The entropy and related thermodynamic properties of methylisocyanate, CH3NCO, have been determined by isothermal calorimetry. The entropy in the ideal gas state at 298.15 K and 1 atmosphere is S m o = 284.3 ± 0.6 J/K · mol. Other thermodynamic properties determined include: the heat capacity from 15 to 300 K, the temperature of fusion (Tfus = 178.461 ± 0.024 K), the enthalpy of fusion (ΔHfus = 7455.2 ± 14.0 J/mol), the enthalpy of vaporization at 298.15 K (ΔHvap = 28768 ± 54 J/mol), and the vapor pressure from fusion to 300 K. Using statistical thermodynamics, the entropy in this same state has been calculated for various assumed structures for methylisocyante which have been proposed based on several spectroscopic and ab initio results. Comparisons between the experimental and calculated entropy have led to the following conclusions concerning historical differences among problematic structural properties: (1) The CNC/CNO angles can have the paired values of 140/180° or 135/173° respectively. It is not possible to distinguish between the two by this thermodynamic analysis. (2) The methyl group functions as a free rotor or near free rotor against the NCO rigid frame. The barrier to internal rotation is less than 2100 J/mol. (3) The CNC vibrational bending frequency is consistent with the more recently observed assignments at 165 and 172 cm−1 with some degree of anharmonicity or with a pure harmonic at about 158 cm−1.

We obtain a set of four-dimensional hyperspherical harmonics in closed form. These harmonics are not only quantized with respect to the rotation group (O2), but are an irreducible basis for the permutation groupS3. An additional symmetry is found which allows us to write hyperspherical harmonics classified with respect to a 12 element groupS3×i×O2. We give a set of three mutually commuting operators whose eigenvalues uniquely characterize each spherical harmonic with respect to degree, symmetry, and angular momentum in the plane.

The importance of excluded volume in determinig the sizes of branched polymers is discussed in terms of several models. First, the branched chain analog of the rotational isomeric model is solved and gives improvement over earlier random flight models but, as was the case for linear chains, the asymptotic dependence of polymer size on monomer number (〈size〉≂ANν) remains the same, except that model differences are found in the preexponential factor A. Next, a hierarchy of subclasses of the full class of generally branched self-avoiding random walks is briefly described and compared to our recent Monte Carlo results on the full problem. Finally, simple heuristic arguments for excluded volume corrections are considered.

The entire asymptotic high temperature expansion of the direct partition function for rotation of a rigid symmetric top is obtained by double application of the Euler–Maclaurin summation formula. The expansion is in the form of an exponential factor times the classical answer times a very simple descending power series in the temperature.

In this note we present selected formalism and results of a full scale hyper-spherical calculation of the quantum statistical mechanical 3-body cluster.

The first two terms of the high energy expansion of the relative density of states for two hard spheres (needed for the computation of the second virial coefficient at high temperature) are obtained directly from the analytic expression for the phase shifts. This is enough to give the classical second virial and the first quantum correction.

The direct and exchange partition functions for a rigid diatomic rotor are obtained via the density matrix. An integral representation which is suitable for finding convergent high temperature expansions is formulated by use of the transformation properties of theta functions. The convergent expansion for the direct partition function is shown to consist of the asymptotic Euler–Maclaurin series with each term corrected by a series of nonanalytic terms. The nonanalytic terms are of the form exp(−Nπ2/σ) and the entire expansion is rapidly convergent at high temperature.

The historic controversy between Van der Waals and Boltzmann concerning the correct way to calculate the fourth virial coefficient of a hard sphere gas is analyzed and put in modern notation. In this form it is easy to see what integrals would have to be evaluated in order to obtain the fifth virial coefficient by this method.

The heat capacity from 15 to 300°K, the heat of fusion (ΔH = 1779.9 ± 1.7 cal/mole), the temperature of fusion (184.368 ± 0.010°K), the vapor pressure, and the heat of vaporization at 285°K (ΔH = 6722 ± 10 cal/mole) have been determined experimentally for tetramethylgermane. The resulting entropy of the ideal gas at 285°K and 1 atm is 89.81 ± 0.15 cal/deg·mole. Agreement between calculated and experimental entropies requires that the potential barrier to rotation of the methyl groups be 750 ± 90 cal/mole. This value is surprisingly low considering the trends of the potential barriers in other methyl compounds of the Group IV elements.

Chemischer Informationsdienst. Organische ChemieVolume 1, Issue 32 Preparative Organic Chemistry ChemInform Abstract: ENTROPIE UND VERWANDTE THERMODYNAMISCHE EIGENSCHAFTEN VON TETRAMETHYLGERMAN ANTONE J. VALERGA, ANTONE J. VALERGASearch for more papers by this authorJOHN E. KILPATRICK, JOHN E. KILPATRICKSearch for more papers by this author ANTONE J. VALERGA, ANTONE J. VALERGASearch for more papers by this authorJOHN E. KILPATRICK, JOHN E. KILPATRICKSearch for more papers by this author First published: August 11, 1970 https://doi.org/10.1002/chin.197032029AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume1, Issue32August 11, 1970 RelatedInformation

The quantum-mechanical second virial coefficients of Lennard-Jones 3He and 4He gases with the De Boer parameters have been obtained over the complete temperature range from near absolute zero to the classical region. A formalism separating the virial into direct (Boltzmann) and exchange (spin and quantum statistics) contributions has been employed. The calculation is based on phase shifts except at the very highest temperatures where a Wigner–Kirkwood method has been used. Examination of the exchange term shows in detail the rapid suppression of the statistical effects with rising temperature, their contribution dropping to less than 0.001 cm3 by 7°K (4He). Comparison of the high-temperature (Boltzmann) results with those obtained by a third-order Wigner–Kirkwood expansion shows excellent agreement down to about 50°K for 4He and 60°K for 3He. The Wigner–Kirkwood expansion is shown to be unsuitable for determining the behavior of the exchange terms. Finally, results are compared with the available experimental data.

The general solution of the problem of expressing the density virial coefficients as polynomials in the pressure virial coefficients (and vice versa) is deduced by a very simple argument. The expressions for both kinds of virial coefficients are deduced as polynomials in the reducible cluster integrals. Some applications are made to the ideal Bosé-Einstein and Fermi-Dirac gases.

where n\, n2, ns are zero or positive integers, n + n2 + ns is even and a, b are real positive numbers. The symbols Ui are used for convenient reference. Integrals (1.1) and (1.2) together with integrals of threefold and fourfold products of associated Legendre functions [7], were used in the calculation of virial coefficients in statistical mechanics [3], [4], [5], [6]. The usual numerical integration techniques such as Simpson's method, Gauss' method, method of indefinite integral of polynomials [7] etc., when applied to integrals with oscillating integrands such as in (1.1) and (1.2) are inefficient. Values of integrals (1.1) and (1.2) can be obtained by transforming them into Mellin-Barnes integrals [1], [2], etc., or Meijer's G-functions [2], and application of the residue calculus as developed by one of the authors in [3] leads to the exact determination of the integrals. As the Mellin-Barnes integrands are rather complicated, one has to do considerable scanning to determine the actual poles. In this paper, the scanning process and the evaluation of the residues at these poles is computer programmed.

We have separated the quantum-mechanical second virial coefficient for hard spheres B into two terms. The first represents the contribution of a Boltzmann gas, and the second is an exchange term embodying the effects of quantum statistics. Numerical computation of B to high precision then allows us to analyze the temperature dependence of the exchange term, which is found to decrease exponentially with temperature, and to determine the asymptotic expansion of the Boltzmann term at high temperatures.

It is shown that Qe, the partition function for a rigid rotor summed over even levels and Qo, summed over odd levels, have exactly the same asymptotic (power series in σ=ℏ2/2IkT) expansion. No information as to differences in thermodynamic properties due to spin and statistics can be obtained from this expansion. The exchange partition function, Qe—Qo, is calculated directly and used to give simple expressions for the differences in thermodynamic properties between the para, ortho, and equilibrium cases.

It is shown that the repulsive core present in realistic two-body potentials and in hard spheres leads to the rapid suppression of the effects of statistics in the second virial coefficient, except at very low temperatures. For hard spheres, an upper bound is obtained which goes down exponentially with temperature when the latter becomes large.