Concentration and reaction velocity

Fig. 3. Relation between substrate concentration and reaction velocity (Mi-chaelis-Menten curve), and reaction velocity and time.

It has been shown that the relationship between substrate concentration and reaction velocity is hyperbolic and that the parameters of this curve are given by two fiictors, and (see Fig. 3). is the concentration at which half-maximal reaction velocity is achieved. The smaller value is for... 

If the substrate concentration and reaction velocity are plotted as the reciprocal of their values, a linear relationship known as the Lineweaver-Burke plot is obtained (Fig. 30.3). 

Only for the intermediate cases - those with velocities in the range of about 100 m yr-1 to 1000 m yr-1 - does silica concentration and reaction rate vary greatly across the main part of the domain. Significantly, only these cases benefit from the extra effort of calculating a reactive transport model. For more rapid flows, the same result is given by a lumped parameter simulation, or box model, as we could construct in REACT. And for slower flow, a local equilibrium model suffices. 

The relationship between substrate concentration ([S]) and reaction velocity (v, equivalent to the degree of binding of substrate to the active site) is, in the absence of cooperativity, usually hyperbolic in nature, with binding behavior complying with the law of mass action. However, the equation describing the hyperbolic relationship between v and [S] can be simple or complex, depending on the enzyme, the identity of the substrate, and the reaction conditions. Quantitative analyses of these v versus [S] relationships are referred to as enzyme kinetics. 

Effect of substrate concentration on reaction velocities for two enzymes enzyme 1 with a small Km, and enzyme 2 with a large Km. 

Substrate concentration is yet another variable that must be clearly defined. The hyperbolic relationship between substrate concentration ([S ) and reaction velocity, for simple enzyme-based systems, is well known (Figure C1.1.1). At very low substrate concentrations ([S] ATm), there is a linear first-order dependence of reaction velocity on substrate concentration. At very high substrate concentrations ([S] A m), the reaction velocity is essentially independent of substrate concentration. Reaction velocities at intermediate substrate concentrations ([S] A"m) are mixed-order with respect to the concentration of substrate. If an assay is based on initial velocity measurements, then the defined substrate concentration may fall within any of these ranges and still provide a quantitative estimate of total enzyme activity (see Equation Cl. 1.5). The essential point is that a single substrate concentration must be used for all calibration and test-sample assays. In most cases, assays are designed such that [S] A m, where small deviations in substrate concentration will have a minimal effect on reaction rate, and where accurate initial velocity measurements are typically easier to obtain. 

The application of the model of Wise et al. (195, 233) to determining deactivation rate constants encounters rather serious experimental limitations in that at low poison concentrations and space velocities, the breakthrough curves are very slow to appear, and the accurate measurement of sulfur concentrations in the ppm or ppb range is difficult. If activity decline of the catalyst for the reaction of interest could be related to the loss of sites by poisoning, a more direct measurement of deactivation rate would be realized. Bartholomew and co-workers (113, 140, 161) extended this model by expressing the rate of deactivation in terms of normalized activity a ... 

Figure II-8 The relationship of substrate concentration and initial velocity in enzymatic reactions.

This instrument was tested in a study of several electron-transfer reactions, including that between IrCL " and Fe(CN)6 [6], = 4.1 x 10 M s . The data treatment according to the equations given above for pseudo-first-order conditions showed the calculated pseudo-first-order rate constants to be a function of flow velocity, the more so the higher the concentrations and reaction rates. This deviation from the expected behavior was rationalized by the effects of mixing and an empirical relationship was found between the true rate constant (k), the measured value (i obs) and flow velocity, Eq. 5,... 

For weakly acidic systems (pH 5-6) in which the accumulation of hydrobromic acid is prevented by buffering agents such as calcium carbonate or benzoic acid salts, more information is available. Isbell and Pigman have made an extensive study of such systems, including a thorough consideration of the effect of the concentration of total bromine, free bromine, hypobromous acid and bromide ion on the velocity of the reaction. The results very definitely showed a direct correlation between free bromine concentration and the velocity of the oxidation. No such correlation could be found with hypobromous acid. The results are shown in Tables VII and VIII. The velocity constants were determined for a- and for 8-D-glucose. In the table for /S-D-glucose, in experiments 2 and 5, the hypobromous acid concentration varied 1 10 but the reaction rate varied 1 3. The variations in free bromine concentration follow the variations in the reaction rate constants and the kf values are based on the assumption that free bromine is the oxidant. The concentration of the oxidant (a in equation 31) is therefore the concentration of free bromine. 

The reaction rate is directly proportional to the concentration of the enzyme if an excess of free substrate molecules is present. Thus, enzyme-substrate interactions obey the mass-action law. For a given enzyme concentration, the reaction velocity increases initially with increasing substrate concentration. Eventually, a maximum is reached, and further addition of substrate has no effect on reaction velocity (v) (Figure 6-4). The shape of a plot of V versus [S] is a rectangular hyperbola and is characteristic of all nonallosteric enzymes (Chapter 7). At low substrate concentrations, the reaction rate is proportional to substrate concentration, with the reaction following first-order kinetics in terms of substrate concentration. 

Because the EI complex readily dissociates, the enzyme is again available for substrate binding. The enzyme s activity declines (Figure 6.8) because no productive reaction occurs during the limited time that the EI complex exists. The effect of a competitive inhibitor on activity is reversed by increasing the concentration of substrate. At high [S], all the active sites are filled with substrate, and reaction velocity reaches the value observed without an inhibitor. 

The -primary salt effect deals with the effect of salt concentration on reaction velocity when the reacting system involves no equilibria which can be displaced by a change in ionic environment. This effect can be very large when both the reacting species are ions, but it is of less importance in acid-base catalysis, where the substrate is almost always an uncharged molecule. To avoid complications due to secondary salt effects, the primary effect is best studied in catalysis by solutions of strong acids and bases, and there exists a large body of experimental data. Some of the main conclusions are as follows ... 

Harcourt and Esson do not use the name affinity , confining the law of mass action to the effect of concentration on reaction velocity, a procedure followed later by van t Hoff (see p. 591). Todhunter said men abandoned their attempts at explanation and finally acquiesced in the name Affinity, as simply a description of the phenomena without further analysis. ... 

Linear stability analysis describes the behavior of a system at near equilibrium. Hamiltonian dynamics show that classical mechanics is invariant to (—t) and (t). In a macroscopic description of dissipative systems, we use collective variables of temperature, pressure, concentration, and convection velocity to define an instantaneous state. The evolution equations of the collective variables are not invariant under time reversal for the reaction ... 

A reversible chemical reaction in this context is between an enzyme and a substrate (a molecule of biological importance). It has been observed that at low substrate concentration, the reaction velocity is proportional to the substrate concentration and the reaction is first order with respect to substrate. Increasing the concentration of the substrate causes the reaction rate to diminish, and there is a point where it is no longer proportional to the substrate concentration and, indeed, the rate of the enzyme-catalyzed reaction becomes constant and independent of substrate concentration. Here, the reaction is zero-order a zero-order reaction is independent of the concentration of the reactant, so a higher concentration of reactants will not increase the rate of reaction) with respect to the substrate and the enzyme is said to be saturated with substrate (saturation). This effect is described by a process in which the enzyme (E) reacts with substrate (S) to form a complex ES, which then breaks down to regenerate the enzyme and products (P). Both reactions are reversible, with the rate constants that are indicated kj-k4. This reaction has been analyzed to give the following ... 

Each reaction is assumed to reside at equilibrium, and all of the reaction velocities in principle depend on the concentrations and reaction temperatures of all involved components. This is why at an equilibrium state we can state that... 

ILLUSTRATIVE EXAMPLE 12.8 A 2 1 molar mixture of ethylene oxide (A) and water was fed to a 10 liter adiabatic tank flow reactor. The flow rate of the solution was 1000 L/h. The initial concentration of ethylene oxide was measured to be 38.3 gmol/L. The temperature at the reactor outlet was found to be 375°F. The heat of reaction, activation energy, average heat capacity and reaction velocity constant for this system are known ... 

When a sample is injected into the carrier stream it has the rectangular flow profile (of width w) shown in Figure 13.17a. As the sample is carried through the mixing and reaction zone, the width of the flow profile increases as the sample disperses into the carrier stream. Dispersion results from two processes convection due to the flow of the carrier stream and diffusion due to a concentration gradient between the sample and the carrier stream. Convection of the sample occurs by laminar flow, in which the linear velocity of the sample at the tube s walls is zero, while the sample at the center of the tube moves with a linear velocity twice that of the carrier stream. The result is the parabolic flow profile shown in Figure 13.7b. Convection is the primary means of dispersion in the first 100 ms following the sample s injection. 

Assay of Enzymes In body fluids, enzyme levels aie measured to help in diagnosis and for monitoiing treatment of disease. Some enzymes or isoenzymes are predominant only in a particular tissue. When such tissues are damaged because of a disease, these enzymes or isoenzymes are Hberated and there is an increase in the level of the enzyme in the semm. Enzyme levels are deterrnined by the kinetic methods described, ie, the assays are set up so that the enzyme concentration is rate-limiting. The continuous flow analyzers, introduced in the early 1960s, solved the problem of the high workload of clinical laboratories. In this method, reaction velocity is measured rapidly the change in absorbance may be very small, but within the capabiUty of advanced kinetic analyzers. 

Fig. 9. Bubble-wake interactions in a gas—Hquid-soHd reactor (a) soHds concentration profile within bubble-wake domain, where A—A and B—B represent planes through the bubble, vortex, and wake (b) projected impact of interactions on reaction rate as function of particle si2e and Hquid velocity, where (—)...

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