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N Is Avogadro's Number; and the Factorial Term Is Due to the Indistinguishability of the Gaslike Molecules

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N Is Avogadro's Number; and the Factorial Term Is Due to the Indistinguishability of the Gaslike Molecules THE SIGNIFICANT STRUCTURE THEORY APPLIED TO SOME LIQUID ROCKET FUELS BY T. RICHARD SCHMIDT,* MU SHIK JHON,* AND HENRY EYRINGt UNIVERSITY OF VIRGINIA AND UNIVERSITY OF UTAH Communicated March 18, 1968 The significant structure theory of liquids' assumes that some molecules pos- sess solidlike, and some gaslike, degrees of freedom and that the relative contri- butions of each type are given by VS/V and (V - V8)/V, respectively. Here V5 is the molar volume of the solid at the melting point, and V is the molar volume of the liquid. This theory has been successfully applied to a number of liquids2 ranging from liquefied gases, hydrocarbons, molten metals, and fused salts to water and liquid mixtures. Recently, Jhon et al.3 applied the theory with success to simple rocket fuels which have found importance in space programs. In this paper, a further valida- tion of the theory is tested for liquid pentaborane and nitromethane. A parti- tion function is developed and the properties of these compounds are calculated successfully. The Partition Function.-According to the significant liquid structure theory, the partition function of a liquid is written as - (Ifsolid fdeg) &NV/V (fgas) N((V-V.)/V) (N(V V)) ! (1) Here fsolid, fdeg, and fgas are, respectively, the partition functions of the solid, of positional degeneracy, and of the gas; N is Avogadro's number; and the factorial term is due to the indistinguishability of the gaslike molecules. Liquid pentaborane and nitromethane have an abnormally high entropy of fusion (6.48 e.u. for pentaborane4 and 9.48 e.u. for nitromethane5). No solid- state transitions4' 5 have been observed and the increase in volume upon melting is not sufficient to permit the rotation of solidlike molecules in the liquid. Ac- cordingly, the rotational term appears only in the gas partition function, and a six-degree Einstein oscillator term appears in the liquid. The internal rotation of the nitro group in the nitromethane molecule requires discussion. According to Pitzer,6 the potential barrier for rotation of the nitro group is very small (1.1 kcal/mole); consequently, a free rotation of the nitro group is assumed in both the solidlike and gaslike states in the liquid. The earlier models7 of pentaborane were assumed to have a hydrocarbon-like structure of low symmetry. However, several experiments8 with electron dif- fraction, X-ray diffraction, and Raman spectroscopy revealed an unusual pyra- midal structure of C4, symmetry. Considering the above facts, and using standard expressions for f80lid, fgas, and Eyring's expression for fdeg, and applying Stirling's approximation for the fac- torial term in equation (1), the partition functions for pentaborane and nitro- methane take the following forms. 387 Downloaded by guest on September 29, 2021 388 CHEMISTRY: SCHMIDT, JHON, AND EYRING PROC. N. A. S. g E/R T 36 fB Hg = {(1 eaIT e6hi_1 1 - 1/kcT {1 + n(x - 1) exp X [-(aE8/nx(-1 )RT) }N]} . {(27rmkT) / eV 87r2(8r3ABC) /2(kT)h/2 h3 N 4h3 36 N[(V-V.)/V] II (la) i=1 1 ee-hv/kT} , and eE.IR 14 1 (87r3IkT)'12 fCH3NO2 = i(1 -e-/T)6 ¶= 1 - ehvi/kT h { + ( ) p X [-aE8/n(x - l)RT]O} . (27rmkT) 3/ eV 87r2(87r3ABC) 1/2(kT) '/2 )~~~ h3 N h3 (87r31kT)l/2 14 N[(V-V,)/VI h i-lII 1 e-h/T . (lb) Here, x = (V/V,); E8, 0, and V, are the heat of sublimation of the solidlike structure, the Einstein characteristic temperature, and the molar volume of the solidlike structure at the melting point; a is a dimensionless proportionality constant; A, B, and C are the principal moments of inertia of the molecule; I is the moment of inertia of the NO2 group; v is the ith mode of the internal molecular vibration frequency; m is molecular mass; N is Avogadro's number; and n is equal to the number of nearest neighbors around a molecule at the melting point. This is given by n = 12(VS/V.) for a close-packed structure. The other symbols have their usual significance. The parametric values a, E8, 0, and Vs are evaluated in the manner described in the appendix to a recent article (J. Chem. Phys., 44, 1465 (1966)). These values and the molecular constants from reliable reference sources being known, all thermodynamic properties plus the surface tension are calculated using the Burroughs B-5500 computer. The input data used are as follows: For liquid B5H9,4 mol wt = 63.172: A = 1.18344 X 10-38 gm cm2, B = 1.18344 X 10-38 gm cm2, C = 1.169172 X 10-38 gm cm2; the vibrational energies in units of cm-' are: 2600, 2600, 1844, 1413, 1126, 985, 799, 1450, 500, 1870, 1387, 900, 782, 2600, 1500, 1100, 738, 470, 2598(2), 1802(2), 1621(2), 1449(2), 1034(2), 822(2), 700(2), 605(2), 568(2). The numbers in parentheses represent the de- generacies. The parametric values obtained are V, = 90.86 cc, 0 = 40.016, a = 0.002206, and E. = 8896 cal. For liquid CH3NO2,9 mol wt = 61.04; A = 67.2 X 10-40 gm cm2, B = 76 X 10-40 gm cm2, C = 137.9 X 10-40 gm cm2, I = 4.86 X 10-40 gm cm2; the vibra- tional energies in units of cm-' are: 476, 599, 647, 921, 1097, 1153, 1384, 1413, 1449, 1488, 1582, 2965, 3048, 3048. The parametric values obtained are V. = 52.057 cc, 0 = 42.96, a = 0.0000342, and E. = 10,703 cal. Results.-(a) Volume and pressure: The Helmholtz free energy A is given by A = -kT lnf. If A is plotted as a function of V for a constant T and a common Downloaded by guest on September 29, 2021 VOL. 60, 1968 CHEMISTRY: SCHMIDT, JHON, AND FYRING 389 TABLE 1. Calculated and observed molar volumes. Temp. Vcalc. Vobs.* (OK) (cc/mole) (cc/mole) A% B5H, 226.41 (mp) 92.63 92.63 0.00 237.00 93.50 93.87 -0.39 247.00 94.32 94.99 -0.70 261.00 95.53 96.74 -1.25 273.00 96.63 98.09 -1.48 292.3 98.57 100.59 -2.00 313.0 100.91 335.00 (bp) 103.75 CH3NO- 244.73 (mp) 52.18 52.18 0.00 275.11 53.19 297.12 54.02 298.0 54.05 53.88 0.31 323.0 55.11 347.46 56.28 362.44 57.07 374.44 (bp) 57.74 59.6 3.12 * For B5H9 see ref. 10, and for CH3NO2, see refs. 11 and 13. tangent is drawn through the points corresponding to the liquid and vapor phases, then the vapor pressure is given by the slope of the common tangent, and the abscissas of the two points of the common tangent indicate the molar volumes of the liquid and vapor. The results obtained are shown in Table 1 and Figure 1 and are compared with observed values. Good agreement was obtained for both cases. (b) Entropy of the liquid and entropy of vaporization: The entropy is calcu- lated from equation (1) by the relation - (aT ( (kT nf)). (2) - Theamlwi o o Extpurmol for 8Ns AprExpmneracH0,02for 1.0- a. 0.4- FIG. I.-Vapor pressure o2- versus temperature. The experimental data (0, A) are obtained from refer- QO- ences 12 and 14, respec- 225 250 275 300 325 350 375 tively. T(OK) Downloaded by guest on September 29, 2021 390 CHEMISTRY: SCHMIDT, JHON, AND EYRING PROC. N. A. S. TABLE 2. Calculated values of the entropy of the liquid and entropy of vaporization. B5H9 CH3NO2 Temp. SL AS, Temp. SL AS, (OK) (e.u.) (exu.) (OK) (e-u.) (e.u.) 226.41 (mp) 35.76 36.11 244.73 (mp) 37.17 40.68 237.00 37.00 34.14 275.11 41.57 35.10 247.00 38.15 32.46 297.12 41.65 31.76 261.00 39.74 30.34 298.0 43.61 31.63 273.00 41.11 28.70 323.0 43.47 28.37 292.3 43.30 26.28 347.46 45.47 25.60 313.0 45.67 24.17 362.44 46.58 24.06 335.0 (bp) 48.19 22.13 374.44 (bp) 47.46 22.91 TABLE 3. Calculated values of heat cappacity, compressibility, and coefficient of thermal expansion. Cv Cp Temp. a X 103 j6 X 10' (cal/gm (cal/gm (OK) (deg-) (atm -1) mole deg) mole deg) -B5H9 226.41 (mp) 0.9454 6.293 19.85 27.07 237.00 0.8694 6.277 20.72 27.18 247.00 0.8841 6.834 21.55 28.00 261.00 0.9325 7.898 22.72 29.37 273.00 0.9844 9.001 23.73 30.61 292.3 1.0799 11.154 25.34 32.64 313.0 1.1972 14.089 27.05 34.83 335.0 (bp) 1.3407 18.151 28.82 37.15 -CEsNO-- 244.73 (mp) 6.0967 1.990 16.02 21.80 275.11 6.7107 2.610 16.78 22.90 297.12 7.336 3.223 17.34 23.83 298.0 7.363 3.250 17.37 23.87 323.0 8.151 4.121 18.01 24.96 347.46 9.012 5.194 18.62 26.03 362.44 9.592 5.989 19.00 26.69 374.44 (bp) 10.090 6.717 19.29 27.22 The entropy of vaporization AS, is equal to S, - St, where SO and St are the en- tropies of the vapor and liquid, respectively.
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