States Steady

At steady state, the rate of production of free radicals equals the rate of consumption, i.e. 

This can be put into equation (4) giving the rate of polymerisation  

From (97) and similar equations for cations and solvent, the steady-state concentration of species i at potential E, obtained for (d/dx) = 0, which gives the insertion isotherm of species i, is 

As noted before, one of the general conditions for equilibrium and steady state is that the forward and backward fluxes of a process are equal (F+ = F ). The specific condition to fulfill a chemical equilibrium is fi+ = n- (Eq. 4.92). Although a chemical equilibrium occurs when two or more reversible processes occur at the same rate, and such a system can be said to be in steady state, a system that is in steady state might not necessarily be in a state of equilibrium, because some of the processes involved are not reversible. A system in a steady state has numerous properties that do not change over time. The concept of steady state has relevance in many fields, in particular thermodynamics. Hence, steady state is a more general situation than dynamic equilibrium. If a system is in steady state, then the recently observed behavior of the system will continue into the f uture. In stochastic systems, the probabilities that various different states will be repeated will remain constant. We will generalize Eq. (2.142) as follows (see Chapter 2.8.4.1 and (4.88))  

In many systems, steady state is not achieved until some time has elapsed after the system is started or initiated. This initial situation is often identified as a transient state, start-up or warm-up period. 

The term steady state is also used to describe a situation where some, but not all, of the state variables of a system are constant. For such a steady state to develop, the system does not have to be a flow system. Therefore, such a steady state can develop in a closed system where a series of chemical reactions take place. Literature on chemical kinetics usually refers to this case, calling it steady-state approximation. Steady-state approximation, occasionally called stationary-state approximation, involves setting the rate of change of a reaction intermediate in a 

Often also called stationary state (Chapter 2.8.4.1). But in physics, especially in quantum mechanics, a stationary state is an eigenstate of a Hamiltonian or, in other words, a state of definite energy. It is called stationary because the corresponding probability density has no time dependence. For all kinds of stationary state in German (and likely in other languages too) there is only one term stationarer Zustand . 

These approximations are frequently used because of the substantial mathematical simpUfications this concept offers. Whether or not this concept can be used depends on the error the underlying assumptions introduce. Therefore, even though a steady state, from a theoretical point of view, requires constant drivers (e. g. constant inflow rate and constant concentrations in the inflow), the error introduced by assuming steady state for a system with non-constant drivers might be negligible if the steady state is approached fast enough (relatively speaking). 

The w+, s are given as functions of the 8 of the two independent intermediates, admitting that the and 0 O) s in Eq. (II.22.f) are given similarly as functions of the three p s, hence as those of ) s of the two independent intermediates. Equation (IV.45) now determines of the two independent intermediates hence the steady state is, in principle, quantitatively determined. It is now exemplified that V approaches V, from below as the V, gets smaller than all other kindred quantities, as generally demonstrated in Section IV,C,4,a. Now eliminating p and p (a)jjBr(a) fj.Qjn jjq (iv.45), we have 

The simplest case to solve is when the concentration stays constant over time in the polymer. If diffusion occurs only along the direction of the x-axis then  

This particular case exists for example in the diffusion of a substance through a film with thickness d (Fig. 7-3) if the concentrations at the two surfaces Cj at x = 0 and c2 at x = d remain constant (stationary case)  

A constant concentration gradient exists in the film perpendicular to the film s surface and consequently there is a constant diffusion flux in the x-axis direction according to Eq. (7-3c) at every location between x = 0 and x = d. Integrating Eq. (7-14) again leads to  

When a system is in equilibrium, the flux is zero. We get from Equation 6.76 1 9c(x) 

the steady-state flux / is a constant, independent of x and t. It depends only on the free energy landscape, duration of the domain, and the diffusion constant of the particle. 

The steady-state concentration profile c(x) is obtained by integrating Equation 6.80, 

Rearrangement of this equation with constant J and D yields 

The particle concentration at x is given by the above equation with the substitution of Equation 6.81 for the constant flux J. It is readily seen from Equations 6.79 and 6.83 that the concentration profiles in the steady state and the equilibrium state are different. 

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Knowing where waste is going is the key to reducing it. When reducing waste from process operations, a steady-state mass balance is not usually comprehensive enough. A balance that takes into account start-up, shutdown, and product changeovers is required. 

In both preceding cases, the demands to the electrolysis unit are limited, since there is no need to keep the silver content in the fixer tank constantly low. A steady state silver concentration in the fixer between 3 and 5 g/1 is acceptable, since this causes no substantial loss of fixation speed. 

Measurement by quasi - constant current (steady - state value of pulse current) providing a compete tuning out from the effect of not only electric but also magnetic material properties. 

Fig. rV-26. Steady-state diffusion model for film dissolution. (From Ref. 293.)... 

For the steady-state case, Z should also give the forward rate of formation or flux of critical nuclei, except that the positive free energy of their formation amounts to a free energy of activation. If one correspondingly modifies the rate Z by the term an approximate value for I results ... 

The basic assumption is that the Langmuir equation applies to each layer, with the added postulate that for the first layer the heat of adsorption Q may have some special value, whereas for all succeeding layers, it is equal to Qu, the heat of condensation of the liquid adsorbate. A furfter assumption is that evaporation and condensation can occur only from or on exposed surfaces. As illustrated in Fig. XVII-9, the picture is one of portions of uncovered surface 5o, of surface covered by a single layer 5, by a double-layer 52. and so on.f The condition for equilibrium is taken to be that the amount of each type of surface reaches a steady-state value with respect to the next-deeper one. Thus for 5o... 

Rate laws have also been observed that correspond to there being two kinds of surface, one adsorbing reactant A and the other reactant B and with the rate proportional to 5a x 5b- For traditional discussions of Langmuir-Hinshelwood rate laws, see Refs. 240-242. Many catalytic systems involve a series of intermediates, and the simplifying assumption of steady-state equilibrium is usually made. See Boudart and co-workers [243-245] for a contemporary discussion of such complexities. 

Derive the steady-state rate law corresponding to the reaction sequence of Eqs. XVIII-40-XVIII-44, that is, without making the assumption that any one step is much slower than the others. See Ref 234. 

In a steady state, with no convection, the two currents must be equal, Ji z)... 

Figure A3.1.4. Steady state heat eonduetion, illustrating the flow of energy aeross a plane at a height z.

Kirkpatrick T R, Cohen E G D and Dorfman J R 1982 Light scattering by a fluid in a nonequilibrium steady state. II. Large gradients Phys. Rev. A 26 995... 

Another possibility is that a system may be held in a constrained equilibrium by external forces and thus be in a non-equilibrium steady state (NESS). In this case, the spatio-temporal correlations contain new ingredients, which are also exemplified in section A3.3.2. 

There are tliree steps in the calculation first, solve the frill nonlinear set of hydrodynamic equations in the steady state, where the time derivatives of all quantities are zero second, linearize about the steady-state solutions third, postulate a non-equilibrium ensemble through a generalized fluctuation dissipation relation. 

A] = b/a (equation (A3.4.145)) is stationary and not [A ] itself This suggests d[A ]/dt < d[A]/dt as a more appropriate fomuilation of quasi-stationarity. Furthemiore, the general stationary state solution (equation (A3.4.144)) for the Lindemaim mechanism contams cases that are not usually retained in the Bodenstein quasi-steady-state solution. 

General first-order kinetics also play an important role for the so-called local eigenvalue analysis of more complicated reaction mechanisms, which are usually described by nonlinear systems of differential equations. Linearization leads to effective general first-order kinetics whose analysis reveals infomiation on the time scales of chemical reactions, species in steady states (quasi-stationarity), or partial equilibria (quasi-equilibrium) [M, and ]. 

Thus, this eigenvalue detenuines the unimolecular steady-state reaction rate constant. 

Figure A3.13.16. Illustration of the level populations (eorresponding to the C-C oseillator states) from various treatments in the model of figure A3.13.15 for C2Hg at a total energy E = (he) 41 000 em and a tlneshold energy = (he) 31 000 em The pomts are mieroeanonieal equilibrium distributions. The erosses result from the solution of the master equation for IVR at steady state and the lines are thennal populations at the temperatures indieated (from [38] quant, is ealeulated with quantum densities of states, elass. with elassieal meehanieal densities.).

Bromide ion acts as an inliibitor through step (9) which competes for HBr02 with the rate detennining step for the autocatalytic process described previously, step (4) and step (5). Step (8) and Step (9) constitute a pseudo-first-order removal of Br with HBr02 maintained in a low steady-state concentration. Only once [Br ] < [Br ] = /fo[Br07]//r2 does step (3) become effective, initiating the autocatalytic growth and oxidation. 

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