Steady State and Quasi-Equilibrium

Consider a simple anodic reaction, such as the oxidation of Cl to CI2, which can occur in the following two steps  

The concentration of the adsorbed intermediate can best be expressed in terms of the partial surface coverage, 0, which is defined as the surface concentration, f, (molcm ) divided by the maximum surface concentration, Fmaxt (expressed, of course, in the same units)  

When a reaction occurs in several consecutive steps, the rate of all steps must be equal at steady state (otherwise the system would not be at steady state). This rate is determined by the slowest step in the sequence, which we refer to as the ratedetermining step (rds). In the preceding example, if fei, the specific rate constant for the step shown by Eq. (6.6), is much smaller than lc2, the specific rate constant for the step shown by Eq. (6.7), the rate of the second step will effectively be limited by the supply of adsorbed intermediates Clads namely, by the rate of the first step. To visualize this situation, consider a potential applied to several resistors in series. The current is determined by the overall resistance, which is simply the sum of the resistances in series = Ri -I- R2 +1 3 + —Rn- If one of these resistances is much larger than all others, it will be dominant, and one has R Rj. 

To use this simple equivalent circuit, it must be realized that the rate constants of the varioixs steps can be represented by the inverse of the corresponding resistances the higher the resistance, the lower the rate constant, and vice versa. Thus, the overall effective rate constant is given by 

It is clear, then, that the overall rate constant is determined by the lowest individual rate constant. We note, in passing, that Eq. (6.10) is similar to Eq. (6.15), which correlates the overall current to the activation- and mass-transport-controlled currents. 

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KINETICS OF NONELEMENTARY REACTIONS QUASI-STEADY-STATE AND QUASI-EQUILIBRIUM APPROXIMATIONS... 

The principles of quasi-steady-state and quasi-equilibrium hypotheses are illustrated in Figures 2.1 and 2.2. If we consider the reversible reaction sequence A R S and assume that R is a rapidly reacting intermediate, its concentration remains at a low, practically constant level during the reaction Figure 2.1 shows the concentrations of A, R, and S in a batch reactor (Chapter 3) as a function of the reaction time. The net generation rate of R is practically zero (Rr = 0), except during a short initial period of time. On the other hand, if one of the reaction steps is very rapid compared with the others, for instance. 

The application of quasi-steady-state and quasi-equilibrium hypotheses will be illustrated with the example reaction... 

The derivation of initial velocity equations invariably entails certain assumptions. In fact, these assumptions are often conditions that must be fulfilled for the equations to be valid. Initial velocity is defined as the reaction rate at the early phase of enzymic catalysis during which the formation of product is linear with respect to time. This linear phase is achieved when the enzyme and substrate intermediates reach a steady state or quasi-equilibrium. Other assumptions basic to the derivation of initial rate equations are as follows ... 

Fig. 35. Computed quasi-steady state and partial equilibrium profiles for standard flame. Conditions as in Fig. 25. Solid lines, q.s.s. profiles broken lines, p.e. profiles (only marked when distinguishable from q.s.s.).

The ocean is also by far the largest reservoir for most of the elements in the atmosphere-biosphere-ocean system. Perturbations caused by our increase in population and industrialization are passing through the ocean, and because the time-scale for ocean circulation is long (about 2000 years) relative to the time-scale of modern society, a new steady-state of quasi-equilibrium will slowly be established. Until that time, local concentrations of toxic chemicals, especially in estuaries and bays with restricted circulation, will be a major concern for mankind. 

Gou, X., Chen, Z., Sun, W., Ju, Y. A dynamic adaptive chemistry scheme with error control for combustion modeling with a large detailed mechanism. Combust. Flame 160, 225-231 (2013) Goussis, D.A. Quasi steady state and partial equilibrium approximations their relation and their validity. Combust. TheOTy Model. 16, 869-926 (2012)... 

Chemical quasi-steady-state and partial equilibrium assumptions in reactive flow modeling... 

Figure 1 Overview of transport models for predicting oral drug absorption. Absorption models are classified into three categories based on their dependence on the spatial and temporal variables. These three categories are quasi-equilibrium, steady-state, and dynamic models. 

Rate determining step (cont.) electrocatalysis and, 1276 methanol oxidation, 1270 in multistep reactions, 1180 overpotential and, 1175 places where it can occur, 1260 pseudo-equilibrium, 1260 quasi equilibrium and, 1176 reaction mechanism and, 1260 steady state and, 1176 surface chemical reactions and, 1261 Real impedance, 1128, 1135 Reciprocal relation, the, 1250 Recombination reaction, 1168 Receiver states, 1494 Reddy, 1163... 

Assume that the diffusion field is in a quasi-steady state and that local equilibrium is maintained at the surface and in the volume at a long distance from the surface, where yv = 0 and yA has the value characteristic of a flat surface. 

A steady state implies that the activity of the daughter and its precursor are equal and do not change in time. However, it is easy to see that in a finite time interval such a state of so-called secular equilibrium can only be approached and never be reached due to the fact that the primordial nucleus can never have a steady state. As a result its daughter product can never reach a steady state, etc. On the other hand, when X/A,, 1 a state of quasi equilibrium may be reached where the activity ratio of mother and daughter nuclei is given by... 

III. PCA was applied to eliminate some parameters from the model fitted through approach II. In consequence, it was assumed that the sites are in quasi-steady state and that AC, EC, MCC and CE are in adsorption equilibrium. It was also assumed that kinetic constants for reactions 4 and 8, and 1 and 2 are not independent and are related through a linear relation. 

The other extreme is to evaluate the remaining terms at time n + 1)((50 the fully implicit or backward differencing approach. It leads to a set of algebraic equations from which the dependent variables at time (n -h 1)( 0 can be calculated. This approach is unconditionally stable (Richtmyer and Morton, 1967), and is the approach used here. We may of course also use other schemes in which intermediate weights are given to the forward and backward differences. These partially implicit schemes lead to improved accuracy. However, if attempts are made to use them on systems of stiff equations, the latter must be treated by asymptotic techniques. In chemical situations such techniques are equivalent to the use of the chemical quasi-steady-state or partial equilibrium assumptions at long times. They will be considered again in Section 9. 

General first-order kinetics also play an important role for the so-called local eigenvalue analysis of more complicated reaction mechanisms, which are usually described by nonlinear systems of differential equations. Linearization leads to effective general first-order kinetics whose analysis reveals infomiation on the time scales of chemical reactions, species in steady states (quasi-stationarity), or partial equilibria (quasi-equilibrium) [M, and ]. 

Transition to steady-state etching. The surface becomes sufficiently disordered to dismpt tlie quasi-equilibrium, and tlie reaction layer becomes a tree stmcture of fluorosilyl chain stmctures tenninated by SiF., groups. 

It is important to realize that the assumption of a rate-determining step limits the scope of our description. As with the steady state approximation, it is not possible to describe transients in the quasi-equilibrium model. In addition, the rate-determining step in the mechanism might shift to a different step if the reaction conditions change, e.g. if the partial pressure of a gas changes markedly. For a surface science study of the reaction A -i- B in an ultrahigh vacuum chamber with a single crystal as the catalyst, the partial pressures of A and B may be so small that the rates of adsorption become smaller than the rate of the surface reaction. 

In solving the kinetics of a catalytic reaction, what is the difference between the complete solution, the steady-state approximation, and the quasi-equilibrium approximation What is the MARI (most abundant reaction intermediate species) approximation ... 

Assuming equilibrium conditions and a linear equilibrium relationship, where Y] = m Xi, and a quasi-steady-state conditions in the gas with dYj/dt = 0 to be achieved, a component balance for the entire two phase system of Fig. 3.56, gives... 

The first procedure is to use the rate equilibrium equations for electrons and ions at the quasi-steady state for a new guess of the particle densities. In the final result, quasi-steady balances of the positive ions and negative ions integrated over... 

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