Stationary state concentrations

Let us compare the ratio of radicals in oxidized 2-propanol and cyclohexanol at different temperatures when oxidation occurs with long chains and chain initiation and termination do not influence the stationary state concentration of radicals. The values of the rate constants of the reactions of peroxyl radicals (kp) with alcohol and decomposition of the alkylhydroxy-peroxyl radical (k ) are taken from Table 7.4 and Table 7.5. 

A different situation in the oxidation of these two alcohols is seen. The hydroperoxyl radical is the main chain propagating species in oxidized 2-propanol the portion of alkylhydroxy-peroxyl radicals in this reaction is less than 2.5%. In oxidized cyclohexanol, on the contrary, the stationary state concentrations of both radicals are close and both of them take important part in chain propagation. 

For series reactions in steady-state operation of a CSTR, it is a matter of determining the stationary-state concentrations of species (product distribution) in or leaving each stage. 

DR. ENDICOTT In fact, the cobalt-eyelam-oxygen adduct can have a very short lifetime depending on the competition reactions. If the cobalt dioxygen adduct is generated in the presence of greater concentrations of CoAA([l4]aneN ), the intermediate lifetime is clearly shortened. The classical studies made by Ralph Wilkins, for instance, were carried out under conditions in which intermediates only achieved small stationary state concentrations. 

TABLE 13-2 Stationary-State Concentration of Oxidants in Photochemical Ozone Generator"... 

At 273 and s 9.02 the polymerization followed first order kinetics. At 225 (Figure 4) the conversion curve was indistinguishable from a zero-order plot up to 40 percent conversion but if the whole curve was examined the internal order was seen to lie between zero and one. At 250 the internal order also lay between zero and one but the fit was nearer to the first-order plot in monomer. The kinetics are consistent with a Bateup-Yerusalimskii mechanism. At low temperatures the limiting condition of Equation 3 is approached. As the temperature rises the stationary state concentration of the complex decreases and the mechanism shifts to its other limit... 

Fig. 1.2. Two of the possible dependences of stationary-state concentration of reactant on flow rate kf for a well-stirred reactor (a) monotonic variation (monostability) (b) multistability, with...

These equations can be rearranged to give the stationary-state concentrations... 

The concentration of A now tends to a value which makes the net inflow rate exactly balance the chemical reaction rate. When this has happened, da/dt = 0, so the concentration becomes steady. This stationary-state concentration can be maintained indefinitely. 

Chemical reactions with autocatalytic or thermal feedback can combine with the diffusive transport of molecules to create a striking set of spatial or temporal patterns. A reactor with permeable wall across which fresh reactants can diffuse in and products diffuse out is an open system and so can support multiple stationary states and sustained oscillations. The diffusion processes mean that the stationary-state concentrations will vary with position in the reactor, giving a profile , which may show distinct banding (Fig. 1.16). Similar patterns are also predicted in some circumstances in closed vessels if stirring ceases. Then the spatial dependence can develop spontaneously from an initially uniform state, but uniformity must always return eventually as the system approaches equilibrium. 

Fig. 1.16. Stationary-state concentration profiles for reaction coupled with diffusion.

The stationary-state concentrations of a and b are given by setting their rates of change simultaneously to zero. Thus... 

In a real experiment the concentration of the reactant falls in time. We should, therefore, establish the way in which bss and ass vary with p. This, in fact, will turn out to be the basis of a particularly convenient approach in which we regard the pseudo-stationary-state concentrations as relatively simple functions of p, rather than the more complex functions of time which we derive later. The stationary-state loci are shown in Fig. 2.1. 

At high concentrations of the reactant, the stationary-state concentration of B is higher than that of A at low p, ass is greater than bss. The two loci must, therefore, cross. This occurs when... 

The stationary-state concentration of A also shows a maximum. Differentiating eqn (2.10) with respect to p, we find the condition for this as... 

The second expression, for bss, is independent of the equilibrium constant and has exactly the same form as that derived for the irreversible system (eqn (2.9)). For the intermediate A, the stationary-state concentration is increased by the reversibility of the steps the first term in eqn (2.30) is that corresponding to the irreversible solution (eqn (2.10)), the second is proportional to the inverse of the equilibrium constant. Thus, as Ke - oo, ass tends smoothly to our previous result. 

Equations (2.37) and (2.38) tend smoothly to the previous results (eqns (2.13) and (2.14)) as the system approaches irreversibility (Ke — oo) for finite Ke, the maximum stationary-state concentration of A increases as Ke decreases and occurs at a higher reactant concentration. The minimum in the locus lies at high reactant concentrations, p oc K J2, and corresponds to a low concentration of the intermediate ass oc K 12. 

The stationary-state concentration / ss of the autocatalyst shows a linear dependence on the reactant concentration p, as shown in Fig. 3.1.The locus for ass(p) shows a maximum ... 

Recalling that ku will usually be small, this can be a relatively large value, occurring at low values of the reactant concentration. For the example data in Table 2.1, ku = 10-2 and hence (ass)max= 5 and is achieved when p = 0.1. Somewhat surprisingly, the maximum stationary-state concentration of the... 

However, we may also see an immediate problem as the reactant concentration n approaches zero, the value of the stationary-state concentration of A becomes infinite, which is physically unrealistic. This problem does not occur if ku is given any non-zero value, no matter how small. 

The dependence of the, pseudo-stationary-state concentration of the intermediate A on p/k shows a maximum value of e-1 when p/k = 1. The time dependence of ass is given by... 

For small y, corresponding loosely to large values for the activation energy, the a5S locus has both the maximum which was displayed with the exponential approximation-and now a minimum which occurs at relatively distant values of h/k. This qualitative change in the locus means that the stationary-state concentration of A increases with high-enough reactant concentra-... 

We may view eqns (4.71)—(4.73) in another way. Choose a system with y < Next choose the dimensionless rate constant k. If k is less than (1 — 4y)e-2, eqn (4.71) can be solved to yield two positive roots 9 and 9. From these values for the stationary-state temperature excess we calculate the reactant concentration required for Hopf bifurcation from eqn (4.72) whilst (4.73) gives the stationary-state concentration of the intermediate A. 

A number of points need to be made about this result. First, unlike the similar relationship (6.11), eqn (6.50) only applies at the stationary state. Secondly, the numerator really contains two contributionsrthe inflow of B, as P0, and the amount of A that has been converted to B, as 1 — ass. The denominator then shows that the stationary-state concentration of the autocatalyst is always less than this. Of course this shortfall between the amount of B present in the reactor and that which has flowed in or been produced merely.reflects the number of such molecules which have then reacted further to produce C. Thus, the denominator increases as the rate... 

The sign of fch, ss is determined by the sign of the factor (a0 + b0 — 3ass), which may be either negative or positive. The sign and value of the relaxation time and the eigenvalue also then depend on the value of the particular stationary-state concentration of A. 

Stationary-state concentrations and extents of reaction for cubic autocataly-... 

Substituting for the stationary-state concentrations from Table 8.1, the determinant becomes... 

Fig. 9.3. Stationary-state concentration profiles aS5(p) for a reaction-diffusion cell with a single cubic autocatalytic reaction (a) D = 0.1157, only small extents of reactant consumption arise (b) D = 0.0633, a higher extent of reactant consumption occurs, particularly towards the centre...

Fig. 9.4. (a) The dependence of the stationary-state concentration of reactant A at the centre of the reaction zone, a (0), on the dimensionless diffusion coefficient D for systems with various reservoir concentrations of the autocatalyst B curve a, / = 0, so one solution is the no reaction states a0i>8 = 0, whilst two other branches exist for low D curves b and c show the effect of increasing / , unfolding the hysteresis loop curve d corresponds to / = 0.1185 for which multiplicity has been lost, (b) The region of multiple stationary-state profiles forms a cusp in the / -D parameter plane the boundary a corresponds to the infinite slab geometry, with b and c appropriate to the infinite cylinder and sphere respectively. 

Typical values for the parameters D and x u might be D = 0.05 and ku = 0.01. We now examine how the stationary-state concentration profiles ass(p) and /Jss(p) depend on the dimensionless concentration of the precursor reactant, p0. Figure 9.10 shows the stationary-state concentrations at the centre of the reaction zone ass(0) and / ss(0) as functions of p0. These loci each draw out a hysteresis loop, with a range of corresponding multiplicity of solutions. 

At low p0, the system has a high stationary-state concentration of A relative to that of the autocatalyst. Typically, both profiles have a maximum at the centre of the reaction zone, p = 0, as shown in Fig. 9.11 (a). High reactant concentrations favour larger concentrations of the autocatalyst B and lower... 

For simplicity, we will consider the case without reactant desorption, K = 0, in which analytical expressions for the stationary-state concentrations can be obtained. With K = 0, the stationary-state condition requires... 

The Hopf bifurcation analysis proceeds as described previously, the required condition being that the trace of the Jacobian matrix corresponding to eqns (12.45) and (12.46) should become equal to zero for some stationary-state concentration given by the lower root from (12.51). (The solution with the upper root corresponds to the middle branch of stationary states for... 

The stationary-state concentrations ass, bss, and css with the above parameter values are... 

On using these two relations we have for stationary state concentration of the radical X",... 

The stationary state concentrations of intermediates in a static system are very small to be detected by common spectroscopic techniques. But if a strong flash oflight is used, a large concentration of the intermediates may be generated which can easily be subjected to spectroscopic analysis. Essentially it is a relaxation method, the flash duration must suitably match the decay constant of the intermediates. The technique was developed by Norrish and Porter in early fifties. 

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