Myung W. Lee

Vapor Pressures of 36Ar and 40Ar. Intermolecular Forces in Solid and Liquid Argon

The vapor pressures of 36Ar and 40Ar have been measured in the temperature range 62–102°K. Measurements in the liquid agree quantitatively with the previous measurements of Clusius et al. which cover the limited range 84–88°K. A significant difference is found between our equilibrium measurements, which have a precision of 0.5% in the logarithm of the vapor pressure ratio, and dynamic measurements of the isotope fractionation in solid–vapor systems. The reduced partition functions of the solid and liquid are calculated from the observed vapor pressure ratios. The reduced partition ratio of the solid can be represented by the equation T2ln(fs / fg) = 63.39 ± 0.34 − (0.124 ± 0.005)T − (95 / 63)(63.4 − 0.124T)2 / T2. The experimental values of ln(fs / fg) are in good agreement with calculations from the improved self‐consistent phonon theory of an anharmonic lattice with a 13–6 potential. Since the quantum effect is small, 〈∇2U〉 is obtained directly from the reduced partition functions. In the liquid range t...

Vapor Pressures of Isotopic Krypton Mixtures. Intermolecular Forces in Solid and Liquid Krypton

The vapor pressures of normal krypton and a krypton sample containing 50% 80Kr have been measured from 94 to 130°K. New vapor pressure equations are derived for solid and liquid normal krypton. The triple‐point pressures and (temperatures) of 80Kr and 84Kr are 547.565 (115.77) and 547.502 (115.78) int. mm Hg (°K), respectively. The triple‐point pressure of normal krypton is 547.52 ± 0.10 int. mm Hg. The reduced partition functions, fc/fg, of 84Kr/80Kr have been derived for the solid and liquid from the vapor pressure data. The solid data lead to a θ0 (D) = 71.8°K in excellent agreement with heat capacity data. The change in ln (fc/fg) on melting is found to be 24% in good agreement with previous results on neon and argon. V 〈 ∇2 U 〉 is derived from the experimental values of ln (fc/fg) in the liquid. The repulsive potential in liquid krypton is less steep than that in neon or argon. Calculations of V 〈 ∇2 U 〉 from neutron scattering measurements show that the exp‐6 potential to be superior to the LJ (12, ...

Phase equilibrium isotope effects in molecular solids and liquids. Vapor pressures of the isotopic carbon dioxide molecules

The vapor pressures of samples of CO2 enriched in 13C and 18O have been compared with normal CO2 over the temperature range 154−217°K. Over the entire temperature range the order of the vapor pressures is P13CO2 > P12CO2 > PC18O2. The results are explained by the small shift in the internal vibrations, which dominate the carbon isotope effect, and hindered rotation in the solid and liquid. While the isotope effect on melting is small for carbon substitution, it is found to be large for oxygen substitution. Hindered rotation is the major contribution to the oxygen isotope effect in the solid, while the hindered translation is the most important contribution to the oxygen isotope effect in the liquid. Model calculations of the logarithm of the reduced partition function ratios of the solid and liquid, ln(fc/fg), are made in an anharmonic version of the Stern−Van Hook−Wolfsberg cell model. These calculations give good agreement with the experimentally determined ln(fc/fg) values for the solid, liquid−vapor i...

Calculation of the mean force constants of the rare gases and the rectilinear law of mean force

The mean energies 〈U〉 and the mean force constants 〈∇2U〉 have been calculated for liquid argon and liquid krypton using the WCA perturbation theory, and for gaseous argon and krypton along the coexistence line by solution of the PY equation for the radial distribution function. Calculations have been made for the Lennard‐Jones, Barker–Henderson, and Maitland–Smith potentials. There is little difference in the values of 〈U〉 and 〈∇2U〉 calculated for the three potentials. The calculated values are in good agreement with experimental data. Correlation of the calculated values of 〈∇2U〉 for the liquid and gas leads to an empirical relationship between 〈∇2U〉 and the density of the phase along the coexistence line. When the latter is combined with the law of rectilinear diameters of the density, one obtains the rectilinear law of mean force, which is in good agreement with experimental data on argon. It is shown that the scaling exponent for the mean force constant 〈∇2U〉l‐〈∇2U〉g is larger than the scaling exponen...

Calculation of the lattice modes of the isotopic CO2 molecules and their reduced partition function ratios

The lattice dynamics of a unit cell with four CO2 molecules have been calculated from the Raman, infrared, and inelastic neutron scattering spectra. The vibrational frequencies of the isotopic 12C16O2, 13C16O2, 12C18O2, and 12C16O18O are calculated in a pseudoharmonic oscillator approximation. From these vibrational frequencies and the gas phase frequencies, the reduced partition ratios of the various isotopic CO2 molecules are calculated in good agreement with experiment. The lattice dynamics calculation are also compared with calculation from a simple cell model. It is shown that the simple cell model is an excellent approximation for the calculation of the reduced partition function ratios as a consequence of the large separation between the eigenvalues of the internal and external modes, the magnitude of the external eigenvalues relative to kT, and the symmetries of the lattice and the molecule.

Approximation of isotopic reduced partition function ratios by the WINIMAX polynomial perturbation method (WIMPER)

Diagonal H, F, and G matrix approximations to the moments of the eigenvalues are studied for a number of polyatomic molecules. For molecules containing hydrogen atoms the diagonal element approximations to each of the first six moments do not differ from the true values by more than 15%. The results further substantiate the simplified method of analysis of the isotope chemistry of hydrogen in terms of stretching and bending forces. It is shown that the coupling of bending and stretching coordinates makes a significant contribution to the motion of ’’central’’ atoms in a molecule. Corrections to the reduced partition function ratios of isotopic molecules corresponding to uncoupled internal coordinates, the diagonal element approximation, are developed through the use of the WINIMAX polynomials (WIMPER). The magnitude of the corrections are related directly to the off‐diagonal F and G matrix elements. Sample calculations show that the diagonal element approximations lead to values of ln (s/s′) f for deuteri...

Generalized finite polynomial approximation (WINIMAX) to the reduced partition function of isotopic molecules

The MINIMAX finite polynomial approximation to an arbitrary function has been generalized to include a weighting function (WINIMAX). It is suggested that an exponential is a reasonable weighting function for the logarithm of the reduced partition function of a harmonic oscillator. Comparison of the error function for finite orthogonal polynomial (FOP), MINIMAX, and WINIMAX expansions of the logarithm of the reduced vibrational partition function show WINIMAX to be the best of the three approximations. A condensed table of WINIMAX coefficients is presented. The FOP, MINIMAX, and WINIMAX approximations are compared with exact calculations of the logarithm of the reduced partition function ratios for isotopic substitution in H2O, CH4, CH2O, C2H4, and C2H6 at 300 °K. Both deuterium and heavy atom isotope substitution are studied. Except for a third order expansion involving deuterium substitution, the WINIMAX method is superior to FOP and MINIMAX. At the level of a second order expansion WINIMAX approximation...

Liquid–vapor isotope fractionation factors in argon–krypton binary mixtures

An equilibrium isotope effect has been studied as a continuous function of the potential field acting on the atom undergoing isotopic exchange. This has been accomplished through a study of the liquid vapor isotope fractionation factors for both, 36Ar/40Ar and 80Kr/84Kr in a series of binary mixtures which span the range between the pure components at 117.5 °K. The 36Ar/40Ar fractionation factor increases (linearly) from (lnα)2.49×10−3 in pure liquid argon to 2.91×10−3 in an infinitely dilute solution in liquid krypton. Conversely, the 80Kr/84Kr fractionation factor decreases (linearly) from (lnα)0.98×10−3 in pure liquid krypton to 0.64×10−3 in an infinetely dilute solution in pure liquid argon. The mean force constants 〈∇2U〉c on both argon and krypton atoms in the mixtures are derived from the respective isotope fractionation factors. It is shown that the mean force constant on argon atoms in infinitely dilute solution in krypton is equal, within experimental error, to the mean force constant on krypton ...

13C/12C and 18O/16O liquid–vapor isotopic fractionation factors in CO as a function of temperature

The carbon and oxygen isotope fractionation factors in the liquid–vapor equilibrium (LVIFF) of carbon monoxide have been determined over the temperature range 81–125° K. The results are in excellent agreement with previous isotopic vapor pressure data which cover the range 68–82° K. The results are analyzed in terms of an anharmonic cell model by the FG matrix method. The anharmonicity is approximated by the use of a temperature dependent F matrix. Analysis of the LVIFF and VPIE data by this method gives good agreement with experiment and the previous theoretical calculation by Gordon of the mean square force, the mean square torque, and the translation rotation coupling at 77.5° K. The present calculations of the mean square force and experimented data on the mean energy in liquid CO are used to derive a set of intermolecular potential parameters for CO using the WCA perturbation theory. If a m‐6 potential is assumed, the best fit of all the data is obtained for m=14, e/k=109° K, and σ=3.858A. The latter...