Pseudo-stationary

MEKC is a CE mode based on the partitioning of compounds between an aqueous and a micellar phase. This analytical technique combines CE as well as LC features and enables the separation of neutral compounds. The buffer solution consists of an aqueous solution containing micelles as a pseudo-stationary phase. The composition and nature of the pseudo-stationary phase can be adjusted but sodium dodecyl sulfate (SDS) remains the most widely used surfactant. 

Breadmore, M. C., Macka, M., and Haddad, P. R., Manipulation of separation selectivity for alkali metals and ammonium in ion-exchange capillary electrochromatography using a suspension of cation exchange particles in the electrolyte as a pseudo stationary phase, Electrophoresis, 20, 1987, 1999. 

Palmer, C.P. and Tanaka, N., Selectivity of polymeric and polymer-supported pseudo-stationary phases in micellar electrokinetic chromatography, /. Chromatogr. A, 792, 105, 1997. 

A. Stationary (mostly pseudo-stationary) methods -+ static measurement B. Non-stationary methods -> dynamic measurement ... 

Poly(acrylic acid) is not soluble in its monomer and in the course of the bulk polymerization of acrylic acid the polymer separates as a fine powder. The conversion curves exhibit an initial auto-acceleration followed by a long pseudo-stationary process ( 3). This behaviour is very similar to that observed earlier in the bulk polymerization of acrylonitrile. The non-ideal kinetic relationships determined experimentally in the polymerization of these two monomers are summarized in Table I. It clearly appears that the kinetic features observed in both systems are strikingly similar. In addition, the poly(acrylic acid) formed in bulk over a fairly broad range of temperatures (20 to 76°C) exhibits a high degree of syndiotacticity and can be crystallized readily (3). 

Besides CZE and NACE, micellar electrokinetic chromatography (MEKC) is also widely used, and ionic micelles are used as a pseudo-stationary phase. MEKC can therefore separate both ionic and neutral species (see Chapter 2). Hyphenating MEKC with ESI/MS is problematic due to the non-volatility of micelles, which contaminate the ionization source and the MS detector, resulting in increased baseline noise and reduced sensitivity. However, MEKC—ESI/MS was applied by Mol et al. for identifying drug impurities in galantamine samples. Despite the presence of non-volatile SDS, all impurities were detected with submicrogram per milliliter sensitivity and could be further characterized by MS/MS. 

Tanaka, Y, and Terabe, S. (1995). Partial separation zone technique for the separation of enantiomers by affinity electrokinetic chromatography with proteins as chiral pseudo-stationary phases. J. Chromatogr. A 694, 277—284. 

Y Tanaka, M Yanagawa, S Terabe. Separation of neutral and basic enantiomers by cyclodextrin electrokinetic chromatography using anionic cyclodextrin derivatives as chiral pseudo-stationary phases. J High Res Chromatogr 19 421-433, 1996. 

Y Ishihama, Y Oda, N Asakawa, Y Yoshida, T Sato. Optical resolution by electrokinetic chromatography using ovomucoid as a pseudo-stationary phase. J Chromatogr A 666 193-201, 1994. 

Fig. 1.8. Consecutive first-order reactions with p0 = 0.1 mol dm-3, fcu = 5 x 10-3 s and k2 = 10"2s (a) the exponential decay of precursor reactant concentration, p (b) growth and decay of intermediate concentrations a(t) and b(t). Also shown in (b), as broken curves, are the pseudo-stationary-state loci, a (t) and b (t), given by eqns (1.31) and...

We are left with the behaviour of the intermediates A and B. A common approach to kinetic models involving relatively reactive species is to apply the pseudo-stationary-state (PSS) hypothesis. 

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. 

Fia. 2.2. Predicted pseudo-stationary-state evolution of the intermediate species concentrations a(t) and b(t), as given by eqns (2.15) and (2.16). Specific numerical values correspond to the rate data in Table 2.1. The time at which the two concentrations become equal and that at which a(t) attains its maximum are indicated. 

If the computations are made with all the data taken from Table 2.1, however, a remarkable difference appears (Fig. 2.4). There is an initial transient behaviour during which the concentrations of the intermediates move quickly to the appropriate pseudo-stationary values. The PSS curves... 

Fig. 2.4. Computed concentration.histories for autocatalytic model with rate constants given exactly as in Table 2.1 (a) exponential decay of precursor (b) intermediate concentrations a(t) and 6(r), showing initial pseudo-stationary-state behaviour but subsequent development of an oscillatory period of finite duration, 1752 s < t < 3940 s.

There is a period of time or, if we prefer, a range of reactant concentration over which the system spontaneously moves away from the pseudo-stationary state. The idea that stationary states may be unstable is not widely appreciated in chemical kinetics but it is fundamental to the analysis of oscillatory systems. 

At this stage, however, we may proceed to the important features of the model s behaviour qualitatively from Fig. 2.4. For some range of time or of reactant concentration the pseudo-stationary state described by eqns (2.15) and (2.16) become locally unstable. Let this range be denoted... 

As well as showing how a and b vary in time, the numerical traces can also be plotted in a different way. If we plot the two concentrations against each other (Fig. 2.5(c)), we find that they draw out a closed curve or limit cycle around which the system circulates. This limit cycle surrounds the unstable pseudo-stationary state appropriate to p. The amplitude of the oscillations is a measure of the size of this limit cycle. 

In addition to the general aims set out at the beginning of this chapter we have discovered a wealth of specific detail about the behaviour of the simple kinetic model introduced here. Most results have been obtained analytically, despite the non-linear equations involved, with numerical computation reserved for confirmation, rather than extension, of our predictions. Much of this information has been obtained using the idea of a pseudo-stationary state, and regarding this as not just a function of time but also as a function of the reactant concentration. Stationary states can be stable or unstable. 

We now turn again to the evaluation of the pseudo-stationary-state responses. 

These have only one true stationary-state solution, ass = pss = 0 (as t - go), corresponding to complete conversion of the initial reactant to the final product C. This stationary or chemical equilibrium state is a stable node as required by thermodynamics, but of course that tells us nothing about how the system evolves in time. If e is sufficiently small, we may hope that the concentrations of the intermediates will follow pseudo-stationary-state histories which we can identify with the results of the previous sections. In particular we may obtain a guide to the kinetics simply by replacing p by p0e et wherever it occurs in the stationary-state and Hopf formulae. Thus at any time t the dimensionless concentrations a and p would be related to the initial precursor concentration by... 

The traces in Fig. 3.9 were computed for a system with an uncatalysed reaction rate constant such that jcu > g, and hence there are no oscillatory responses in the corresponding pool chemical equations. For ku < we may also ask about the (time-dependent) local stability of the pseudo-stationary state. The concentration histories may become unstable to small perturbations for a limited time period. For sufficiently small e this should occur whilst the group p0e cr lies within the range... 

Fig. 3.9. Comparison of exact concentration histories for species A, oc(t), with pseudo-stationary-state form (broken curves) showing the influence of the precursor decay rate constant c when the uncatalysed reaction rate is relatively large, k = i (a) s= 10 3 (b) = 10 2 (c) e= 10 1 ...

The second significant difference between the predictions and the actual results is that oscillations survive beyond the time. This arises because the pseudo-stationary state has focal character just after the second Hopf bifurcation (i.e. the slowly varying eigenvalues i1>2 are complex conjugates with now negative real parts) so there is a damped oscillatory return to the locus. In Fig. 3.10(a) this can be seen after t 3966, whilst t = 3891. 

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