s number and C, = 2,(r/ry)n. With U given by eq 2, the quantity in large brackets in eq ICis cm2/dyn, ( a - ao)/ao = equal to -7, PO = 1.18 X 0.0303, a0 = 9.1510, and ro = a o / d = 6.4746 A. For the high-temperature, cubic close-packed structure c6 = 14.45392 and C12 = 12.31125.12 The static equilibrium nearest-neighbor distance ro is given by ro6 = 2ClzB/C&
(3)
a
Simultaneous solution of eq 2 and 3, with ro = 6.4745 and U(r = 6.6652 A) = -15441 cal/mol, gives A = 22.44 X erg cm6, B = 97.05 X 10-lo1 erg cm12, or for the depth of the potential well 4 = 1.298 X erg (Jk = 940.0 K), u = 5.924 %I, and U(ao), the cohesive energy at the static equilibrium separation, is -15.848 kcal/mol. The correction for the effects of anharmonicity is based upon an explicit functional form for the volume dependence of the cohesive energy. The choice of 12 for the repulsive exponent (or indeed representation of the repulsive part of the potential energy as an inverse power of r) is conventional, but without firm foundation in experiment and theory. However, these results are not very sensitive to the choice of repulsive exponent. If 18 were used instead of 12, the calculated value of PO would increase by 0.3% and the static cohesive energy would increase by 0.050 kcal/mol. The results are not greatly different from those obtained by Breitling, Jones, and Boyd2 (U(0 K) = 15.1 kcal, Elk = 915 K). The determination of g(v) for the high-temperature, cubic phase by the inelastic scattering of neutrons makes it possible to avoid the assumption of ref 2 that the heat capacities of the cubic and orthorhombic phases of adThe Journalof Physical Chemistry, Vol. 79, No. 24, 1975
Asim K. Das and Kiron K. Kundu
2604
amantane are the same; a correction for thermal expansion, zero-point dilation, and dynamic contributions to the bulk modulus and the data given here in conjunction with the results of Boydg probably constitute a somewhat more complete account of the thermodynamics of vaporization. The values of A calculated from the Kirkwood-Mueller erg cm6),the London formula (A = 3mc2ax = 6.35 X erg cm6), or the Slaterformula (A = 3a21/4 = 3.06 X = 6.3 X Kirkwood formula (A = 3eha3/2N1f2/8~m1f2 where m is the mass of the electron, e the electronic charge, c the velocity of light, a the molecular polarizability (1.65 X cm3), x the diamagnetic susceptibility (15.6 X cm3), I the ionization potential 9.31 eV,13 and N the number of valence electrons, are not in good agreement with experiment. The changes in the frequencies of the internal modes of adamantane on vaporization are surprisingly small. In a relatively weakly bound crystal such as ethane, the CH bending and stretching frequencies increase by 10-15 cm-l on passage from the solid to the vapor phase.16 The infrared bandwidth due to dipolar interaction may 2)2/(3nm)2(N/ be expressed as17 uf - up = ( n m 2 a)(adaQ)2, where nm, the index of refraction at optical frequencies, is 1.568 and N , the density in molecules/cm3 is 4.73 X loz1. The value of ah/aQ for the strongest CH2 stretching vibration of adamantane as estimated from the ap/aQ for u3 in methane,ls is 85 esu g-ll2, and the calculated V I - ut is then 2 cm-l. The splittings are negligible for all other fundamental modes. The frequencies listed in Table I
+
are thus center frequencies rather than transverse optical frequencies. The source of considerable breadth (-40 cm-l at 298 K) observed in the hydrogen stretching modes in the solid must then be sought elsewhere, probably in multiphonon processes. Acknowledgment. This research was supported in part by the National Science Foundation.
References and Notes (1)C. E. Nordman and D. L. Schmitkens, Acta Crystallogr., 18, 764 (1965); J. Donohue and S. H. Goodman, ibid., 22,352 (1967). (2)S. M. Breitling, D. Jones, and R. H. Boyd, J. Chem. Phys., 54, 3959 (1971). (3)T. Luty, Acta Phys. Pol. A, 40,37,49. 63 (1971). (4)R. Stockmeyer and H. Stiller, Phys. Status Solidi, 27, 269 (1966);19, 781 (1967). (5)R. G. Snyder and J. H. Schachtschneider, Spectrochim. Acta, 21, 169 (1964). (6)R. H. Boyd, S. N. Sanwal, S. Shary-Tehrany, and D. McNally, J. Phys. Chem., 75, 1264 (1971). (7)P. J. Wu, L. Hsu, and D. A. Dows, J. Chem. Phys., 54, 2714 (1971). (8) W. K. Bratton and I. Szilard, J. Org. Chem., 32, 2019 (1967). (9)R. H. Boyd, private communication. (10)M. Born and K. Huang, "Dynamical Theory of Crystal Lattices", Oxford University Press, London, 1954. (11) E. F. Westrum, Jr., J. Phys. Chem. Solids, 18, 83(1961). (12)J. 0.Hirschfelder, C. F. Curtis, and R. B. Bird, "Molecular Theory of Gases and Liquids", Wiley, New York, N.Y., 1954. (13)J. W. Raymonda, J. Chem. Phys., 58,3912(1972). (14)Aldrich Chemical Co. (15)N. J. Larsen, Rev. Sci. Instrum., 39, l(1968). (18)S.Tejada, Ph.D. Thesis, University of Washington, 1963. (17)C. Haas and D. F. Hornig, J. Chem. Phys., 26,707 (1957). (18)E. Ruf, M.S. Thesis, University of Minnesota; J. Heicklen, Spectrochim. Acta, 17,201 (1961).
Thermodynamics of the Ionization of Water in Urea-Water Mixtures and the Structuredness of the Solvents Asim K. Das and Klron K. Kundu" Physical Chemistry Laboratories, Jadavpur University, Calcutta 700032, India (Received May 6, 1974; Revised Manuscript Received April 1, 1975)
The ionization constants (sKw)of water in aqueous solutions of urea containing 11.52, 20.31, 29.64, and 36.83 wt % urea have been determined a t five different temperatures (10-30') from emf measurements of the cell Pt, Hz(g, 1 atm)lNaOH (ml), NaCl (rnz) solvenqAgC1-Ag. The standard free energies, entropies, and enthalpies of ionization of water were calculated from these data. These results, coupled with literature data for the free energies of transfer of H+, AGt'(H+), from water to urea-water mixtures and the relative activities of water, yield values of AGt'(OH-),, and AGto(OH-). Relative magnitudes of these quantities suggest that aqueous urea solutions become less acidic but more basic than water. Analysis of the relative entropic contributions, T(6ASo), to the ionization process in aqueous solutions of methanol, ethanol, 2-propanol, and 2-methyl-2-propanol suggest that alcohol molecules in water-rich regions shift the bulky/dense water equilibria to the left while urea molecules shift them to the right by forming urea-water clusters in the region 2-7 m urea.
Introduction The importance of the overall autoprotolysis constant of any mixed solvent, composed of at least one amphiprotic component, to the acid-base properties in the solvent is well known. The temperature e f f e c t ~ l -on ~ the autoprotolThe Journal of Physical Chemistry, Vol. 79, No. 24, 1975
ysis of amphiprotic solvents contain information regarding solvent structure. In view of the effects of urea4 on the pH of protein solution, solubilities of amino acids, protein denaturation and its dependence on the struct~redness~ of aqueous urea so-
