Virial Corrections Research Articles

The solubility and isotopic fractionation of gases in dilute aqueous solution. I. Oxygen

A very precise and accurate new method is described for determination of the Henry coefficient k and the isotopic fractionation of gases dissolved in liquids. It yields fully corrected values for k at essentially infinite dilution. For oxygen the random error for k is less than 0.02%, which is an order of magnitude better than the best previous measurements on that or any other gas. Extensive tests and comparison with other work indicate that systematic errors probably are negligible and that the accuracy is determined by the precision of the measurements. In the virial correction factor (1+λPt), where Pt is the total pressure of the vapor phase, the coefficient λ for oxygen empirically is a linear decreasing function of the temperature over the range 0–60°C. The simple three-term power series in 1/T proposed by Benson and Krause, $$\ln k = a_0 + a_1 /T + a_2 /T^2 $$ provides a much better form for the variation of k with temperature than any previous expression. With a0=3.71814, a1=5596.17, and a2=−1049668, the precision of fit to it of 37 data points for oxygen from 0–60°C is 0.018% (one standard deviation). The three-term series in 1/T also yields the best fit for the most accurate data on equilibrium constants for other types of systems, which suggests that the function may have broader applications. The oxygen results support the idea that when the function is rewritten as $$\ln k = - (A_1 + A_2 ) + A_1 \left( {\frac{{T_1 }}{T}} \right) + A_2 \left( {\frac{{T_1 }}{T}} \right)^2 $$ it becomes a universal solubility equation in the sense that A2 is common to all gases, with T1 and A1 characteristic of the specific gas. Accurate values are presented for the partial molal thermodynamic function changes for the solution of oxygen in water between the usual standard states for the liquid and vapor phases. These include the change in heat capacity, which varies inversely with the square of the absolute temperature and for which the random error is 0.15%. Analysis of the high-temperature data of Stephan et al., in combination with our values from 0–60°C, shows that for oxygen the fourterm series in 1/T, $$\ln k = - 4.1741 + 1.3104 \times 10^4 /T - 3.4170 \times 10^6 /T^2 + 2.4749 \times 10^8 /T^3 $$ where p=kx and p is the partial pressure in atmospheres of the gas, probably provides the best and easiest way presently available to calculate values for k in the range 100–288°C, but more precise measurements at elevated temperatures are needed. The new method permits direct mass spectrometric comparison of the isotopic ratio34O2/32O2 in the dissolved gas to that in the gas above the solution. The fractionation factor α=32k/34k varies from approximately 1.00085 (±0.00002) at 0°C to 1.00055 (±0.00002) at 60°C. Although the results provide the first quantitative determination of α vs. temperature for oxygen, it is not possible from these data to choose among several functions for the variation ofInα with temperature. If the isotopic fractionation is assumed to be due to a difference in the zero-point energy of the two species of oxygen molecules, the size of the solvent cage is calculated to be approximately 2.5 A. The isotopic measurements indicate that substitution of a34O2 molecule for a32O2 molecule in solution involves a change in enthalpy with a relatively small change in entropy.

Virial scaling corrections to Koopmans’ ionization potential in positive ions of He, Ne, and Ar series

The ionization potentials calculated for the positive ions isoelectronic with He, Ne, and Ar, respectively, using the virial scaling method are found to be in good agreement with the results obtained from the ΔSCF method.

Virial scaling corrections to Koopmans' ionization potentials in atoms

Relaxation corrections to Koopmans' approximation for atomic ionization potentials are derived on the basis of the virial theorem. They require no information in addition to that available in the Hartree-Fock computation for the neutral system, and considerably improve the agreement with more accurate ionization potentials.

Thermodynamic study of theHe4monolayer adsorbed on Grafoil

A calorimetric and vapor-pressure study of the behavior of $^{4}\mathrm{He}$ adsorbed on Grafoil in the temperature range 4 to 15 K is reported. These results have been combined with preexisting low-temperature heat-capacity data to form a complete thermodynamic description of the film. The Grafoil substrate evidently consists almost entirely of basal-plane graphite, with only a small fraction of energetically distinct adsorption sites. Techniques of semiempirical modelling are introduced which make it possible to eliminate the effects both of these inhomogeneities and of the formation of second and higher layers, so that a detailed picture may be formed of the behavior of the $^{4}\mathrm{He}$ first layer on an ideal graphite substrate. The binding energy of a $^{4}\mathrm{He}$ atom on the graphite substrate is reported to be 143 \ifmmode\pm\else\textpm\fi{} 2 K, with a first excited state at 89 \ifmmode\pm\else\textpm\fi{} 3 K. A lattice-gas ordering transition occurs near 3 K at a coverage corresponding to one $^{4}\mathrm{He}$ atom for every three carbon hexagons in the graphite surface, and is used as a fiducial for calibrating our system relative to those of previous investigators. At both higher and lower densities the film seems little influenced by the crystal structure of the substrate. Particular attention is given to the high-coverage low-temperature region, which has been identified as a two-dimensional (2D) solid bounded by a melting-phase transition. The available data are combined with two-dimensional elastic theory to establish the presence of low-lying transverse sound modes, thus confirming the solidlike character of this medium. The behavior of the thermodynamic variables of state through the observed transition is also presented. The theoretical implications of the existence of a 2D solid are discussed, together with a dislocation theory of melting which gives rise to the well-known three-dimensional Lindemann empirical melting formula without an adjustable parameter. It is shown that at very low coverage the film does not form a low-density solid as previously suggested, but is rather dominated by substrate inhomogeneities. At intermediate temperatures and coverages the film is a fluid which is accurately described if a leading-order quantum virial correction is made to the 2D ideal-gas equation of state.

Phases ofHe3andHe4Monolayer Films Adsorbed on Basal-Plane Oriented Graphite

Heat capacities of ${\mathrm{He}}^{3}$ and ${\mathrm{He}}^{4}$ monolayers adsorbed on Grafoil graphite substrates at low temperatures are qualitatively different from previous results using other adsorbents. The present study consists of a detailed survey of many samples conducted in two similar but independent calorimeters: the fractional coverages ranged from 5 to 115% of a completed first layer and the temperature range extended from 0.04 to above 10\ifmmode^\circ\else\textdegree\fi{} K. Several distinct thermodynamic regimes are seen, some of which correspond closely with well-known theoretical models as well as others that had not been predicted. At moderate densities and temperatures the behavior resembles that of two-dimensional gases. Departures from ideality are correlated with quantum-mechanical virial corrections for interacting molecules. In this range of coverages the ${\mathrm{He}}^{4}$ specific heats begin to rise at $T\ensuremath{\lesssim}{3}^{\ensuremath{\circ}}$ K, forming strong rounded peaks near 1\ifmmode^\circ\else\textdegree\fi{}K. ${\mathrm{He}}^{3}$, on the other hand, shows a monotonic decrease with falling temperature until $T\ensuremath{\simeq}{0.2}^{\ensuremath{\circ}}$ K, then (for a narrow range of coverages) reversing the trend to form rounded maxima near 100 m\ifmmode^\circ\else\textdegree\fi{}K. At the lowest temperatures the ${\mathrm{He}}^{3}$ appears to enter a two-dimensional ($2D$) Fermi-liquid regime. At higher coverage both ${\mathrm{He}}^{3}$ and ${\mathrm{He}}^{4}$ undergo second-order phase transitions to regular arrays in registry with the substrate. In the critical region that heat capacities have symmetric peaks of logarithmic shape, with coefficients in close quantitative agreement with exact $2D$ Ising models. Above critical density the ordering peaks disappear and $2D$ liquid and solid behavior is seen. ${\mathrm{He}}^{4}$ at high density and low $T$ is Debye-like, with characteristic temperatures ${\ensuremath{\Theta}}_{2D}$ equal to ${\ensuremath{\Theta}}_{3D}$ of hcp solid ${\mathrm{He}}^{4}$ of the same interatomic spacing. At higher $T$ the $2D$ solid appears to transform to a $2D$ fluid by a continuous process, and pronounced heat-capacity peaks associated with the transformation are located at temperatures near the melting points of hcp ${\mathrm{He}}^{4}$ having the same interatomic spacing. At very low coverage the $2D$ gas character gives way to a regime resembling a low-density $2D$ solid.

Self-Trapped States of an Electron in a Structurally Disordered System

Electrons injected into structurally disordered materials may occupy free-electron-like states in a conduction band, band-tail states, some of which will be nondiffusing and are thus localized in the Anderson sense, and, finally, self-trapped states. Eggarter has recently described the first two types of states with specific application to He gas. In the present work, the self-trapped states are discussed. Specific calculations are carried out for He, ${\mathrm{H}}_{2}$, ${\mathrm{N}}_{2}$, and Ne in the ideal-gas approximation. Virial corrections are demonstrated for He gas at 4.2 K. Available experimental results, for the four materials mentioned, are discussed in the context of the theory presented and it is shown that for certain regimes of density and temperature self-trapped states exist and play an important role in determining the observed transport parameters. Further calculations and experiments are suggested to probe in detail the characteristics of all the states available to electrons in the type of disordered materials discussed.

Molecular weights of homogeneous coliphage DNA's from density-gradient sedimentation equilibrium

The problem of obtaining meaningful molecular weights for high molecular weight DNA samples by sedimentation equilibrium in a density gradient has been solved by correcting the theory for thermodynamic non-ideality. Several different methods of virial corrections are suggested and examined to determine the best method of obtaining molecular weights at infinite dilution. A pronounced virial effect is experimentally demonstrated for high molecular weight DNA's. The concentration distributions at sedimentation equilibrium are studied as a function of DNA concentration for T7, T5 and T4 coliphage DNA's and the results extrapolated to infinite dilution to obtain molecular weights for these DNA's. The values obtained by this technique are 23.2 × 10 6 for T7, 65.7 × 10 6 for T5 and 104 × 10 6 daltons for T4 DNA. These results are in agreement with recent hydrodynamic predictions but are in disagreement with many literature values. A brief review of the literature is presented to elucidate these discrepancies.

Die Vielteilchen-T-Matrix und ihre Anwendung in der Theorie realer Gase von mittlerer Dichte /

Abstract The improved Landau equation governing nonequilibrium behaviour of quantum gases of moderate density is shown to yield the exact second virial coefficient B(T). After simplifying the collisional operator the linearized stress tensor is evaluated to give the lowest order virial correction to the shear viscosity.

Structure of the H Function for a Hard-Sphere Gas

We consider, for a spatially uniform hard-sphere gas, the generalization of the connection between the H function and the thermodynamic entropy to that limit where gas imperfections are taken into account through the first virial correction term. Our choice of a generalized H function is based on the expansion of the canonical entropy density given by Nettleton and Green. We show that this choice allows us to give a physical significance to the condition that Boltzmann's H function be convex as a function of time.

Transport Coefficients in Moderately Dense Gases

The first virial corrections for shear viscosity η and thermal conductivity λ are calculated, starting from the BOLTZMANN-LANDAU equation of a quantum gas in binary collision approximation. The first order CHAPMAN-ENSKOG solution is used. With a very simplified collisional operator numerical values are got and compared with experiments. The result is that the kinetic equation including the interaction effects in quasi-free motion is able to explain the empirical order of magnitude of dη/dn.

Eigenvalues for the equation of species conservation with heterogeneous reaction

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Sound Absorption and Velocity in Water Vapor

Kirchhoff'f tube-wall corrections and virial real-gas corrections adequately account for experimental values for the unsaturated vapor at 100° and 250°C.