Stage III Maximum Rate and Steady State. Definition. To express the overall rate of a sequence of reactions, a special mathematical treatment is often used, known as the steady state treatment. This is based on the assumption that the concentration of certain intermediate compounds or complexes is never large, that their concentration rises at the beginning of the reaction and soon reaches a constant (or steady) value, and that, at this point, the rate of change in the concentration, dc/dt, can be assumed to be zero. If the overall rate of reaction depends on the concentration of this intermediate, then the rate will have reached its maximum at this time. [Pg.327]

Steady state kinetic measurements on an enzyme usually give only two pieces of kinetic data, the KM value, which may or may not be the dissociation constant of the enzyme-substrate complex, and the kcM value, which may be a microscopic rate constant but may also be a combination of the rate constants for several steps. The kineticist does have a few tricks that may be used on occasion to detect intermediates and even measure individual rate constants, but these are not general and depend on mechanistic interpretations. (Some examples of these methods will be discussed in Chapter 7.) In order to measure the rate constants of the individual steps on the reaction pathway and detect transient intermediates, it is necessary to measure the rate of approach to the steady state. It is during the time period in which the steady state is set up that the individual rate constants may be observed. [Pg.77]

All the time-dependent terms in Equation 50 assembled between the braces are independent of the oxygen consumption rate, v. This means that the rate at which the concentration C at any depth approaches a steady-state value (dC/dt) is independent of the zero order reaction rate constant. The time-dependent terms contain, however, the eddy diffusion coefficient and advection velocity, and the rate of approach to steady-state is therefore dependent on these two physical characteristics of the environment. [Pg.69]

When there is a constant source of a reacting chemical species in the water column or at its boundaries (e.g., water-air and/or water-sediment interface) then, by a rule of thumb, a steady-state may be attained within a period of time equal to a few half-lives of the species. In detail, a steady-state concentration is attained after infinitely long time. The time required for the concentration to come close to the steady-state value at any point in the water column depends on its distance from the source, transport properties of the medium (i.e., its diffu-sivity and distribution of advective velocities), and the rates of the reactions removing the species from the water. A concentration of 95% of a steady-state value may be arbitrarily taken as sufficiently close to a steady-state and indicating that the transient state has effectively come to an end. The time required to attain this concentration level (i.e., when C = 0.95C ) at some point of a concentration-depth profile will be referred to as the time to steady-state. By way of generalization, a chemical species with a constant half-life would attain a steady-state concentration at any point in the water column sooner when the distance [Pg.60]

Calculate and plot the time-dependent species profiles for an initial mixture of 50% H2 and 50% Cl2 reacting at a constant temperature and pressure of 800K and 1 atm, respectively. Consider a reaction time of 200ms. Perform a sensitivity analysis and plot the sensitivity coefficients of the HC1 concentration with respect to each of the rate constants. Rank-order the importance of each reaction on the HC1 concentration. Is the H atom concentration in steady-state [Pg.72]

By a parametric study we mean a presentation of the way in which the performance of the system depends on the choice of a parameter. If we want to make a parametric study of the effect of variations of, say, temperature, which affects k but nothing else, this form is ideal, because k is present in Da but in no other parameter. We perform a quick computation of the conversion (%) and time to reach 99% of steady state (min), for U = 0, 0 = 12 s, and k = 6.2 X 10 7 exp — 14000/T s 1 (this last datum is a rate constant taken from L. D. Schmidt s The Engineering of Chemical Reactions (Oxford, 1997) for a reaction involving butadiene). Any spreadsheet program will immediately give a table of results such as [Pg.7]

Vectors, such as x, are denoted by bold lower case font. Matrices, such as N, are denoted by bold upper case fonts. The vector x contains the concentration of all the variable species it represents the state vector of the network. Time is denoted by t. All the parameters are compounded in vector p it consists of kinetic parameters and the concentrations of constant molecular species which are considered buffered by processes in the environment. The matrix N is the stoichiometric matrix, which contains the stoichiometric coefficients of all the molecular species for the reactions that are produced and consumed. The rate vector v contains all the rate equations of the processes in the network. The kinetic model is considered to be in steady state if all mass balances equal zero. A process is in thermodynamic equilibrium if its rate equals zero. Therefore if all rates in the network equal zero then the entire network is in thermodynamic equilibrium. Then the state is no longer dependent on kinetic parameters but solely on equilibrium constants. Equilibrium constants are thermodynamic quantities determined by the standard Gibbs free energies of the reactants in the network and do not depend on the kinetic parameters of the catalysts, enzymes, in the network [49]. [Pg.407]

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