Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “equilibrium modeling”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 595 records · Page 33
All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

GEOSH: Ideal Gas Chemical Equation of State

We present the framework and methodology for the new Los Alamos National Laboratory (LANL) G as chemical E quation O f S tate at H igher temperatures code (GEOSH) which aims to accurately model the behavior of chemically complex gaseous mixtures in equilibrium. Assuming the ideal gas approximation, GEOSH leverages the recursive nature of the Saha ionization and molecular equations in order to eliminate the molecular and ionic degrees of freedom, thereby reducing the problem size to the number of atomic species plus one for the free electrons if ions are included. This approach allows the chemical species, both molecular and ionic, of the mixture to be expressed in terms of the abundances of the elemental species. As a result, the GEOSH framework achieves a reduction in computational expense, increased processing speed, and the capability to efficiently model large-scale chemical networks. This report provides the necessary physical background and theoretical foundations for the GEOSH code, accompanied by benchmarking studies.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2024

High-Energy Collision of Quarks and Mesons in the Schwinger Model: From Tensor Networks to Circuit QED

With the aim of studying nonperturbative out-of-equilibrium dynamics of high-energy particle collisions on quantum simulators, we investigate the scattering dynamics of lattice quantum electrodynamics in 1 + 1 dimensions. Working in the bosonized formulation of the model and in the thermodynamic limit, we use uniform-matrix-product-state tensor networks to construct multiparticle wave-packet states, evolve them in time, and detect outgoing particles post collision. This facilitates the numerical simulation of scattering experiments in both confined and deconfined regimes of the model at different energies, giving rise to rich phenomenology, including inelastic production of quark and meson states, meson disintegration, and dynamical string formation and breaking. We obtain elastic and inelastic scattering cross sections, together with time-resolved momentum and position distributions of the outgoing particles. Furthermore, we propose an analog circuit-QED implementation of the scattering process that is native to the platform, requires minimal ingredients and approximations, and enables practical schemes for particle wave-packet preparation and evolution. Furthermore, this study highlights the role of classical and quantum simulation in enhancing our understanding of scattering processes in quantum field theories in real time.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2026

Distinct Kinetic Signatures of Photodesorption from Metal Nanoparticles

Visible photon fluxes can influence the rate and selectivity of heterogeneously catalyzed reactions on metal nanoparticle surfaces. Models describing the influence of photon fluxes have typically introduced photon flux dependent apparent thermal kinetic parameters (reaction orders, activation energies, binding energies, etc.). This has relied on empirical fitting of reaction rate data, making mechanistic interpretations of how photon fluxes influence elementary step rates challenging and inconsistent with fundamental descriptions of photochemistry on metal surfaces developed from surface science studies. Using the CO adsorption–desorption quasi-equilibrium reaction on Pt/Al 2 O 3 catalysts as a model system, we measured steady state adsorbed CO (CO*) coverages under isothermal and isobaric (1 mbar CO) conditions as a function of temperature (473–573 K) and of 440 nm photon flux ((0.1–5.2) × 10 3 # hv Pt site –1 s –1 ) using in situ IR spectroscopy. Steady state CO* coverage on Pt was photon flux dependent with increasing photon flux causing decreasing coverage, consistent with photons driving CO* desorption rates faster than thermal CO* desorption rates. However, photon flux dependent CO* coverages were essentially temperature independent, inconsistent with models that describe photon effects using perturbations to apparent thermal kinetic parameters. Instead, 120 steady state CO* coverages as a function of temperature and photon flux are quantitatively described by a kinetic model in which the overall desorption rate is a summation of independent thermal and photon induced CO* desorption rates. Site-resolved analysis reveals distinct kinetic parameters for photon driven desorption of CO* from well-coordinated, under-coordinated, and highly under-coordinated Pt sites, with temperature-dependent apparent quantum efficiencies (AQE) consistent with temperature dependence of vibrational quanta distribution of adsorbed CO. The rigorous kinetic rate laws for independent photon and thermal driven pathways allow for predictive modeling of the influence of photon fluxes on the rates of CO* desorption under catalytic conditions. Further, the analysis provides evidence that steady state continuous wave photon fluxes can drive desorption/adsorption reactions on metal surfaces out of thermal equilibrium, reconciling surface science observations of molecular photodesorption with applied catalysis. The work establishes a general kinetic framework to be considered for photon driven processes on metals, and defines catalyst, reaction, and photon flux characteristic design principles for breaking Sabatier limitations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

A Consistent BGK Model with Velocity-Dependent Collision Frequency for Gas Mixtures

Abstract We derive a multi-species BGK model with velocity-dependent collision frequency for a non-reactive, multi-component gas mixture. The model is derived by minimizing a weighted entropy under the constraint that the number of particles of each species, total momentum, and total energy are conserved. We prove that this minimization problem admits a unique solution for very general collision frequencies. Moreover, we prove that the model satisfies an H-Theorem and characterize the form of equilibrium.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

The Delplot kinetic method applied to systems with adsorbates: Hydrodeoxygenation of benzofuran on a bimetallic CoPd phosphide catalyst supported on KUSY

The Delplot method provides a means of analyzing conversion and selectivity data to derive a reaction sequence and has been applied so far to gas-phase or liquid-phase species. This study analyses the applicability of the method for catalytic reactions and considers for the first time adsorbed intermediates. The method is applied to a contact time study of the hydrodeoxygenation (HDO) of benzofuran at 0.5 MPa and 350 °C over a catalyst consisting of bimetallic CoPd phosphide supported on a potassium ion-exchanged ultra-stable Y (KUSY) zeolite. Both simple and detailed reaction networks were derived. The simple networks considered only gas-phase species and were modeled by a first-order sequence of steps, whereas the detailed network took into account adsorbed intermediates and was simulated using a rake mechanism. The networks were compared using F-statistics, which accounted for the differences in the number of fitting parameters. The detailed network gave a better fit to the experimental data because it represented a more realistic description of the transformation. A suggested sequence from the Delplot method was consistent with the simple network but not with the detailed network. The lack of applicability of the Delplot analysis to the network with adsorbates was linked to overly large equilibrium constants, a determination supported by Delplot fitting for a model sequence. Further, this study indicated the inapplicability of the Delplot method in determining reaction sequences that involve adsorbed species with large equilibrium constants for formation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Simulations of vertical displacement oscillatory modes and global Alfvén Eigenmodes in JET geometry

Vertical Displacement Oscillatory Modes (VDOM), with frequency in the Alfvén range, are natural modes of oscillation of magnetically confined laboratory plasmas with elongated cross-section. These axisymmetric modes arise from the interaction between the plasma current, which is in equilibrium with currents flowing in external coils, and perturbed currents induced on a nearby conducting wall. The restoring force exerted by these perturbed currents on the vertical motion of the plasma column leads to its oscillatory behavior. An analytic model for VDOM was proposed based on an idealized 'straight tokamak' equilibrium with uniform equilibrium current density. This article introduces the first numerical simulations of VDOM in a realistic JET tokamak configuration, using the extended-MHD code NIMROD and drawing comparisons with Global Alfvén Eigenmodes (GAE). The results show qualitative agreement with analytic predictions regarding mode frequency and radial structure, supporting the identification of VDOM as a fundamental oscillation mode in tokamak plasmas. VDOM and GAE are modeled in a representative JET discharge, where axisymmetric perturbations with toroidal mode number n = 0 driven unstable by fast ions were observed. The two modes are examined separately using a forced oscillator within the NIMROD code, which enables a comparison of their characteristics and helps identify the experimentally observed mode possibly as a GAE.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

New Empirical Formulation for the Sublimational Breakup of Graupel and Dendritic Snow

Abstract Ice fragments are generated by sublimation of ice particles in subsaturated conditions in natural clouds. Conceivably, such sublimational breakup would be expected to cause ice multiplication in natural clouds. Any fragment that survives will grow to become ice precipitation that may sublimate and fragment further. As a first step toward assessing this overlooked process, a formulation is proposed for the number of ice fragments from sublimation of ice particles for an atmospheric model. This is done by amalgamating laboratory observations from previously published studies. The concept of a “sublimated mass activity spectrum” for the breakup is applied to the dataset. The number of ice fragments is determined by the relative humidity over ice and the initial size of the parent ice particles. The new formulation applies to dendritic crystals and heavily rimed particles only. Finally, a thought experiment is performed for an idealized scenario of subsaturation with in-cloud descent. Scaling analysis yields an estimate of an ice enhancement ratio of about 5 (10) within a weak deep convective downdraft of about 2 m s −1 , for an initial monodisperse population of dendritic snow (graupel) particles of 3 L −1 and 2 mm. During descent, there is a dynamic equilibrium between continual emission of fragments and their depletion by sublimation. A simplified bin microphysics parcel model exhibits this dynamical quasi equilibrium, consistent with the thought experiment. The fragments have average lifetimes of around 90 and 70 s for dendrites and graupel, respectively. Sublimational breakup is predicted to cause significant secondary ice production.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Replica symmetry breaking in spin glasses in the replica-free Keldysh formalism

We show that the algebra of Parisi ultrametric matrices is recovered by the real-time, replica-free, Dyson-Keldysh equations of infinite-range quantum spin glasses in the late time glassy limit. This connects to earlier results on classical and quantum systems showing how ultrametricity emerges from the persistent slow aging dynamics of the glass phase. The stationary spin glass state thereby spontaneously breaks thermal symmetry, or the Kubo-Martin-Schwinger relation of a state in global thermal equilibrium. We describe the Keldysh path integral of the infinite-range Ising model in transverse and longitudinal fields, and in the context of the Landau expansion of the action functional, show how the long-time limit connects to the full replica symmetry breaking obtained in the equilibrium formalism. We also illustrate our formalism by applying it to the spherical quantum p p -spin model, which only exhibits one-step replica symmetry breaking.

Lang, Johannes (ORCID:0000000283533392)↗
DOE OSTI2022

Comparing and contrasting In-Vial and full-scale systems for sparging volatile analytes

Here, in-vial sparging was demonstrated as an effective, practical alternative to a full-scale sparging system for supporting the analysis of volatile constituents. Using elemental mercury (Hg 0 ) and toluene as representative purgeable analytes, the mass removal for various sparge configurations was measured and a reduced order model was developed and validated. In the primary experiments, Hg 0 in the sparge gas was trapped on activated carbon or gold, thermally desorbed, and quantified using atomic absorption or atomic fluorescence spectroscopy. Toluene experiments using the same in-vial sparge apparatus and sparge parameters were performed to demonstrate the applicability of the reduced order model to a broad range of compounds. Toluene removal was tracked by measuring the remaining toluene in sparged aliquots using Ultraviolet-visible (UV–Vis) spectroscopy. For the sparging, flow rates varied from 25 to 75 mL/min for periods from 0 to 30 min. Sparge performance, mass removal as a function of time, and sparge gas volume were measured for both in-vial and full-scale systems. A model based on dimensionless Henry's Law coefficient, normalized sparge gas volume, and fractional extent of equilibrium matched the experimental data for both compounds and provides a practical tool for future applications. For the conditions tested in this study, the calibrated model indicated that the sparge gas in the in-vial system reached approximately 33% of its equilibrium value before exiting the water surface, while a full-scale system reached approximately 100%. The tests validated the quality, reproducibility, and predictability of sparging performance for both full scale and in-vial sparge systems. Related factors such as waste generation, worker risk, and labor were also assessed. Full scale sparge systems provide the advantage of lower detection levels due to larger sample volume, while the in-vial sparge systems provide advantages for most other factors; including automatability, reducing secondary wastes, lessening the need to clean and check the sparge apparatus, and lowering labor and costs. The data and associated reduced order model support continued development and deployment of in-vial sparge platforms as a practical option for analysis of purgeable analytes such as volatile organic compounds and volatile metals/organometallics.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

TOUGH3/EOS7MG (EOS7MG) v1.1

EOS7MG is a new TOUGH3 module for modeling water, brine, and multiple gas mixtures in the subsurface multiphase flow system. The module allows mixture of as more as three gas components selecting from a list of predefined gases (CH4, C2H6, C3H8, H2S, CO2, N2, NO2, H2, and AIR). EOS7MG was developed based on EOS7C and inherited the general modeling capabilities of the TOUGH3 platform. It uses cubic equation of states for gas density, viscosity, and enthalpy calculation. The module provides the selection options for Peng-Robinson, Redlich-Kwong, or Soave-Redlich-Kwong equations of state. The subsurface flow and transport of gas and aqueous phase mixtures over a wide range of pressures and temperatures is modeled with a multiphase Darcy's Law and an accurate chemical equilibrium approach as originally implemented in EOS7C is used for calculation of the solubilities of different mass components. EOS7MG was designed to allow flexible number of components based on the number of gases included in the model. The potential application of this module may include subsurface gas storage, energy storage, CCS, enhanced gas recovery, and more.

Zhang, Keni↗
DOE OSTI2023

Water dissociation at the water–rutile TiO 2 (110) interface from ab initio-based deep neural network simulations

The interaction of water with TiO 2 surfaces is of crucial importance in various scientific fields and applications, from photocatalysis for hydrogen production and the photooxidation of organic pollutants to self-cleaning surfaces and bio-medical devices. In particular, the equilibrium fraction of water dissociation at the TiO 2 –water interface has a critical role in the surface chemistry of TiO 2 , but is difficult to determine both experimentally and computationally. Among TiO 2 surfaces, rutile TiO 2 (110) is of special interest as the most abundant surface of TiO 2 ’s stable rutile phase. While surface-science studies have provided detailed information on the interaction of rutile TiO 2 (110) with gas-phase water, much less is known about the TiO 2 (110)–water interface, which is more relevant to many applications. In this work, we characterize the structure of the aqueous TiO 2 (110) interface using nanosecond timescale molecular dynamics simulations with ab initio-based deep neural network potentials that accurately describe water/TiO 2 (110) interactions over a wide range of water coverages. Simulations on TiO 2 (110) slab models of increasing thickness provide insight into the dynamic equilibrium between molecular and dissociated adsorbed water at the interface and allow us to obtain a reliable estimate of the equilibrium fraction of water dissociation. We find a dissociation fraction of 22 ± 6% with an associated average hydroxyl lifetime of 7.6 ± 1.8 ns. These quantities are both much larger than corresponding estimates for the aqueous anatase TiO 2 (101) interface, consistent with the higher water photooxidation activity that is observed for rutile relative to anatase.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Entanglement Hamiltonian of Many-Body Dynamics in Strongly Correlated Systems

A powerful perspective in understanding nonequilibrium quantum dynamics is through the time evolution of its entanglement content. Yet apart from a few guiding principles for the entanglement entropy,to date, much less is known about the refined characteristics of entanglement propagation. In this paper we unveil signatures of the entanglement evolving and information propagating out of equilibrium, from the view of the entanglement Hamiltonian. We investigate quantum quench dynamics of prototypical Bose-Hubbard model using state-of-the-art numerical technique combined with conformal field theory. Before reaching equilibrium, it is found that a current operator emerges in the entanglement Hamiltonian, implying that entanglement spreading is carried by particle flow. In the long-time limit the subsystem enters a steady phase, evidenced by the dynamic convergence of the entanglement Hamiltonian to the expectation of a thermal ensemble. Importantly, the entanglement temperature in steady state is spatially independent,which provides an intuitive trait of equilibrium. These findings not only provide crucial information on how equilibrium statistical mechanics emerges in many-body dynamics, but also add a tool to exploring quantum dynamics from the perspective of the entanglement Hamiltonian.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Active Thermochemical Tables: the thermophysical and thermochemical properties of methyl, CH 3 , and methylene, CH 2 , corrected for nonrigid rotor and anharmonic oscillator effects

The thermophysical properties (isobaric heat capacity, entropy, enthalpy increment) of two prominent radicals, methyl, CH 3 , and methylene, CH 2 , were computed using the Nonrigid Rotor Anharmonic Oscillator (NRRAO) approach and compared to their RRHO counterparts, demonstrating significant differences between the results from the two approaches. Methylene presents a typical case in which the NRRAO thermophysical properties have significantly higher values than their RRHO counterparts at higher temperatures. In the case of methyl, the positive anharmonicity of the umbrella motion causes an opposite effect, and the NRRAO corrected thermophysical properties have lower values than their RRHO counterparts. The NRRAO corrected thermophysical properties, in turn, affect the resulting thermochemical properties. Here, two reactions important in combustion modelling were tested: the recombination of methyl radical with hydrogen atoms to form methane, and the recombination of two methyl radicals to form ethane. The related NRRAO equilibrium constants differ significantly from their RRHO analogs, and the consequences for chemical modelling are discussed. Also reported are the most current ATcT enthalpies of formation for CH n (n = 4-0) species and for C 2 H 6 , together with the tightly related sequential bond dissociation enthalpies along with the CHn series.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2020

Experimental Investigation of Uranium Volatility during Vapor Condensation

The predictive models that describe the fate and transport of radioactive materials in the atmosphere following a nuclear incident (explosion or reactor accident) assume that uranium-bearing particulates would attain chemical equilibrium during vapor condensation. Here, we show that kinetically driven processes in a system of rapidly decreasing temperature can result in substantial deviations from chemical equilibrium. This can cause uranium to condense out in oxidation states (e.g., UO 3 vs UO 2 ) that have different vapor pressures, significantly affecting uranium transport. To demonstrate this, we synthesized uranium oxide nanoparticles using a flow reactor under controlled conditions of temperature, pressure, and oxygen concentration. The atomized chemical reactants passing through an inductively coupled plasma cool from ~5000 to 1000 K within milliseconds and form nanoparticles inside a flow reactor. The ex situ analysis of particulates by transmission electron microscopy revealed 2–10 nm crystallites of fcc-UO 2 or α-UO 3 depending on the amount of oxygen in the system. α-UO 3 is the least thermodynamically preferred polymorph of UO 3 . The absence of stable uranium oxides with intermediate stoichiometries (e.g., U 3 O 8 ) and sensitivity of the uranium oxidation states to local redox conditions highlight the importance of in situ measurements at high temperatures. Therefore, we developed a laser-based diagnostic to detect uranium oxide particles as they are formed inside the flow reactor. Our in situ measurements allowed us to quantify the changes in the number densities of the uranium oxide nanoparticles (e.g., UO 3 ) as a function of oxygen gas concentration. Our results indicate that uranium can prefer to be in metastable crystal forms (i.e., α-UO 3 ) that have higher vapor pressures than the refractory form (i.e., UO 2 ) depending on the oxygen abundance in the surrounding environment. This demonstrates that the equilibrium processes may not dominate during rapid condensation processes, and thus kinetic models are required to fully describe uranium transport subsequent to nuclear incidents.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Assessment of SCALE and MELCOR for a generic pebble bed fluoride high-temperature reactor

This paper presents the development and application of SCALE and MELCOR models for a fluoride salt-cooled high-temperature reactor (FHR) based on publicly available specifications. SCALE version 6.3beta15 was used to generate power distributions and decay heat curves, and MELCOR version 2.2.18019 was used to calculate the thermal hydraulic response of an assumed FHR primary system. Here, an approach was developed for determining the equilibrium state of the core using depletion of a core slice model and blending of fuel compositions at different burnups to provide three-dimensional fuel composition in the core. Results compared between a core with entirely fresh fuel and one with an equilibrium fuel composition revealed that the equilibrium core led to lower steady-state temperatures but slower cooldown during a loss of flow accident (LOFA). A sensitivity study was then conducted to explore the transient response of our FHR system to variations in thermal hydraulic parameters of the system using the equilibrium core. The inlet temperature, graphite thermal conductivity, and SCRAM time all provided significant control over peak fuel and coolant temperatures. Uncertainties in radionuclide decay data in SCALE were used to perform a decay heat sensitivity study, and we found that uncertainties in decay heat led to negligible impact on peak temperatures during the course of the transient. In all cases evaluated, the observed peak fuel temperatures remained approximately 700 K below anticipated failure limits for the LOFA.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2021

Modeling the electrochemical behavior and interfacial junction profiles of bipolar membranes at solar flux relevant operating current densities

A 1-dimensional, multi-physics model that accounts for the migration and diffusion of solution species, electrostatics, and chemical reactions, in particular water dissociation (WD), at bipolar membrane (BPM) interfaces was developed to study the electrochemical behavior of bipolar membranes (BPMs) at solar flux relevant operating current densities (tens of mA cm -2 ). Significant partial current densities for WD were observed at BPM voltages much less than the equilibrium voltage, e.g., 59 mV × ΔpH from both experiments and modeling. The co-ion leakage across the BPM at pH differentials accounted for the early presence of the partial current density for WD. Two distinctive electric field dependent WD pathways, the un-catalyzed pathway and the catalyzed pathway, were quantitatively and parametrically studied to improve the turn-on potential of the BPM. The catalyzed pathway accounted for the majority of the partial current density for WD at low voltages, while the un-catalyzed pathway dominated the WD at relatively high voltages. Significant WD was observed only within the interfacial CL (<5 nm), in which a large electric field was present. To improve the electrochemical behavior and the turn-on potential of BPMs, the impacts of the pKa of the immobilized WD catalysts, the electric-field dependent rate constant, the thickness of the catalyst layer and fixed charge density in BPMs on the partial current densities for WD were studied systematically. In addition, the electrochemical behavior and concentration profiles of BPMs in a buffered electrolyte were studied and contrasted with those in an un-buffered electrolyte from both modeling and experiments.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Machine learning nonequilibrium electron forces for spin dynamics of itinerant magnets

Abstract We present a generalized potential theory for conservative as well as nonconservative forces for the Landau-Lifshitz magnetization dynamics. Importantly, this formulation makes possible an elegant generalization of the Behler-Parrinello machine learning (ML) approach, which is a cornerstone of ML-based quantum molecular dynamics methods, to the modeling of force fields in adiabatic spin dynamics of out-of-equilibrium itinerant magnetic systems. We demonstrate our approach by developing a deep-learning neural network that successfully learns the electron-mediated exchange fields in a driven s-d model computed from the nonequilibrium Green’s function method. We show that dynamical simulations with forces predicted from the neural network accurately reproduce the voltage-driven domain-wall propagation. Our work also lays the foundation for ML modeling of spin transfer torques and opens a avenue for ML-based multi-scale modeling of nonequilibrium dynamical phenomena in itinerant magnets and spintronics.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Modified Energy Span Analysis of Catalytic Parallel Pathways and Selectivity

Mechanistic modeling provides vital insights into catalytic reactions. To analyze complex reaction networks with parallel pathways, we leverage the graph theory approach of the Energy Span Model (ESM) to develop a modified energy span analysis (MESA). A new method of cycle plots is proposed to perform reaction pathways analysis visually. We demonstrate this method on two published models: one describing carbon monoxide oxidation and the other simulating ethylene conversion to propanal via hydroformylation or ethane via hydrogenation. Fundamental insights explain kinetic observables, such as a reactant’s negative reaction order. General principles are revealed, such as rate-determining surface species being outside the primary reaction flux cycle and pathway selectivity being a purely kinetic property when reaction conditions are not near equilibrium. Lastly, we demonstrate MESA’s consistency with published microkinetic modeling results, highlighting this technique’s extension of the ESM to heterogeneous catalysts using collision theory to describe adsorption steps and concentration effects.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗