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Equilibria states

In the preceding derivation, the repulsion between overlapping double layers has been described by an increase in the osmotic pressure between the two planes. A closely related but more general concept of the disjoining pressure was introduced by Deijaguin [30]. This is defined as the difference between the thermodynamic equilibrium state pressure applied to surfaces separated by a film and the pressure in the bulk phase with which the film is equilibrated (see section VI-5). [Pg.181]

In considering isotherm models for chemisorption, it is important to remember the types of systems that are involved. As pointed out, conditions are generally such that physical adsorption is not important, nor is multilayer adsorption, in determining the equilibrium state, although the former especially can play a role in the kinetics of chemisorption. [Pg.698]

The populations, /Q, appear on the diagonal as expected, but note that there are no off-diagonal elements—no coherences this is reasonable since we expect the equilibrium state to be time-independent, and we have associated the coherences with time. [Pg.233]

It follows that there are two kinds of processes required for an arbitrary initial state to relax to an equilibrium state the diagonal elements must redistribute to a Boltzmaim distribution and the off-diagonal elements must decay to zero. The first of these processes is called population decay in two-level systems this time scale is called Ty The second of these processes is called dephasmg, or coherence decay in two-level systems there is a single time scale for this process called T. There is a well-known relationship in two level systems, valid for weak system-bath coupling, that... [Pg.233]

In analogy to the constant-pressure process, constant temperature is defined as meaning that the temperature T of the surroundings remains constant and equal to that of the system in its initial and final (equilibrium) states. First to be considered are constant-temperature constant-volume processes (again Aw = 0). For a reversible process... [Pg.346]

A statistical ensemble can be viewed as a description of how an experiment is repeated. In order to describe a macroscopic system in equilibrium, its thennodynamic state needs to be specified first. From this, one can infer the macroscopic constraints on the system, i.e. which macroscopic (thennodynamic) quantities are held fixed. One can also deduce, from this, what are the corresponding microscopic variables which will be constants of motion. A macroscopic system held in a specific thennodynamic equilibrium state is typically consistent with a very large number (classically infinite) of microstates. Each of the repeated experimental measurements on such a system, under ideal... [Pg.384]

If //is 00 (very large) or T is zero, tire system is in the lowest possible and a non-degenerate energy state and U = -N xH. If eitiier // or (3 is zero, then U= 0, corresponding to an equal number of spins up and down. There is a synnnetry between the positive and negative values of Pp//, but negative p values do not correspond to thennodynamic equilibrium states. The heat capacity is... [Pg.403]

The equilibrium state for a gas of monoatomic particles is described by a spatially unifonn, time independent distribution fiinction whose velocity dependence has the fomi of the Maxwell-Boltzmaim distribution, obtained from equilibrium statistical mechanics. That is,/(r,v,t) has the fomi/" (v) given by... [Pg.666]

We now show that when H is constant in time, the gas is in equilibrium. The existence of an equilibrium state requires the rates of the restituting and direct collisions to be equal that is, that there is a detailed balance of gain and loss processes taking place in the gas. [Pg.685]

Wlien H has reached its minimum value this is the well known Maxwell-Boltzmaim distribution for a gas in themial equilibrium with a unifomi motion u. So, argues Boltzmaim, solutions of his equation for an isolated system approach an equilibrium state, just as real gases seem to do. Up to a negative factor (-/fg, in fact), differences in H are the same as differences in the themiodynamic entropy between initial and final equilibrium states. Boltzmaim thought that his //-tiieorem gave a foundation of the increase in entropy as a result of the collision integral, whose derivation was based on the Stosszahlansatz. [Pg.685]

The current frontiers for the subject of non-equilibrium thennodynamics are rich and active. Two areas dommate interest non-linear effects and molecular bioenergetics. The linearization step used in the near equilibrium regime is inappropriate far from equilibrium. Progress with a microscopic kinetic theory [38] for non-linear fluctuation phenomena has been made. Carefiil experiments [39] confinn this theory. Non-equilibrium long range correlations play an important role in some of the light scattering effects in fluids in far from equilibrium states [38, 39]. [Pg.713]

Radiation probes such as neutrons, x-rays and visible light are used to see the structure of physical systems tlirough elastic scattering experunents. Inelastic scattering experiments measure both the structural and dynamical correlations that exist in a physical system. For a system which is in thennodynamic equilibrium, the molecular dynamics create spatio-temporal correlations which are the manifestation of themial fluctuations around the equilibrium state. For a condensed phase system, dynamical correlations are intimately linked to its structure. For systems in equilibrium, linear response tiieory is an appropriate framework to use to inquire on the spatio-temporal correlations resulting from thennodynamic fluctuations. Appropriate response and correlation functions emerge naturally in this framework, and the role of theory is to understand these correlation fiinctions from first principles. This is the subject of section A3.3.2. [Pg.716]

Here L is the Onsager coefficient and the minus sign (-) indicates that the concentration flow occurs from regions of high p to low p in order that the system irreversibly flows towards the equilibrium state of a... [Pg.720]

There are many examples in nature where a system is not in equilibrium and is evolving in time towards a thennodynamic equilibrium state. (There are also instances where non-equilibrium and time variation appear to be a persistent feature. These include chaos, oscillations and strange attractors. Such phenomena are not considered here.)... [Pg.731]

A homogeneous metastable phase is always stable with respect to the fonnation of infinitesimal droplets, provided the surface tension a is positive. Between this extreme and the other thennodynamic equilibrium state, which is inhomogeneous and consists of two coexisting phases, a critical size droplet state exists, which is in unstable equilibrium. In the classical theory, one makes the capillarity approxunation the critical droplet is assumed homogeneous up to the boundary separating it from the metastable background and is assumed to be the same as the new phase in the bulk. Then the work of fonnation W R) of such a droplet of arbitrary radius R is the sum of the... [Pg.754]

The exponential fiinction of the matrix can be evaluated tln-ough the power series expansion of exp(). c is the coliinm vector whose elements are the concentrations c.. The matrix elements of the rate coefficient matrix K are the first-order rate constants W.. The system is called closed if all reactions and back reactions are included. Then K is of rank N- 1 with positive eigenvalues, of which exactly one is zero. It corresponds to the equilibrium state, witii concentrations r detennined by the principle of microscopic reversibility ... [Pg.790]

The scan rate, u = EIAt, plays a very important role in sweep voltannnetry as it defines the time scale of the experiment and is typically in the range 5 mV s to 100 V s for nonnal macroelectrodes, although sweep rates of 10 V s are possible with microelectrodes (see later). The short time scales in which the experiments are carried out are the cause for the prevalence of non-steady-state diflfiision and the peak-shaped response. Wlien the scan rate is slow enough to maintain steady-state diflfiision, the concentration profiles with time are linear within the Nemst diflfiision layer which is fixed by natural convection, and the current-potential response reaches a plateau steady-state current. On reducing the time scale, the diflfiision layer caimot relax to its equilibrium state, the diffusion layer is thiimer and hence the currents in the non-steady-state will be higher. [Pg.1927]

Relationships from thennodynamics provide other views of pressure as a macroscopic state variable. Pressure, temperature, volume and/or composition often are the controllable independent variables used to constrain equilibrium states of chemical or physical systems. For fluids that do not support shears, the pressure, P, at any point in the system is the same in all directions and, when gravity or other accelerations can be neglected, is constant tliroughout the system. That is, the equilibrium state of the system is subject to a hydrostatic pressure. The fiindamental differential equations of thennodynamics ... [Pg.1956]

A simple, non-selective pulse starts the experiment. This rotates the equilibrium z magnetization onto the v axis. Note that neither the equilibrium state nor the effect of the pulse depend on the dynamics or the details of the spin Hamiltonian (chemical shifts and coupling constants). The equilibrium density matrix is proportional to F. After the pulse the density matrix is therefore given by and it will evolve as in equation (B2.4.27). If (B2.4.28) is substituted into (B2.4.30), the NMR signal as a fimction of time t, is given by (B2.4.32). In this equation there is a distinction between the sum of the operators weighted by the equilibrium populations, F, from the unweighted sum, 7. The detector sees each spin (but not each coherence ) equally well. [Pg.2100]

Sohaublin S, FIdhener A and Ernst R R 1974 Fourier speotrosoopy of non-equilibrium states. Applioation to CIDNP, Overhauser experiments and relaxation time measurements J. Magn. Reson. 13 196-216... [Pg.2113]

Bain A D and Martin J S 1978 FT NMR of non-equilibrium states of oomplex spin systems I. A Liouville spaoe desoription J. Magn. Reson. 29 125-35... [Pg.2113]

In practice, colloidal systems do not always reach tlie predicted equilibrium state, which is observed here for tlie case of narrow attractions. On increasing tlie polymer concentration, a fluid-crystal phase separation may be induced, but at higher concentration crystallization is arrested and amorjihous gels have been found to fonn instead [101, 102]. Close to the phase boundary, transient gels were observed, in which phase separation proceeded after a lag time. [Pg.2688]

The two main difficulties facing tlie experimenter are (i) how to detect binding, and (ii) how to ensure tliat the system under investigation is truly in an equilibrium state. [Pg.2826]

Transient, or time-resolved, techniques measure tire response of a substance after a rapid perturbation. A swift kick can be provided by any means tliat suddenly moves tire system away from equilibrium—a change in reactant concentration, for instance, or tire photodissociation of a chemical bond. Kinetic properties such as rate constants and amplitudes of chemical reactions or transfonnations of physical state taking place in a material are tlien detennined by measuring tire time course of relaxation to some, possibly new, equilibrium state. Detennining how tire kinetic rate constants vary witli temperature can further yield infonnation about tire tliennodynamic properties (activation entlialpies and entropies) of transition states, tire exceedingly ephemeral species tliat he between reactants, intennediates and products in a chemical reaction. [Pg.2946]

Complex chemical mechanisms are written as sequences of elementary steps satisfying detailed balance where tire forward and reverse reaction rates are equal at equilibrium. The laws of mass action kinetics are applied to each reaction step to write tire overall rate law for tire reaction. The fonn of chemical kinetic rate laws constmcted in tliis manner ensures tliat tire system will relax to a unique equilibrium state which can be characterized using tire laws of tliennodynamics. [Pg.3054]

The forces in a protein molecule are modeled by the gradient of the potential energy V(s, x) in dependence on a vector s encoding the amino acid sequence of the molecule and a vector x containing the Cartesian coordinates of all essential atoms of a molecule. In an equilibrium state x, the forces (s, x) vanish, so x is stationary and for stability reasons we must have a local minimizer. The most stable equilibrium state of a molecule is usually the... [Pg.212]

Naturally, the study of non-cquilibriti m properties in vo Ives different criteria alth ough the equilibrium stale and cvoltilion towards the equilibrium state may be important. [Pg.316]

In Ihc canonical, microcanonical and isothermal-isobaric ensembles the number of particles is constant but in a grand canonical simulation the composition can change (i.e. the number of particles can increase or decrease). The equilibrium states of each of these ensembles are cha racterised as follows ... [Pg.321]


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A Comparison of Steady State Procedures and Equilibrium Conditions in the Reversible Reaction

Absorption Columns or High Dimensional Lumped, Steady State and Equilibrium Stages Systems

Amorphous equilibrium supercooled liquid state

Approach to equilibrium and steady state

Associated equilibrium state

Bifurcation of an equilibrium state with one zero exponent

Brownian motion in the equilibrium state

CHEMICAL EQUILIBRIUM OF SIMPLE SYSTEMS IN THE IDEAL GAS STATE

Chemical Equilibrium and Steady State

Chemical Equilibrium—a Dynamic Steady State

Chemical equilibrium A state of dynamic

Chemical equilibrium states

Chemical reaction equilibrium state

Closed system equilibrium state

Completely unstable equilibrium state

Component-equilibrium dead state

Convergence to the Equilibrium State

Corresponding states principle phase equilibrium calculations

Critical equilibrium state

Differentiability equilibrium thermodynamic state

Dynamic equilibrium solid-state diffusion

Dynamic equilibrium state

Effects of pressure changes on the equilibrium state in gaseous reactions

Electrochemical Equilibrium State

Electron models equilibrium state equations

Entropy change equilibrium state and

Equilibria Involving Ions of the Intermediate Oxidation State

Equilibria in the Excited State

Equilibrium A state of dynamic balance

Equilibrium Rate Constants. Transition-State Method

Equilibrium States, Pathways, and Measurements

Equilibrium Statistical Distribution of Diatomic Molecules over Vibrational-Rotational States

Equilibrium Theory of Reaction Rates The Transition-state Method

Equilibrium complexes, ground-state

Equilibrium condition steady state comparison

Equilibrium constant standard state

Equilibrium constant steady state kinetics

Equilibrium constant transition state

Equilibrium constants from a modified Redlich-Kwong equation of state

Equilibrium constants states

Equilibrium density operator, coherent states

Equilibrium glassy state

Equilibrium intermediate oxidation state

Equilibrium point, oxide-solution state

Equilibrium solid-state diffusion

Equilibrium state and

Equilibrium state approximation

Equilibrium state defined

Equilibrium state for

Equilibrium state metastable

Equilibrium state of reactants and products

Equilibrium state parameters, electron models

Equilibrium state specification

Equilibrium state stable

Equilibrium state unstable

Equilibrium state, amorphous solids, glass

Equilibrium state, amorphous solids, glass transition

Equilibrium state, predictions

Equilibrium states and reversibility

Equilibrium states and thermodynamic potentials

Equilibrium states between structure elements in solids

Equilibrium steady-state

Equilibrium system, intensive state

Equilibrium versus Steady State

Excited state, thermal equilibrium

Excited states equilibrium geometries

Existence of non-equilibrium indifferent states

Exponentially stable equilibrium state

Far-from-equilibrium state

Fluid systems, phase equilibrium state

Focus equilibrium state

Galvanic cells equilibrium state

Ground State Protonation Equilibria of the AvGFP Chromophore

Growth and equilibrium (stationary state)

Infinitely degenerate equilibrium state

Isolated equilibrium state

Isomeric reactions equilibrium state

Kinetics and Equilibria of Excited State Protonation Reactions

Liquid Equilibrium Using the Equations of State Method

Liquid-crystalline polymer equilibrium states

Local equilibrium state, hypothesis

Metastable Versus Equilibrium States

Molecular states equilibrium populations

Monomolecular RNA Two-state Conformational Equilibria

Multiphase system, equilibrium state

Node equilibrium state

Non-equilibrium glassy state

Non-equilibrium state

Non-equilibrium stationary state

Non-equilibrium steady state

Non-equilibrium steady states and cycle kinetics

Phase Equilibrium State

Phase diagram equilibrium states

Phase equilibria (reduced equation of states)

Phase transitions equilibrium states

Phase-space transition states relative equilibrium

Polymer network systems equilibrium swollen state

Principle of actual gas and steady-state equilibrium

Quasi-equilibrium state

Quasi-equilibrium states, localized

Quasi-stationary-state and partial equilibria approximations

Quenching, from high temperature equilibrium states

Radioactive Equilibrium and Steady State

Rapid Equilibrium and Steady-State Hypothesis

Rapid equilibrium or steady-state

Rapid spin equilibrium in solid state

Repelling equilibrium state

Rough equilibrium states

Saddle equilibrium state

Saddle-focus equilibrium state

Saddle-node equilibrium state

Semi-stable equilibrium state

Separation in a Macroscopic Sample Equilibrium State Diagram

Simple equilibrium state

Simple system equilibrium state

Simulation of the Rouse Relaxation Modulus — in an Equilibrium State

Solving for the equilibrium state

Specification of the equilibrium state

Spin-state equilibrium

Stable complex equilibrium state

Stable focus equilibrium state

Stable node equilibrium state

Standard-state Free Energies, Equilibrium Constants, and Concentrations

State of equilibrium

States global equilibrium

States local equilibrium

Steady State and Quasi-Equilibrium

Steady States Far from Equilibrium Autocatalysis

Steady state far from equilibrium

Steady state kinetics ionic equilibria

Steady state kinetics of reversible effectors and ionic equilibria

Steady state of equilibrium

Steady state or equilibrium

Steady state secular equilibrium

Steady-state (equilibrium) conditions

Structurally stable equilibrium state

Succession of equilibrium states

The Equilibrium Populations of Molecular States

The Number and Stability of Equilibrium States in Closed Systems

The equilibrium state

The steady-state and partial-equilibrium approximations

Thermal equilibrium state

Thermal equilibrium, between different spin states

Thermodynamic Description of the Equilibrium State

Thermodynamic equilibrium state

Thermodynamic equilibrium, stationary state

Transition state theory , development equilibrium

Transition state theory equilibrium hypothesis

Transition state theory relative equilibrium

Transition-state theory equilibrium

Two-State Equilibrium Modulated by an External Field

Unstable focus equilibrium state

Vapor-Liquid Equilibrium Based on Equations of State

Vapor-Liquid Equilibrium Modeling with Two-Parameter Cubic Equations of State and the van der Waals Mixing Rules

Water equilibrium state

What Is a State of Equilibrium

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