Stationary state treatment

The explicit mathematical treatment for such stationary-state situations at certain ion-selective membranes was performed by Iljuschenko and Mirkin 106). As the publication is in Russian and in a not widely distributed journal, their work will be cited in the appendix. The authors obtain an equation (s. (34) on page 28) similar to the one developed by Eisenman et al. 6) for glass membranes using the three-segment potential approach. However, the mobilities used in the stationary-state treatment are those which describe the ion migration in an electric field through a diffusion layer at the phase boundary. A diffusion process through the entire membrane with constant ion mobilities does not have to be assumed. The non-Nernstian behavior of extremely thin layers (i.e., ISFET) can therefore also be described, as well as the role of an electron transfer at solid-state membranes. 

ITulanicki and Levenstam 1Z5) already suggested a model with a diffusion layer inside the solution. The stationary-state treatment of Ilushenko and Mirkin in... 

Application of stationary state treatment for ArS02 and ArS02CH2CHCgHs produces a complex rate law which reduces at low Cu(II) concentrations to... 

By applying the stationary state treatment to the formation of radicals, a value for [RJ could be obtained. Substitution of this expression into Eqn. 24 resulted in the final form of the deposition rate. 

From stationary-state treatment, the concentration of the radical intermediate [X-]ss is expressed as (Ia = intensity of absorption 4> = quantum yield)... 

With these assumptions, the stationary-state treatment gives for the stationary-state concentration of H atoms the same result as before [Eq. (XIII.2.7)] ... 

This is ill more or less reasonable agreement with other work " in the field, although because of the difficulties already discussed concerning the validity of the stationary-state treatment in this system, all these results must be taken with reservations. 

By applying the stationary-state treatment to (H), (CH3), and (CaHa) we find... 

The stationary-state treatment of this more complex chain reaction can be used to derive an expression that governs the total concentration of all radicals in the system, (H) + (CH3) + (C2IT6). Since the propagation reactions merely replace one radical by another but do not affect the total concentration of radicals, we can write for the stationary state for all radicals that the sum of all initiation reactions is equal to the sum of all termination reactions. This leads to the equation ... 

It can be seen that the condition for a first-order reaction depends not only on the ratio of rate constants kz and k2 but also on the concentration of X and HOH. It may thus be supposed that, even when kz ) > A 2, it will still be possible to have X sufficiently greater than H2O to observe the competition of reactions 2 and 3. If we apply the stationary-state treatment to the scheme XVI. 1.2, assuming that h h and ignoring the back reaction 6,... 

Taking reactions (i)—(v), together with the surface destruction of H and O atoms, the stationary state treatment gives the complete first and second limit explosion boundary as... 

With such very hydrogen-rich mixtures the partial stationary state treatment becomes valid for [OH] and [O], and eqn. (54) is identical with eqn. (29) if surface termination of H and O atoms are omitted from the latter by putting Pi = 0. Equation (54a) is the basis of the ignition delay correlation t, [O2] = constant at constant temperature used by Schott and Kinsey [102]. 

Using a stationary state treatment for CO then gives... 

A stationary-state treatment of eqs. (33)-(40) yields the equation [Total yield of hydrocarbons]... 

The formation of cyclohexene with insufficient energy to isomerize, reaction (52), is postulated to account for the observation that a plot of the yield of cyclohexene at low pressures extrapolated to zero pressure gives a positive intercept corresponding to about 50%. Ignoring reactions (54), (55), and those that depend directly on the products formed in these reactions, a stationary-state treatment of the mechanism yields the equation... 

A stationary-state treatment gives the following expression for the product quantum yield O (molecules of product formed per quantum of light absorbed, that is moles of product per einstein of light absorbed), as a function of the concentration of B... 

So far, instabilities arising in reacting systems have been ascribed wholly to thermal imbalance. Only very simple kinetic schemes have been invoked (and the stationary-state treatments adopted have, by their neglect of reactant consumption, r arded all kinetic equations as pseudo-zero order). Thermal treatments were never able to explain contemporary experimental studies of the hydrogmi-oxygmi reaction, and in the 1920 s and 1930 such theories were virtually edipsed by isothermal branched-chain theory which originated around the same time as an alternative explanation of the instability of gas reactions. 

Later developments included allowance for the spatial variation of radical concentration, and stationary-state treatments were put forward which were later echoed by their thermal parallels developed by Frank-Kamenetskii. 

Reaction conditions (i) and (ii) resemble the Semenov classifications of stable and unstable behaviour. For (iii), the reaction conditions are called parametrically sensitive. With absolute control of system parameters, any degree of self-heating can be produced and a complete range of maximum temperature excesses attained up to the adiabatic flame temperature. Physically such exact control is impossible and althou in the laboratory we should expect to see occasional temperature rises of the order of 100 K, repeatable non-explosive temperatures will be practically bounded by the steady-state limits. For simple systems, therefore, stationary-state treatments are still of great value. First, they impose a stability bound, inasmuch as conditions stable under stationary-state theory always remain stable under... 

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