Steady-state rate laws

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. 

Derive the steady-state rate law. To what form does this reduce if fc-i[Fe"] /t2[Vin] ... 

Steady-state kinetics. The reaction of methylthiamine (MT+) in the presence of a large excess of SO3 and of 4-thiopyridone (= ArS-) is believed to follow the mechanism shown here,15 in which A" and B are steady-state intermediates. Derive the steady-state rate law. 

Steady-state mechanism. Consider the oxidation of RufNHj) by CL, which is believed to occur by the scheme shown below at constant pH. Imagine that one does a series of experiments with [Ru(NHs)g+ ] [O2]. Derive the steady-state rate law. Could these experiments equally well have had the reverse inequality of concentrations Should [RulNH.O ] also be adjusted (how and why) What apparent rate constant could be obtained from the concentration conditions that you consider optimum How would you design a longer series of experiments, and what rate constants could be obtained from the data If the data were examined graphically, what quantities would be displayed on the axes to obtain linear plots, and how would the rate constants be obtained from them ... 

Derive the steady-state rate law for each, and the expression for the product ratio, [RN3]/[ROH], in terms of the rate constants and [Nj ]. Use the following data (in 25 percent aqueous dioxane at 36.2 °C) to decide between them by means of graphs or calculations ... 

A second scheme is equally consistent with the kinetic data. The steps and the steady-state rate law are as follows ... 

One more case remains to be considered for composite rate constants. This is the one in which the denominator contains a summation. A mechanism and steady-state rate law with this feature are... 

If elementary rate laws are assumed for each step, and if Nc. is essentially constant over a short time, a (pseudo-) steady-state rate law can be developed ... 

This means that we again use k3[EX] to represent v). The term ( 2 + ks)/ has the same units as the constant Ks obtained in the rapid equihbrium treatment. In honor of Leonor Michaehs, we use the designation (or Mi-chaelis constant) to represent this quotient, and the steady state rate law reduces to ... 

As we have seen, the catalytic cycle flux provides a useful metric for analyzing enzyme kinetics. In this section, we analyze the turnover time for catalytic cycles and show that the quasi-steady rate law arises from the mean cycle time [151]. In addition, we show that for arbitrary mechanisms for a single-substrate reaction, the steady state rate law can always be expressed using the Michaelis-Menten form... 

For each of these idealized models there is a stationary state. For a continuous open system, this is the steady state. Rate laws and steady material flows arc required to define the steady state. For a closed system, equilibrium is the stationary state. Equilibrium may be viewed as simply the limiting case of the stationary state when the flows from the surroundings approach zero. The simplicity of closed-system models at equilibrium is in the rather small body of information required to describe the time-invariant composition. We now turn our attention to the principles of chemical thermodynamics and the development of tools for the description of equilibrium states and energetics of chemical change in closed systems. 

This is the classic Michaelis- Menten model of enzyme catalysis from which simple steady-state rate laws may be derived. 

The steady-state rate law based on the above mechanism is... 

Identification of the product R-TRAP and fitting of the kinetic data to the steady-state rate law of Equation 8 serve to confirm the mechanism and to evaluate k, and k 1 deduced from... 

Equation 14.33 gives the steady-state approximation for the concentration of A. Substituting this into the rate law (Equation 14.32) gives the steady-state rate law... 

The mechanisms of enzymic action are discussed briefly in a paperback in the Oxford Biology Readers series, while a new book on enzyme kinetics has covered a number of topics, including one-substrate mechanisms, the steady-state rate law, methods of graphical and statistical assessment, inhibitors and activators, homotropic branching and linear mechanisms, co-operative interactions, etc. ... 

The observed rate laws for the oxidation of mercaptoethanol by methylene blue under different reaction conditions are consistent with the steady-state rate laws derived from the proposed mechanism. If step 2 of the reaction sequence, namely formation of the disulfide radical anion from a thiyl radical and a sulfide ion, is the rate limiting step of the chain sequence, and therefore only termination by 6 (coupling of thiyl radicals) occurs, the derived rate law for the reaction is Eq. 11. This rate law, which takes.into account the distribution of the thiol and sulfide as determined by the of the thiol and the acidity of the medium, predicts that the observed rate law would be half order in MB and three halves order in mercaptoethanol. This is the rate law expected, however, only if the concentration of MB " is sufficiently high so that the second term in the denominator is neglible. At pH s below the pH maximum, and at a sufficiently... 

The steady-state rate of C02 formation increases continuously with increasing temperature up to a maximum as shown in Fig. 35. In this range the CO coverage under reaction conditions decreases continuously due to progressive desorption and becomes practically zero at the maximum rate, max (774). As a consequence the rate of oxygen adsorption increases continuously. The rate law is approximately given by... 

Equation (66) (Mazon law in electrical engineering) was used [6] to calculate steady-state rates of complex fermentation reactions. Recently [7], the correctness of this equation has been confirmed. 

The expression on the right-hand side of eqns. (112) and (113) is usually written down as a kinetic law for a simple step consisting of two elementary (direct and inverse) reactions satisfying the law of mass action. As a rule, the steady-state rate for a complex reaction does not fit this expression. It appears that this natural type is satisfied by Wj(n Tj) rather than the steady-state rate W. This value is experimentally observed ( W and t, from the steady-state and non-steady-state experiments, respectively). This value must have been given some special term. 

Kinetic modeling of diesel autothermal reforming is extremely complicated. Diesel fuel consists of a complex variable mixture of hundreds of hydrocarbon compounds containing paraffins, isoparaffins, naphthenes, aromatics, and olefins. To simplify the model, a steady-state power law rate expression for the diesel reforming over each type of catalyst used in this study was developed. A linearized least-squares method of data analysis was used to determine the power law parameters from a series of diesel ATR experiments. The power law rate model for diesel autothermal reaction may be written as ... 

Fick s first law provides a method for calculation of the steady state rate of diffusion when D can be regarded as constant during the diffusion process, and the concentration is a function only of the geometric position inside the polymer. However, concentration is often a function of time as well as of position. We said Equation 14.9 describes a steady state flow, but how does the system reach this steady state The unsteady state flow, or transient state, is described by Fick s second law. For a one-dimensional diffusion process, this can be written as... 

Actually we should not be surprised at the fact that steady-state rate studies are not decisive for determination of the number (or nature) of the intermediates in the mechanism. By the very nature of the steady-state assumption, the intermediates are virtually impossible to detect experimentally. What then is the value of carrying out steady-state rate experiments For one thing, information about the structural specificity of the enzyme can often be obtained by varying the substrate for another, in more complicated mechanisms, possible reaction pathways can be inferred from the form of the rate law. Also of considerable interest to the kineticist is the fact that knowledge of the steady-state kinetic constants allows the determination of a lower bound for all the rate constants in the mechanism. For example, in the case of a reaction mechanism of the type we are considering with n reaction intermediates (where n is an arbitrary number), the following inequalities can be shown to prevail [2] ... 

Steady state is achieved if, at a given temperature, the constant pressures ph and pi(< ph) are maintained at opposing membrane interfaces, respectively. For example, the following relation can be derived from Pick s first law, cf. Eq. (61.1), for the steady-state rate of gas permeation /s through unit area of a planar, isotropic, and homogeneous membrane of effective thickness 8, i.e., when v — 1 and 0 = 1 ... 

Since for a single-route mechanism the numerator of the rate does not depend on the details of the mechanism, it can be written directly based on the overall reaction according to the mass-action law. In such a fast derivation of the steady-state rate equation, steps (7) and (8) are omitted. 

In contrast, the rate law for the nearly steady-state rate (interval II, conversion 5 to 50%) of polymerization in emulsion (2,47-50) is... 

Fourier s law A relationship that states that the steady-state rate of heat transfer by conduction is proportional to the cross-sectional area perpendicular to the direction of flow and to the temperature gradient of the path of conduction ... 

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. 

Material and energy balances are based on the conservation law, Eq. (7-69). In the operation of liquid phase reactions at steady state, the input and output flow rates are constant so the holdup is fixed. The usual control of the discharge is on the liquid level in the tank. When the mixing is adequate, concentration and temperature are uniform, and the effluent has these same properties. The steady state material balance on a reacdant A is... 

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