Steady-state assumptions

The result of Section 8.4.1 may be generalized. If the rate constant for the initial step is small compared to the rate of the subsequent steps, the concentration of the intermediate products is similar to the one found for B above. A steady state will appear in the concentrations of the intermediary products. The time derivatives are so small that they may be assumed equal to zero. This is a useful approximation in order to solve the complicated kinetics of many photochemical and biological reactions. 

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The ratio describes the relative reactivity of polymer chain M toward monomer M and monomer M2. Likewise, describes the relative reactivity of polymer chain M2 toward M2 and M. With a steady-state assumption, the copolymerisation equation can be derived from the propagation steps in equations 3—6. 

On the basis of the pseudo-steady state assumption, the net rate of disappearanee is zero, therefore... 

The interpretations of Michaelis and Menten were refined and extended in 1925 by Briggs and Haldane, by assuming the concentration of the enzyme-substrate complex ES quickly reaches a constant value in such a dynamic system. That is, ES is formed as rapidly from E + S as it disappears by its two possible fates dissociation to regenerate E + S, and reaction to form E + P. This assumption is termed the steady-state assumption and is expressed as... 

Given the relationships between E, S, and I described previously and recalling the steady-state assumption that d[E /dt= 0, from Equations (14.14) and (14.16) we can write... 

Application of a steady state assumption (cqs, 33-35) enables derivation of the composition relationship (eq. 36). 

Case D is an interesting one, since fcimp and kpt, but not ss, are good approximations to A3. With the values given, [I] is not much smaller than [A], but is about one-half of it. That is, the A I step rapidly equilibrates with substantial conversion of A to I. This is not consistent with the ordinary steady-state assumption, so it comes as no surprise that kss is an invalid approximation compared to A3 itself. 

Assuming that chain-breaking occurs exclusively by step 6, derive the steady-state rate equation for -r/[H2C204]/c/f. Make the steady-state assumption for Cl and CiO , and assume long chains (that is, [Fe2+] added [C121). The answer will involve R. Repeat the derivation for chain termination with step 7. 

Effects of Temperature on Ionic Reactions in TD/D2 CH4/ CD4. Observation that the methanium ion proton (deuteron) transfer sequence fails to exhibit a temperature coefficient within experimental uncertainties leads unavoidably to the conclusion that none of the reactions from 1 to 12 requires thermal activation between —78° and 25°C. From Equations I, II, III, appropriate steady state assumptions, and representing both neutralization steps by kX2, we find that... 

In a case of where the radical steady-state assumption can be made, the reactor heat balance can be written in dimensionless form at constant pressure (l )... 

The inlet monomer concentration was varied sinusoidally to determine the effect of these changes on Dp, the time-averaged polydispersity, when compared with the steady-state case. For the unsteady state CSTR, the pseudo steady-state assumption for active centres was used to simplify computations. In both of the mechanisms considered, D increases with respect to the steady-state value (for constant conversion and number average chain length y ) as the frequency of the oscillation in the monomer feed concentration is decreased. The maximum deviation in D thus occurs as lo 0. However, it was predicted that the value of D could only be increased by 10-325S with respect to the steady state depending on reaction mechanism and the amplitude of the oscillating feed. Laurence and Vasudevan (12) considered a reaction with combination termination and no chain transfer. 

Setting the derivatives with respect to time in Equations (Al) and (A2) equal to zero (i.e. invoking the steady state assumption), solving for [R°c] and [R1.J, and substituting into Equation (A3) gives ... 

Assuming that this process runs under steady state conditions, as for an industrial flow reactor with a constant inflow of reactants and a constant outflow of products, the concentration of the intermediate will be constant, as expressed in the steady-state assumption ... 

Application of the steady state assumption yields [O] overall rate becomes... 

Derive the rate expression for the formation of CH4 by using the steady state assumption. 

Exercise 2.6 Steady State Assumption in the Kinetics of Chain Reactions... 

Applying steady-state assumptions, the rate equation for the reaction mechanism described by Equations 17.6, 17.7, and 17.8 can be obtained ... 

A paper by Ozturk, Palsson, and Dressman (OPD), reporting a refinement of the MMSH model, did create some controversy. OPD developed a film model with reaction in spherical coordinates and applied quasi-steady-state assumptions to the boundary conditions at the solid surface [11], They theorized that the flux of all species at the solid surface must be zero, except for HA, or the other species (A-, H+, OH ) would penetrate the solid surface. A debate by correspondence in the Letters to the Editor columns of Pharmaceutical Research ensued [12,13], The reader is invited to evaluate which author s arguments are more convincing. What is difficult to evaluate is whether the OPD model produces dissolution results which are different from those which would be predicted using the MMSH model cast in comparable spherical geometry. Simply, these authors never graphically demonstrate how their model predictions compare to the MMSH model. Algebraically, the solutions to both models appear comparable. 

Under the steady-state assumption, the fraction of dose absorbed, Fa, is 1 - C0J... 

If a steady-state assumption is applied (total mass entering a spherical shell must equal the mass leaving the shell), the result is... 

The mathematical solution to moving boundary problem involves setting up a pseudo-steady-state model. The pseudo-steady-state assumption is valid as long as the boundary moves ponderously slowly compared with the time required to reach steady state. Thus, we are assuming that the boundary between the salt solution and the solid salt moves slowly in the tablet compared to the diffusion... 

Clearly, /t0(f) is the total number and is the total volume of the agglomerates in the charge. The latter is approximately independent of time in view of the quasi-steady-state assumption in Eq. (46). Kapur (K3) has shown that an asymptotic similarity solution to Eq. (47) exists in the following form ... 

Some economies are possible if equilibrium is assumed between selected compartments, an equal fugacity being assignable. This is possible if the time for equilibration is short compared to the time constant for the dominant processes of reaction or advection. For example, the rate of chemical uptake by fish from water can often be ignored (and thus need not be measured or known within limits) if the chemical has a life time of hundreds of days since the uptake time is usually only a few days. This is equivalent to the frequently used "steady state" assumption in chemical kinetics in which the differential equation for a short lived intermediate species is set to zero, thus reducing the equation to algebraic form. When the compartment contains a small amount of chemical or adjusts quickly to its environment, it can be treated algebraically. 

Because the steady-state assumption leads to the equilibrium relation for the bromine atom concentration (4.2.20), it does not matter what mechanism is assumed to be responsible for establishing this equilibrium. Alternative elementary reactions for the initiation and termination processes, which give rise to the same equilibrium relationship, would also be consistent with the observed rate expression for HBr formation. For example, the following reactions give rise to the same equilibrium ... 

In addition to the quasi-steady state assumption, the other assumptions required to arrive at equation (1) are 1. the aerosol itself does not coagulate 2. there is a fully developed concentration gradient around each aerosol particle and 3. the concentration of unattached radon progeny atoms is much greater than the concentration of aerosol particles (in order that concentration gradients of radon progeny atoms may exist). This last assumption is usually not valid since the radon progeny concentration is usually much less than the aerosol concentration. 

As with most assumptions and approximations, those professors who do not ignore this entirely will undoubtedly think that you should at least know that it s an assumption. What the steady-state assumption actually... 

Very often, the quasi steady-state assumption for the hydrogen liquid phase concentration is proposed (Eq. (30)) ... 

Schachter, H. (1972). The use of the steady-state assumption to derive kinetic formulations for the transport of a solute across a membrane. In Metabolic Transport, ed. Hokin, L. E., Metabolic Pathways. Vol. 6, Series ed. Greenberg, D. M., Academic Press, New York, pp. 1-15. 

In the previous chapters the data reconciliation problem was analyzed for systems that could be assumed to be operating at steady state. Consequently, only one set of data was available. In some practical situations, the occurrence of various disturbances generates a dynamic or quasi-steady-state response of the process, thus nullifying this steady-state assumption. In this chapter, the notions previously developed are extended to cover these cases. 

We assumed that the adduct formed between the substrate and the active form of the catalyst obeys the steady-state assumption. Equations (6.225) to (6.227) are thus replaced by... 

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