Takanobu Ishida

Application of Finite Orthogonal Polynomials to the Thermal Functions of Harmonic Oscillators. I. Reduced Partition Function of Isotopic Molecules

The utility as well as limitations of the Taylor series expansion of the quantum‐mechanical partition function of a harmonic oscillator is discussed. Finite orthogonal polynomials are suggested as a basis for the approximation of the thermodynamic functions of an assumbly of harmonic oscillators. Shifted Chebyshev polynomials of the first kind are used to obtain finite expansions of arbitrary order of the reduced partition functions of isotopic molecules. The resulting series is similar, except for modulating coefficients, term by term to the Bernoulli series obtained from the Taylor expansion. The modulating coefficients of each term in the Chebyshev expansion approach unity as the order of the expansion increases. Thus, the Chebyshev series developed here approaches the Bernoulli series asymptotically. The Chebyshev expansion converges much faster than the Bernoulli expansion. The radius of convergence can be made arbitrarily large in contrast to the limit of 2π which exists for the Bernoulli expansion....

Vapor Pressure of the Isotopic Ethylenes. IV. Liquid Ethylene‐d3 and ‐d4

The purification of ethylene gas samples (200–400 cc NTP) to a chemical purity of 99.9997% is described. The vapor pressures of liquid ethylene‐d3 and ethylene‐d4 have been measured and compared with that of C2H4 in the temperature range 115°–180°K. The reduced partition function ratios of liquid C2HD3/C2H4 and C2D4/C2H4 compared to the ideal‐gas reference state are derived from the vapor‐pressure measurements. The experimental reduced partition function ratios of the liquid are in satisfactory agreement with cell model calculations of Stern, Van Hook, and Wolfsberg. The latter calculations have been extended to include interaction force constants between the internal and external modes as well as the external modes among themselves. The experimental partition function ratios show significant curvature in a plot of T lnfe / fgvs 1 / T in contrast to the almost linear behavior inherent in the harmonic oscillator model. Analysis of the curvature shows that the mean square intermolecular force in liquid ethy...

Vapor Pressure Isotope Effects in Liquidand Solid Ammonia

The H/D and 14N/15N vapor pressure isotope effects in liquid and solid ammonia have been measured at temperatures between 163 K and 243 K. The isotopic vapor pressure data have been fitted to T ln (P′P) = A/T- B for the liquid/liquid, liquid/solid and solid/solid ranges of temperature. The triple points are; 195.41 K (45.49 torr) for 14NH3, 198.96 K (48.35 torr) for 14ND3, and 195.58 K (48.83 torr) for 15NH3. The isotopic difference in the vapor pressures of NH3 and ND3 at temperatures between 195.41 K and 198.96 K is nearly independent of temperature within the present experimental uncertainty. The phase ratios of the reduced partition function ratios, fliq/fgas and fsol/fgas, deduced from these results are well represented by Tin (fc/fg) = A/T-B. Molecular forces in the liquid and solid ammonias are discussed using a simple cell and a 4-molecular unit cell model, respectively. The librational motions in the liquid are almost as highly hindered as they are in the solid, but the directionality of the external forces on nitrogen atoms in liquid ammonia is not as well defined as in the solid.

Zero-Point Energy Shifts in Hydrides of the Elements upon H/D Isotope Substitutions and Their Periodicity

Periodicities are established among the zero-point energy shifts, δ (ZPE), of AHn (n =1-4) type hydrides of the elements upon mono H/D isotope substitutions. δ (ZPE) for diatomic hydrides is correlated with the position of the element A in the periodic table. The contribution of the A-H stretching force constant to δ (ZPE) is to a good approximation a linear function of the square root of the force constant, irrespective of the molecular type. A similar correlation is also found between the square root of the H-A-H bending force constant and its contribution to δ (ZPE).