C. A. Ward

Statistical rate theory of interfacial transport. I. Theoretical development

After assuming that the transport of molecules between phases at thermal equilibrium results primarily from single molecular events, the expression for the rate of molecular transport between phases is developed by using a first order perturbation analysis of the Schrodinger equation and the Boltzmann definition of entropy. This leads to an Einstein‐type relation with the constant of proportionality being the average rate of exchange between microscopic states of different molecular distributions. A hypothesis is introduced which leads to the conclusion that this exchange rate is unchanged as the system moves through the molecular distributions leading to equilibrium, and to it being equal to the molecular rate of exchange between phases in the final equilibrium state. This allows a complete expression for the rate of molecular transport between phases to be developed. The validity of the hypothesis can be examined by comparing the predictions that follow from the derived rate expressions with the availab...

The rate of gas absorption at a liquid interface

To determine the diffusion coefficient of a gas in a liquid the usual procedure is to expose the liquid interface for a period of time, measure the amount of gas absorption, and from this infer the diffusion coefficient. The usual assumption to make at the liquid–gas interface is that equilibrium is established immediately upon exposure of the liquid to the gas. This assumption has led to the illogical consequence of the inferred diffusion coefficient depending upon the period of time for which the liquid interface is exposed to the gas. We use the Einstein expression for the probability of a transition in a system to derive an expression for the rate of gas absorption. When this derived rate is used to infer the diffusion coefficient, the same value is found from experimental techniques with extreme values of the exposure time. The accuracy of the predicted gas absorption rate supports the view that the Einstein relation is not limited to predicting fluctuations about an equilibrium state.

Critical state of bubbles in liquid‐gas solutions

According to homogeneous nucleation theory, the spontaneous formation of a bubble occurs in a liquid when a thermodynamic fluctuation of sufficient magnitude occurs to form a bubble of at least the critical size. Here we report the results of an experimental study of bubbles near the critical state. We examine both the stability of this state and the predicted value of the critical radius. Photographs of two bubbles simultaneously present in a liquid, one slightly larger than the critical size and the other slightly smaller, are presented. Subsequent photographs show the larger one to grow and the smaller one to dissolve. Hence the critical state is unstable and the critical radius lies between the initial radii of the two bubbles. A second series of tests was conducted to examine more carefully the value of the critical radius.

Temperature programmed desorption: A statistical rate theory approach

The equation traditionally used to interpret temperature programmed desorption (TPD) spectra, the Polanyi–Wigner equation, does not contain explicitly the coverage and temperature dependence necessary to predict TPD spectra in several important systems including CO–Ni(111). Herein, the statistical rate theory (SRT) approach is used to formulate equations for temperature programmed desorption which are then used to examine TPD spectra reported in the literature for CO–Ni(111). The molecular and material properties for the CO–Ni(111) system have been previously established. One experimental spectrum has been chosen to determine the apparatus constants. The material properties and the apparatus constants are then used in the SRT equations to predict the eight additional TPD spectra for different initial coverages. A critical comparison can then be made between the theory and these eight experimental spectra, since no fitting constants were used in these eight cases. The results show that there is clearly qua...

Stability of bubbles in a closed volume of liquid‐gas solution

The conditions are examined under which a single bubble and a number of bubbles are in equilibrium within a closed volume of liquid that is maintained at constant temperature and pressure. It is predicted that depending on the amount of gas present in the volume, there may be no equilibrium state for the bubble or bubbles, one equilibrium size, or two possible equilibrium sizes. In the latter case, it is also predicted that the equilibrium state corresponding to the larger bubble size is a stable equilibrium state. This is in contrast to the case of an unbounded volume of liquid where there is the possibility of only one equilibrium state for a bubble, and this state is unstable. The predicted stability for a bubble in a closed volume was examined experimentally, and agreement was found between the measurement and the prediction. A striking result is the reduction in the stable equilibrium size with the number of bubbles present. In particular, micron‐sized bubbles can be shown to be in stable equilibrium...

Statistical rate theory of interfacial transport. II. Rate of isothermal bubble evolution in a liquid–gas solution

The statistical rate theory approach is used to derive the expression for the rate of gas absorption by a liquid. This process involves two sequential rates−the rate of transport from the gas to the surface and the rate of transport from the surface to the bulk liquid. According to the statistical rate theory, the rate limiting step is the rate of transport from the surface. After deriving the rate expression for the rate limiting step, it is incorporated in an integral equation approach for predicting the rate of evolution of a bubble evolving isothermally in a liquid–gas solution. This approach accounts for the movement of the bubble boundary in the diffusion problem. Statistical rate theory leads to a complete expression for the rate of gas absorption; thus by comparing the predicted rate of bubble evolution with a set of measurements, one can investigate the validity of the statistical rate theory. This comparison is carried out and the predictions are found to be in close agreement with the experiments throughout the experimental period.

Statistical rate theory description of beam-dosing adsorption kinetics

Absolute rate theory and the sticking probability approach have been previously examined as possible means of predicting the rate of adsorption. However, when applied to examine adsorption kinetics, they have been found not to contain the coverage and pressure dependence required for several important systems including CO–Ni(111). Statistical rate theory (SRT) is being developed with the objective of predicting the rate of molecular (or atomic) transport across the interface between macroscopic phases in terms of experimentally controllable variables and material properties of the two phases. Previous applications of SRT to adsorption have been limited to systems for which both the gas phase pressure and the temperature could be assumed to be constant. Herein, the SRT approach is extended to systems in which the number of molecules in the system (and hence the gas phase pressure) is not constant. To examine this extension, SRT is used to formulate the equations governing the rate of adsorption in isotherm...

A statistical rate theory study of interface concentration during crystal growth or dissolution

The concentration at the interface of a growing or dissolving crystal has been previously found experimentally to be different from the equilibrium value; however, a method for predicting this concentration has not been available. We report an investigation that uses statistical rate theory to obtain the expression for the rate of molecular transport across the interface of a growing or dissolving crystal. This expression is in terms of the concentration at the interface, and serves as the boundary condition for the convective diffusion equation. The solution of this system of equations contains two constants that appear in the expression for the equilibrium exchange rate. To evaluate the validity of the system of equations obtained, KAlum dissolution in a rotating disk apparatus was examined using data that had been previously reported. In the series of experiments, saturated, aqueous solutions of KAlum were prepared, and the temperature changed to form undersaturated solutions. KAlum{111} crystals were ...

Statistical rate theory of interfacial transport. III. Predicted rate of nondissociative adsorption

The statistical rate theory approach is used to derive the expression for the rate of adsorption of a nondissociating gas on a solid surface. By comparing the predicted adsorption with the available experimental results, one of the central assumptions in the statistical rate theory can be examined. On the basis of these experimental results, with which the theoretical predictions are found to be in close agreement, the assumption would appear to be valid. By being in close agreement with the experimental results throughout the experimental period, the adsorption rate expression derived from the statistical rate theory is an improvement over the presently available theories.

Temperature discontinuity at the surface of an evaporating droplet

In a series of experiments, a temperature discontinuity has been found to exist at the surface of an evaporating water droplet. Statistical rate theory has been used to predict the pressure in the vapor to within the experimental uncertainty during each of the experiments. While the qualitative trend of the D2 law is observed to be consistent with the measurements, it underpredicts the measured rate of evaporation by 21%–37%. When the temperature discontinuity is taken into account in the D2 law, the difference between the predicted and measured values is at most 7%. The results suggest that the rate limiting process in the experiments is not diffusion in the gas phase, as is assumed in the D2 law, but is the interface kinetics.