The Pseudo-Steady-State Approximation

After initial collision, the adduct will decompose rapidly if an energetically accessible channel exists. For such reactions the activation energy E is often negative, meaning that as temperature increases the rate coefficient decreases. The explanation is that, with virtually no barrier, all collisions have sufficient energy to react. However, less energetic molecules at lower temperatures actually allow the initial adduct to have more time to rearrange itself so as to decompose to the products. 

Many chemical reactions involve very reactive intermediate species such as free radicals, which, as a result of their high reactivity, are consumed virtually as rapidly as they are formed and consequently exist at very low concentrations. The pseudo-steady-state approximation1 (PSSA) is a fundamental way of dealing with such reactive intermediates when deriving the overall rate of a chemical reaction mechanism. 

It is perhaps easiest to explain the PSSA by way of an example. Consider the unimolecular reaction A — B + C, whose elementary steps consist of the activation of A by collision with a background molecule M (a reaction chaperone) to produce an energetic A molecule denoted by A, followed by decomposition of A to give B + C  

Note that A may return to A by collision and transfer of its excess energy to an M. The rate equations for this mechanism are 

The reactive intermediate in this mechanism is A. The PSSA states that the rate of generation of A is equal to its rate of disappearance physically, what this means is that A is so reactive that, as soon as an A molecule is formed, it reacts by one of its two paths. Thus the PSSA gives 

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Therefore, we need to find approximate methods for simultaneous reaction systems that will permit finding analytical solutions for reactants and products in simple and usable form. We use two approximations that were developed by chemists to simplify simultaneous reaction systems (1) the equilibrium step approximation and (2) the pseudo-steady-state approximation... 

Let us next apply the pseudo-steady-state approximation to see if it agrees with the experimentally observed rate. We write the mass balance on CN2O2 ... 

Thus we see that the pseudo-steady-state approximation gives orders of the reaction as the thermodynamic equilibrium approximation, the only difference being the definition of the rate constant... 

Thus both approximations predict rate expressions that agree with experimental data as long as the second term in the denominator of the pseudo-steady-state approximation is small. 

To show that the above rate expression can yield this rate expression, let us apply the pseudo-steady-state approximation on [CH3] and [CH3CO] and see what it predicts. Mass balances on all species yield. 

We will use the pseudo-steady-state approximation on [CH3] and on [CH3CO]. These... 

Applying the pseudo-steady-state approximation to A we obtain dCA ... 

Textbooks state that the pseudo-steady-state approximation will be valid if the concentration of a species is small. However, one then proceeds by setting its time derivative equal to zero (]/t/f = 0) in the batch reactor equation, not by setting the concentration (CH3CO ) equal to zero. This logic is not obvious from the batch reactor equations because setting the derivative of a concentration equal to zero is not the same as setting its concentration equal to zero. 

However, when we examine the CSTR mass balance, we see that the pseudo-Steady-state approximation is indeed that the concentration be small or that [CH3CO ]/t = 0. Thus by examining the CSTR version of the mass-balance equations, we are led to the pseudo-steady-state approximation naturally. This is expressly because the CSTR mass-balance equations are developed assuming steady state so that the pseudo-steady-state approximation in fact implies simply that an intermediate species is in steady state and its concentration is small. 

As for the quasi (pseudo)-steady-state case, the basic assumption in deriving kinetic equations is the well-known Bodenshtein hypothesis according to which the rates of formation and consumption of intermediates are equal. In fact. Chapman was first who proposed this hypothesis (see in more detail in the book by Yablonskii et al., 1991). The approach based on this idea, the Quasi-Steady-State Approximation (QSSA), is a common method for eliminating intermediates from the kinetic models of complex catalytic reactions and corresponding transformation of these models. As well known, in the literature on chemical problems, another name of this approach, the Pseudo-Steady-State Approximation (PSSA) is used. However, the term "Quasi-Steady-State Approximation" is more popular. According to the Internet, the number of references on the QSSA is more than 70,000 in comparison with about 22,000, number of references on PSSA. 

Consider a straight tubular runner of length L. A melt following the power-law model is injected at constant pressure into the runner. The melt front progresses along the runner until it reaches the gate located at its end. Calculate the melt front position, Z(f), and the instantaneous flow rate, Q t), as a function of time. Assume an incompressible fluid and an isothermal and fully developed flow, and make use of the pseudo-steady-state approximation. For a polymer melt with K = 2.18 x 10 N s"/m and n = 0.39, calculate Z(t) and Q(t)... 

The enolate ions are unstable intermediates, hence the pseudo steady state approximation can be applied to these intermediates, resulting in a kinetic model in which only stable components figure. It also can be proven (ref.5) that such a model will be mathematically equivalent to the one as follows from the network presented in figure 1. 

When the hydrate growth rate (dm/dt) is measured by the rate of gas consumption (drii/dt) the pseudo-steady-state approximation is made. That is, at any instant the rate of gas consumption by the hydrate is assumed equal to the rate of gas consumption from the gas phase. Frequently, experimenters monitor the amount of gas needed to keep the pressure constant in the hydrate vessel so that the driving force remains constant. In such cases, the rate of gas consumption from a separate supply reservoir is measured. 

To illustrate this approximation, let us consider a pressure flow in which the driving-force pressure drop varies with time. We set dp/dt and d /dt in the equations of continuity and motion, respectively, equal to zero and proceed to solve the problem as if it were a steady-state one, that is, we assume AP to be constant and not a function of time. The solution is of the form v = v(x, A P t), geometry, etc.). Because AP was taken to be a constant, v is also a constant with time. The pseudo-steady-state approximation pretends that the foregoing solution holds for any level of AP and that the functional dependence of v on time is v(x,-, t) = v(x,-, AP(t), geometry, etc.). The pseudo-steady state approximation is not valid if the values of A(pv)/At (At being the characteristic time of fluctuation of AP) obtained using this approximation contribute to an appreciable fraction of the mean value of the applied AP. 

There are three terms here because there are two ways to lose A and one way to form it. The reactive intermediate in this system of reactions is A. The pseudo-steady-state approximation states that the rate of formation of A is equal to its rate of loss in other words, [A ] does not change over time. Thus,... 

This analysis does not account for the heat required to heat the liquid filled core to a new temperature which is nearly equal to the liquid surface temperature. This amount of heat is small compared to the heat of evaporation. Again the pseudo-steady state approximation has been used for similar reasons. A summary of the derived equations for the drying time when transport in the pores is the rate determining step are given in Table 14.2. 

Although the pseudo steady state approximation provides a useful tool for estimating some aspects of gasifier dynamics, it does not provide the means to examine the full range of dynamic behavior that one would expect to find for a gasifier. Therefore, a different approach has been taken here in that a nonlinear... 

One further note, the University of Delaware gasifier model used in the pseudo steady state approximation assumes that the gas and solids temperatures are the same within the reactor. That assumption removes an important dynamic feedback effect between the countercurrent flowing gas and solids streams. This is particularly important when the burning zone moves up and down within the reactor in an oscillatory manner in response to a step change in operating conditions. 

Let us compute the rate of formation of OH radicals from reactions 1-4. 0(1D) is sufficiently reactive that the pseudo-steady-state approximation can be invoked for its concentration. Thus... 

Since the oxygen atom is so reactive that it disappears by reaction 2 virtually as fast as it is formed by reaction 1, one can invoke the pseudo-steady-state approximation and thereby assume that the rate of formation is equal to the rate of disappearance ... 

One approach commonly used in the integration of such chemical kinetics problems is the pseudo-steady-state approximation (PSSA) (see Chapter 3). For example, instead of solving a differential equation for short-lived species like O, OH, and N03, one calculates and solves the corresponding PSSA algebraic equations. For example, McRae et al. (1982a) estimated that nine species (O, RO, OH, R02, N03, RCO, H02, HN04, and N2Os) with characteristic lifetimes less than 0.1 min in the environment of interest could be... 

However, if we were to evaluate the right-hand side numerically we would find that it is very close to zero. Physically, this means that the oxygen atom is so reactive that it disappears by reaction 5.2 virtually as fast as it is formed by reaction 5.1. In dealing with highly reactive species such as the oxygen atom, it is customary, as noted in Chapter 3, to invoke the pseudo-steady-state approximation (PSSA) and thereby assume that the rate of formation is exactly equal to the rate of disappearance, for example. 

A transformation of the dependent variables Cjt, and Cs allowed DelBorghi, Dunn, and Bischoff [9] and E>udukovic [25] to reduce the coupled set of partial differential equations for reactions first-order in the fluid concentration and with constant porosity and diffusivity, into a single partial differential equation. With the pseudo-steady-state approximation, this latter equation is further reduced to an ordinary differential equation of the form considered in Chapter 3 on diffusion and reaction (sk Problem 4.2). An extensive collection of solutions of such equations has been presented by Aris [7]. 

The model equation is again Eq. 4.2-1 in which the time derivative is set zero, as implied by the pseudo-steady-state approximation ... 

In order to express the concentrations of available active sites C and surface intermediates Ca> and Cq. as a function of the concentrations of the observable species Ca and Cq and the total concentration of active sites, the pseudo steady-state approximation is applied to the net rates of formation of the surface intermediates, that is ... 

With kx = k2Ct- From equation 68, it is clear that the assumption of the occurrence of a RDS in addition to the pseudo-steady-state approximation for the surface intermediates results in a significant simplification of the expressions for the surface intermediate concentrations and, hence, for the rate of the global reaction. Similar equations can be derived for the reactant adsorption or product desorption as RDS. 

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