Pseudo-steady state modes

A mode is said to be in pseudo-steady state if the amount of each of the intermediates remains small compared to the amounts of the principal components 

Such a mode would not be a good approximation at the reaction start and end, because neither of the two conditions [7.1] will be met in a closed system. At the reaction start, the intermediaries that are not at the origin appear and therefore their concentration is not constant and negligible compared with that of the products. Likewise these intermediates should vanish at the reaction end. We will see that these conditions are often acceptable, however, with respect to measurement accuracy. 

Uniqueness of Ihe reaction speed in pseudo-steady state mode 

If the system is in the pseudo-steady state mode, every intermediate will be at a constant amoimt over time and therefore the stoichiometry relation will be preserved throughout the reaction. Thus, we can state the following theorem  

Theorem 7.1.- The extent and speed of a reaction in pseudo-steady state mode is independent of the component chosen to define them. 

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In general, these two extents are different at a given time, except, as we will see later, for pseudo-steady state modes (see section 7.4.3). 

For each one of these methods, if the stoichiometric conditions of the reaction are observed at every moment (we will see in section 12.2 that this is the case with pseudo-steady state modes), the expressions binding experimental data with fractional extents or rates are simplified. 

We will see that some couplings are very useful in the comprehension of mechanisms, particularly for checking a pseudo-steady state mode (see section... 

A majority of the reactions considered derive from the qnasi-chemical reactions listed in Chapter 2 (see section 2.5). For each type of reaction, we will give the elementary steps that proceed in the same zone and their volirmtnal speed. We will also introduce the voluminal speed of the equivalent step whose jnstification is given by the theorem of local pseudo-steady state mode (see section 7.9). 

Table 7.1. Terms of a pseudo-steady state mode of diffusion under a gradient...

In case of the diffusion of a charged particle, under the simultaneous action of a gradient of concentration and an electric field, the reactivity within the framework of a pseudo-steady state mode, according to relation [5.47], is given by ... 

Linear mechanisms in pseudo-steady state modes... 

In the majority of the practical cases and taking into accoimt the precision and the reproducibility of measurements, we can be satisfied, at least after a certain time, of a category of solutiorrs described as pseudo-steady state modes. In the case of diffusions of particles charged under electric field, the approximatiorrs of a null total cmrent, local electric neutrality and of an electric mobility (in general those of the interstitial iorts or vacancies) small compared to the other (in general the one of the electrons or electron holes), i.e. approximations that we used with section 5.5.3, are sufficient. 

A mode will be known as a pseudo-steady state mode if the amourrts of matter of all the intermediate species are small compared to those of the reactants and products and are practically constarrt, that is ... 

The second condition of the pseudo-steady state mode results in At At... 

It is observed that if the mode is already stationary, as we will find it in the pure modes (see section 7.5.2), the pseudo-steady state mode leads to the approximation ... 

Theorem of the equality of rates of a linear mechanism in pseudo-steady state modes... 

For the linear mechanisms in pseudo-steady state mode, the following theorem can be given ... 

But according to the definition of the multiplying coefficients, the last members of the equalities of this type are null whatever be the i and we have a pseudo-steady state mode, which shows our reciprocal. 

Theorem.- In a linear mechanism in pseudo-steady state modes, extents, fractional extents, speeds or rates are identical whatever the principal component of the total reaction chosen for their definitions. 

The common value of all these rates will be called the rate of the reaction in the pseudo-steady state mode it is thus ... 

For equations of a linear mechanism in pseudo-steady state modes, we will simply use the linear mechanism represented by the steps [7.Et.bl] to [7.Et.bn+l]. 

According to the properly of a pseudo-steady state mode, we can write the rate in the two forms ... 

The first considers pseudo-steady state modes not with two rate-determining steps but with three or more. It is shown in this case that we can generalize the slowness theorem and extend the sum of slownesses to all pure modes (provided that all the multiplying coefficients of the selected steps are equal). 

Some mixed modes do not have the properly to be completely pseudo-state modes. We will show how to deal with such a problem, which is the possibly the simplest among the non-pseudo-steady state modes. 

Equivalent reaction of a linear subset in local pseudo-steady state mode... 

Theorem.- A subset of steps of a linear mechanism in pseudo-steady state mode occurring in the same zone and meeting certain conditions such as the presence of main reactants and products with variable concentratiorrs, can formally be replaced by the reaction summons steps of the snbset, called the eqrrivalent reaction of the subset, to express the variations of the reactivity as a function of concentrations. 

We already applied this result, directly to replace the set of the steps of diffusion by a single elementary reaction (Chapter 5). We also applied it for some reactions of interfaces (Chapter 4) which made it possible to draw up Table 4.1 by considering the three pseudo-steady state modes of the system of the two elementary steps ... 

The pseudo-steady state modes for a reaction, if the whole of the reaction is held in only a single zone or several zones whose dimensions always remain equal to each other, even if these sizes vary with time (we will encounter this case in the process of nucleation of a new solid phase on the surface of an initial solid phase, see chapter 8, the mode being a pseudo-steady state one). 

We can apply the approximation of mixed pseudo-steady state modes and the space functions of the two rate-determining steps keeping values constantly identical with each other in time. The rate is then given by equation [7.58] ... 

A linear mechanism with a general pseudo-steady state mode if the space functions are equal to each other at any time and with all equal multiplying coefficients. The total reactivity is given by equation [7,63] it is independent of time at constant temperature, pressure, and concentration and the reactance is separable. 

Pseudo-steady state mode with a rate-determining step... 

Examine a pseudo-steady state mode with a rate-determining step i. (the mechanism is assumed to be linear). We know that all the other steps p i) are practically at equilibrium and Gibbs energy of each one of them is nnll ... 

In all the pseudo-steady state modes of nucleation, as the space function always has the same value for the whole of the steps, the theorem of the equahty of the speeds (see section 7.4.2) can be applied to the reactivities and thus... 

A solution for non-pseudo-steady state mode of the model of condensation, assuming steps of creation of the precursor at equilibrium, can be obtained. The total system is supposed to be far from equilibiimn what enables us to neglect speeds of the opposite reactions of the elementary steps. 

From where comes the solution of the reactivity of nucleation far from equilibrium, in pseudo-steady state mode with step i as the rate-determining step ... 

General pseudo-steady state modes on potential nuclei... 

In the particular case of pseudo-steady state modes, then (see sections 7.4.3 and 7.4.4) ... 

If it is a pseudo-steady state mode, taking into account the ejqtression of z (equation [7.36]), we get ... 

We notice that the fractional extent can be expressed, in a pseudo-steady state mode, in two manners according to whether we are interested in the volume of A disappeared or B manufactured. Thus, we can write ... 

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