Kinetic Theory Values Research Articles

Verification of Onsager's reciprocal relations for evaporation and condensation using non-equilibrium molecular dynamics

Non-equilibrium molecular dynamic (NEMD) simulations have been used to study heat and mass transfer across a vapor–liquid interface for a one-component system using a Lennard-Jones spline potential. It was confirmed that the relation between the surface tension and the surface temperature in the non-equilibrium system was the same as in equilibrium (local equilibrium) [A. Røsjorde, D.W. Fossmo, S. Kjelstrup, D. Bedeaux, B. Hafskjold, J. Colloid Interface Sci. 232 (2000) 178]. Interfacial transfer coefficients were evaluated for the surface, which expressed the heat and mass fluxes in temperature and chemical potential differences across the interfacial region (film). In this analysis it was assumed that the Onsager reciprocal relations were valid [A. Røsjorde, S. Kjelstrup, D. Bedeaux, B. Hafskjold, J. Colloid Interface Sci. 240 (2001) 355]. In this paper we extend the number of simulations such that we can calculate all four interface film transfer coefficients along the whole liquid–vapor coexistence curve. We do this analysis both for the case where we use the measurable heat flux on the vapor side and for the case where we use the measurable heat flux on the liquid side. The most important result we found is that the coupling coefficients within the accuracy of the calculation are equal. This is the first verification of the validity of the Onsager relations for transport through a surface using molecular dynamics. The interfacial film transfer coefficients are found to be a function of the surface temperature alone. New expressions are given for the kinetic theory values of these coefficients which only depend on the surface temperature. The NEMD values were found to be in good agreement with these expressions.

Transfer coefficients for evaporation

An analysis of the recent experimental results by Fang and Ward is given. They found a temperature jump almost upto 10 K across the evaporating surface of water, octane and methylcyclohexane. We use nonequilibrium thermodynamics to obtain the appropriate boundary conditions at the surface. Interfacial transfer coefficients, appearing in these boundary conditions, can then be determined from the experiments. We present them in a form useful for engineering calculations. A comparison is made with the predictions for the transfer coefficients from kinetic theory. For the three materials the kinetic theory values were found to be 30 to 100 times larger than those found from the experiment. It is explained why this gives a liquid surface, which is colder than the adjacent vapor, contrary to the prediction by the kinetic theory. The relative magnitude that we find for the interfacial transfer coefficients, suggests that the condensation coefficient of the kinetic theory decreases with increasing internal degrees of freedom in a molecule. However, a lower value of this coefficient is not sufficient to explain the small value of the transfer coefficients. As a possible explanation for this, we forward the hypothesis that the single-particle collision model used in the kinetic theory for this phenomenon, should be modified to account for multiparticle events.

Transport properties of the hard ellipsoid fluid

Transport properties of isotropic fluids composed of hard ellipsoids of revolution are studied using molecular dynamics simulation. The self-diffusion coefficient, the shear viscosity, and the thermal conductivity are evaluated for a range of densities and elongations and are compared with the results from an Enskog kinetic theory for nonspherical bodies. The full anisotropic pair correlation function, which is required input in an Enskog kinetic theory, can be obtained from simulation or can be approximated. If the pair correlation function is taken as isotropic on the contact surface, with a contact value derived from an accurate equation of state, the resulting kinetic theory transport properties agree to within a few percent of those calculated on the basis of the exact pair correlation function. The simulation and the kinetic theory values for the shear viscosity and the thermal conductivity show the same qualitative behavior, i.e., increasing with density and with particle nonsphericity. Quantitatively, there is good agreement at low densities (up to 30% of closest packing); at higher densities (60% of closest packing), deviations from Enskog theory are larger than and in the opposite direction to those seen for hard spheres. The Stokes–Einstein and Debye relations are tested and indicate a transition from a kinetic theory region towards the hydrodynamic limit as density increases.

The temperature dependence of the half widths of some self-and foreign-gas-broadened lines of methane

The temperature dependence of the half width of four lines of the 2 v 3 band of methane has been measured. The broadeners used were H 2, He, N 2, and CH 4. The temperature coefficients, defined by the relation γ = γ 0( T T 0 ) n , have been determined for all of the broadeners and, in some cases, depart significantly from a kinetic theory value of -0.5. Where possible, the half-widths were determined by two independent methods. All measurements were made at pressures below 2 atm absolute to reduce interfering effects from the wings of neighboring lines.

Microwave Collision Diameters II. Theory and Correlation with Molecular Quadrupole Moments

Observed microwave collision diameters are interpreted as arising exclusively from transitions induced during the collision. Collision diameters greater than kinetic theory values arise from dipole-dipole or dipole-quadrupole interactions. The former case affords a quantitative test of the theory. In the latter case measurements of collision diameters yield quadrupole moment measurements which are correlated with molecular structure.

The Effective Cross Section of Positive Alkali Ions with Respect to Gas Molecules

The absorption of positive sodium ions with velocities of 40 to 400 equivalent volts has been studied in hydrogen, nitrogen and oxygen. The nature of the collision processes has, in each case, been studied to determine the relative importance of neutralization, retardation or small angle scattering. The mean free paths and effective cross section of the sodium ions with respect to the gas molecules is found to be a function of the pressure of the gas, the velocity of the ion and the nature of the gas-ion combination. The effective cross section of each gas-ion combination approaches the kinetic theory value at low velocity of the ion while at high velocities it approaches a minimum considerably smaller than the kinetic theory value. At any given velocity over the range studied, the effective cross section of the gas-ion combination increases in the order hydrogen, nitrogen, oxygen. The values for nitrogen and oxygen are essentially, within experimental errors, equal. An improved furnace for the production of positive ions has been developed.

A theory of electro-kinetic effects in solution; reactions between ions and polar molecules

The experimental values of the velocity coefficient k for bimolecular reactions may be summarized in an equation of the form k = Z . P . e -E A /RT (1) where P and E A are empirical constants, specific for each reacting system, and Z is the kinetic theory value for the frequency of binary collisions, namely, Z = N 0 /1000 r 1,2 2 n 1 n 2 { 8π k T (1/m 1 + 1/m 2 )}. (2) The physical significance of the term P may be realized by considering the hypothetical case where the Arrhenius energy of activation E A is identical with the true energy of activation E'. Since the theoretical expression for the maximum velocity of the bimolecular process is k = Z . e -E'/RT , (3) it is seen that P measures the absolute probability that a binary collision of sufficient energy leads to chemical reaction. In general, however, E A and E' are not identical, and the interpretation of the term P is not so simple.

The Scattering of Hydrogen and Helium Beams in Mercury Vapor

The scattering of beams of hydrogen molecules and helium atoms in mercury vapor has been investigated in the angular region from 5\ifmmode^\circ\else\textdegree\fi{} to 65\ifmmode^\circ\else\textdegree\fi{} at beam temperatures of 115\ifmmode^\circ\else\textdegree\fi{}, 300\ifmmode^\circ\else\textdegree\fi{} and 590\ifmmode^\circ\else\textdegree\fi{}K. The results are expressed in terms of the number of molecules scattered per incident molecule, per cm path, per ${10}^{\ensuremath{-}3}$ mm of mercury at 0\ifmmode^\circ\else\textdegree\fi{}C per unit solid angle in the direction $\ensuremath{\theta}$ to the original beam. The general shape, the absolute intensity and the change in shape of the helium scattering curves with change in beam temperature is in moderately good quantitative agreement with the quantum theory of scattering of rigid spheres as developed by Massey and Mohr. The scattering of hydrogen molecules, especially at low temperatures, shows some deviation from the results predicted by the same theory. In the case of both hydrogen and helium it is necessary to assume collision radii larger than the corresponding kinetic theory values in order to obtain the best agreement between the calculated and observed curves.

Some Experiments in Positive Ion Kinetics

Mean free path and mean energy loss per collision of positive ions of potassium and caesium in helium, neon, argon, hydrogen and nitrogen.—The motion of certain positive ions through gases was investigated in an experimental tube of simple design. The ions were obtained from the potassium and caesium catalyst sources developed at the Fixed Nitrogen Laboratory, and their behavior was studied in helium, neon, argon, hydrogen, and nitrogen. It was found that the results were best explained on the basis of a free path approximately equal to the kinetic theory value, and on an average fraction of energy lost per collision considerably less than would be expected on ordinary considerations. It is suggested that the explanation of this is to be found in a more accurate consideration of the fields of force involved at these encounters.

The Motion of Electrons in Hydrogen and Helium

Mobilities of electrons in ${\mathrm{H}}_{2}$ and He at 760 mm pressure have been measured by the Rutherford alternating potential method for fields ranging from 0.5 to 46 volts/cm. The limiting mobilities for zero field are calculated to be, for ${H}_{2}$, 6,700 cm/sec per volt/cm and for He, 10,840 cm/sec per volt/cm.Energy loss of an electron on impact in ${\mathrm{H}}_{2}$ and He.---K. T. Compton's theory of electronic mobilities is shown to fit the observed variation of mobility with field strength provided it be assumed that the fraction of its energy an electron loses on impact is, in ${H}_{2}$, 2.16, and in He, 8.6 times what it would be were the collisions elastic.Mean free path of electrons in ${\mathrm{H}}_{2}$ and He.---From the limiting values of the mobilities at zero field it is calculated that the mean free path of electrons in thermal equilibrium with the gas is for ${\mathrm{H}}_{2}$ at 1 mm pressure, 0.0408 cm, and for He at 1 mm pressure, 0.066 cm. These values are approximately half the kinetic theory values.

Probability of Ionization of Gas Molecules by Electron Impacts

Ionization in various gases by electrons of energy 0-325 volts.---Electrons from a hot filament were projected through an ionization chamber $B$ containing gas at low pressure (.001 to.005 mm) and into a trap. Positive ions produced in $B$ were collected on fine wires arranged in the form of a squirrel cage $C$ and maintained at a potential lower than $B$ by an amount ${V}_{r}$. Corrections were made for the facts (1) that the wires $C$ lowered the potential at the center of $B$ and hence retarded the electrons passing through $B$ by a maximum amount of $0.4\ifmmode\times\else\texttimes\fi{}{V}_{r}$, and (2) that if ${V}_{r}$ was numerically less in value than the accelerating potential ${V}_{\ensuremath{\alpha}}$, some primary electrons reached $C$. The pressures were measured with a McLeod gauge to 1 percent and were corrected for the heating effect of the filament, which was determined by using the apparatus as an ionization manometer to be about 26\ifmmode^\circ\else\textdegree\fi{}. The resulting curves for $N$, the number of ions per cm path produced by an electron moving through a gas at 1 mm pressure, are believed correct within a few percent. In agreement with previous results of Hughes and Klein and others, the curves each rise to a maximum. The maximum values found and the corresponding voltages are as follows: The values for $P(max)$, the corresponding probability that a collision will result in ionization of the hit molecule, were computed from $N(max)$ assuming the kinetic theory values for electronic mean free paths. These results are compared with previous experimental values. Ionization was found to begin at the ionizing potential in each case, except that with ${\mathrm{N}}_{2}$ weak ionization was observed at 10 volts. This may possibly be associated with the presence of traces of active nitrogen.

The Motion of Electrons in Nitrogen

Mobilities of electrons in nitrogen have been determined in fields ranging from 1 volt/cm to 55 volts/cm at atmospheric pressure. For low fields the mobility has a value of 18,000 cm/sec. but decreases rapidly for higher fields. The results agree for low fields with Compton's mobility equation, viz., $u=\frac{a}{{[1+{(1+B{x}^{2})}^{\frac{1}{2}}]}^{\frac{1}{2}}}$, but in order to obtain full agreement it must be assumed that the mean free path of the electron decreases as the velocity of the electron increases and also that the impacts of electrons with nitrogen molecules are not completely elastic. The mean free path of the electrons in nitrogen at a pressure of 760 mm and in thermal equilibrium with the molecules has been determined from the limiting mobility as the field in which the mobility is determined approaches zero. The value thus found is.000131 cm. This is about 2.7 times the kinetic theory value.

Mobilities of Electrons in Gases

Motion of electrons through gases.---The theoretical equation for mobility constant obtained in a recent paper, is put in the form $K=2.54({10}^{5}){l}_{0}{\left\{1+{[1+1.355({10}^{6})M{{l}_{0}}^{2}{(\frac{E}{p})}^{2}]}^{\frac{1}{2}}\right\}}^{\ensuremath{-}\frac{1}{2}}$ where ${l}_{0}$ is electronic mean free path at 1 mm pressure, $M$ is molecular weight relative to H, $E$ is the electrical field in volts/cm, and $p$ is pressure in mm Hg; and it is tested by comparison with recent experimental data obtained by Loeb and by Townsend and co-workers. Although the equation contains no arbitrary constant if ${l}_{0}$ is taken from kinetic theory, the agreement with experiments is good in the case of ${\mathrm{H}}_{2}$ and within a small factor in the case of He, ${\mathrm{N}}_{2}$ and A, for values of $\frac{E}{p}$ less than the critical value which is characteristic of each gas and equal to about 20 for ${\mathrm{H}}_{2}$, 1.3 for ${\mathrm{N}}_{2}$, 0.5 for A, and >.4 for He. Elasticity of collisions. In explanation it is suggested that when electrons collide at speeds greater than the critical for the gas, the collisions are no longer perfectly elastic, as assumed by the theory. In the region of elastic collisions, the equation may be used to compute the actual equivalent mean free paths for elastic spheres from the measured values of $K$, for comparison with the kinetic theory values of ${l}_{0}$. This free path in ${\mathrm{H}}_{2}$ is close to the kinetic theory value, in He it is a little less, while in ${\mathrm{N}}_{2}$ and A it is greater and varies with the speed, being greater at small speeds. These free paths are compared with values determined in other ways and the differences are attributed to differences in the meaning attached to the term collision. The data for ${\mathrm{O}}_{2}$ and C${\mathrm{O}}_{2}$ are insufficient to give the values of the critical $\frac{E}{p}$ but indicate that these are lower than for the other gases.