Zero-order rate equation

THE CHANGE OF CONCENTRATION WITH TIME We learn that rate equations can be written to express how concentrations change with time and look at several examples of rate equations zero-order, first-order, and second-order reactions. 

Figure 11.1 illustrates the behavior of Equation 11.6. By the assumption of rapid equilibrium the rate determining step is the unimolecular decomposition. At high substrate composition [S] KM and the rate becomes zero-order in substrate, v = Vmax = k3 [E0], the rate depends only on the initial enzyme concentration, and is at its maximum. We are dealing with saturation kinetics. The most convenient way to test mechanism is to invert Equation 11.6... 

We now determine the system parameters by evaluating Eq. (64). First, although it is not necessary to limit our considerations to the saturated-surface, zero-order case, we do so to simplify the analysis of high-conversion systems. [We earlier assumed in connection with Eq. (61) that the surface is saturated and that a is constant.] Equation (62) indicates that the total rate is zero order when (kj -I- k3 -I- ks) is small in comparison to the rest of the denominator. Thus, since b = k2 + k + k )/L, h = 0 in the zero-order case, and the (b/P x) term can be removed from Eq. (64). 

CHEMICAL KINETICS RATE SATURATION MICHAELIS-MENTEN EQUATION ZERO-ORDER REACTIONS ORDER OF REACTION MOLECULARITY... 

You must choose the form of the rate-law expression or the integrated rate equation —zero, first, or second order—that is appropriate to the order of the reaction. These are summarized in Table 16-2. One of the following usually helps you decide. 

The reaction was carried out by dissolving gaseous HCl in a stirred vessel containing the alcohol. The resulting concentration-time data could be correlated with a rate equation half-order in alcohol concentration. However, the rate constant was found to vary with the gas (HCl) flow rate into the reactor, suggesting that the observed rate was influenced by the resistance to diffusion of dissolved HCl in the liquid phase. A method of analysis which took into account the diffusion resistance indicated that the chemical step was probably first order in dissolved HCl and zero order with respect to lauryl alcohol. 

The location of the maximum rate point (zero order dependence point) and its value are highly affected by the rate equation parameters. A parametric study has shown that ... 

Hence at high peroxide concentration when. KIH2O2]2 ) > 1 the rate is zero order. The fall off in rate at the lower peroxide concentrations also follows from this equation. Analogous equations can be written for HMoCV-... 

We now discuss the lifetime of an excited electronic state of a molecule. To simplify the discussion we will consider a molecule in a high-pressure gas or in solution where vibrational relaxation occurs rapidly, we will assume that the molecule is in the lowest vibrational level of the upper electronic state, level uO, and we will fiirther assume that we need only consider the zero-order tenn of equation (BE 1.7). A number of radiative transitions are possible, ending on the various vibrational levels a of the lower state, usually the ground state. The total rate constant for radiative decay, which we will call, is the sum of the rate constants,... 

This equation describes the steady-state, or zero-order, release of the dmg. When the dmg completely dissolves, its concentration within the system begins to dilute, and the release rate foUows a parabohc decline with time (102). Acutrim (ALZA Corp.), dehvering phenylpropanolamine hydrochloride [154-41 -6] for appetite suppression, is an example of an elementary osmotic pump. 

Related to the preceding is the classification with respect to oidei. In the power law rate equation / = /cC C, the exponent to which any particular reactant concentration is raised is called the order p or q with respect to that substance, and the sum of the exponents p + q is the order of the reaction. At times the order is identical with the molecularity, but there are many reactions with experimental orders of zero or fractions or negative numbers. Complex reactions may not conform to any power law. Thus, there are reactions of ... 

Consider a reaetion involving a reaetant sueh dial A —> produets. The rate equations eorresponding to a zero, first, seeond, and third order reaetion together with their eorresponding units are ... 

This means that the eonversion is proportional to time. Eigure 3-4 shows plots of the zero order rate equations. Examples of zero order reaetions are the intensity of radiation within the vat for photoehemieal reaetions or the surfaee available in eertain solid eatalyzed gas reaetions. 

Improbable as a zero-order reaction may seem on the basis of what has been said thus far, let us consider the possibility of this rate equation ... 

Integration between the limits of c = c° when r = 0 and c = c when t = t gives the integrated zero-order rate equation. 

We can reach two useful conclusions from the forms of these equations First, the plots of these integrated equations can be made with data on concentration ratios rather than absolute concentrations second, a first-order (or pseudo-first-order) rate constant can be evaluated without knowing any absolute concentration, whereas zero-order and second-order rate constants require for their evaluation knowledge of an absolute concentration at some point in the data treatment process. This second conclusion is obviously related to the units of the rate constants of the several orders. 

Linear differential equations with constant coefficients can be solved by a mathematical technique called the Laplace transformation . Systems of zero-order or first-order reactions give rise to differential rate equations of this type, and the Laplaee transformation often provides a simple solution. 

The rate equation is first-order in acetone, first-order in hydroxide, but it is independent of (i.e., zero order in) the halogen X2. Moreover, the rate is the same whether X2 is chlorine, bromine, or iodine. These results can only mean that the transition state of the rds contains the elements of acetone and hydroxide, but not of the halogen, which must enter the product in a fast reaction following the rds. Scheme VI satisfies these kinetic requirements. 

Throughout this section the hydronium ion and hydroxide ion concentrations appear in rate equations. For convenience these are written [H ] and [OH ]. Usually, of course, these quantities have been estimated from a measured pH, so they are conventional activities rather than concentrations. However, our present concern is with the formal analysis of rate equations, and we can conveniently assume that activity coefficients are unity or are at least constant. The basic experimental information is k, the pseudo-first-order rate constant, as a function of pH. Within a senes of such measurements the ionic strength should be held constant. If the pH is maintained constant with a buffer, k should be measured at more than one buffer concentration (but at constant pH) to see if the buffer affects the rate. If such a dependence is observed, the rate constant should be measured at several buffer concentrations and extrapolated to zero buffer to give the correct k for that pH. 

Schmid s observation of the dependence of the reaction rate on the square of the concentration of nitrous acid was interpreted by Hammett (1940, p. 294) as due to the rate-limiting formation of dinitrogen trioxide, N203. The consequent attack of the amine by N203 was postulated to be faster therefore the concentration of the amine has no influence on the overall rate (zero order with respect to amine). Similarly, Hammett regards the second factor of Schmid s equation for diazotization in the presence of hydrochloric or hydrobromic acid as the result of the formation of nitrosyl halide. 

Hofer et al. [671] observed that the decompositions of Ni3C and Co2C (the iron compounds melt) obeyed the zero-order equation for 0.3 < a < 0.9 (596-628 K and E = 255 kJ mole-1) and 0.2 < a < 0.75 (573-623 K and E = 227 kJ mole-1), respectively. The magnitudes of the rate coefficients for the two reactions were closely similar but the nickel compound exhibited a long induction period and an acceleratory process which was not characteristic of the reaction of the cobalt compound. Decomposition mechanisms were not discussed. 

The reaction of Si02 with SiC [1229] approximately obeyed the zero-order rate equation with E = 548—405 kJ mole 1 between 1543 and 1703 K. The proposed mechanism involved volatilized SiO and CO and the rate-limiting step was identified as product desorption from the SiC surface. The interaction of U02 + SiC above 1650 K [1230] obeyed the contracting area rate equation [eqn. (7), n = 2] with E = 525 and 350 kJ mole 1 for the evolution of CO and SiO, respectively. Kinetic control is identified as gas phase diffusion from the reaction site but E values were largely determined by equilibrium thermodynamics rather than by diffusion coefficients. 

Finally, although rare, we mention the occurrence of zero-order reactions. The special case of a pseudo-zero order reaction arises if a reactant is present in large excess, and the reaction does not noticeably change the concentration of the reactant. The differential and integral rate equations for a zero-order reaction R —> P are... 

In kinetics, reactions are classified as being first, second, third, etc. order depending on the way the rate of the reaction is related to the concentration terms in the rate equation. If the rate of reaction is apparently independent of concentration, the reaction is said to be of zero order. 

So far, what has been examined is the effect of the concentrations of the reactants and the products on the reaction rate at a given temperature. That temperature also has a strong influence on reaction rates can be very effectively conveyed by considering the experimentally found data on the formation of water from a mixture of hydrogen and oxygen. At room temperature the reaction will not take place hence the reaction rate is zero. At 400 °C it is completed in 1920 h, at 500 °C in 2 h, and at 600 °C the reaction takes place with explosive rapidity. In order to obtain the complete rate equation, it is also necessary to know the role of temperature on the reaction rate. It will be recalled that a typical rate equation has the following form ... 

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