First-Order Reaction Kinetics

Referring back to the rate equation for a first-order reaction (Equation A1.2), we have a differential equation for which the derivative of the variable ([S]) is proportional to the variable itself. Such a system can be described by an infinite series with respect to time  

If we set t= 1, this infinite series converges to the value 2.718271.. .. This number is a universal constant of nature (analogous to it) and is given the special symbol e. Thus the infinite series can be expressed as 

As an illustration of first-order kinetics, let us consider the simple dissociation of a binary enzyme-inhibitor complex ( 7) to the free enzyme (E) and the free inhibitor (/), 

Here k, is the rate constant for this dissociation. By the law of mass action, we know that the rate of dissociation will be directly proportional to the concentration of El complex, with -k, being the constant of proportionality (the minus sign denotes the fact that the concentration of El is diminishing over time). Thus the rate equation for this dissociation reaction is given by 

Thus a plot of ln([ 7]t/[ 7]0) as a function of time will be linear with a slope of 

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For combustion of simple hydrocarbons, the oxidation reactions appear to foUow classical first-order reaction kinetics sufficiently closely that practical designs can be estabUshed by appHcation of the empirical theory (8). For example, the general reaction for a hydrocarbon ... 

The second type of coalescence arises from the rupture of films between adjacent bubbles [Vrij and Overbeek, y. Am. Chem. Soc., 90, 3074 (1968)]. Its rate appears to follow first-order reaction kinetics with respect to the number of bubbles [New, Proc. 4th Int. Congr. Suif. Active Substances, Brussels, 1964, 2, 1167 (1967)] and to decrease with film thickness [Steiner, Hunkeler, and Hartland, Trans. In.st. Chem. Fng., 55, 153 (1977)]. Many factors are involved [Biker-man, Foams, Springer-Verlag, New York, 1973 and Akers (ed.). Foams, Academic, New York, 1976]. 

Fig. 5.21. First-order reaction kinetics mechanism without inhibitions.

The results of x-ray structure analysis and neutron diffraction, as well as spectroscopic experiments (J(HSi) = 70.8 Hz for 30), can be interpreted in the sense mentioned above. The observed reactivity of 30 is also consistent with this view, the coordinated silanes can be displaced smoothly by phosphines, according to first-order reaction kinetics. 

The more usual pattern found experimentally is that shown by B, which is called a sigmoid curve. Here the graph is indicative of a slow initial rate of kill, followed by a faster, approximately linear rate of kill where there is some adherence to first-order reaction kinetics this is followed again by a slower rate of kill. This behaviour is compatible with the idea of a population of bacteria which contains a portion of susceptible members which die quite rapidly, an aliquot of average resistance, and a residue of more resistant members which die at a slower rate. When high concentrations of disinfectant are used, i.e. when the rate of death is rapid, a curve ofthe type shown by C is obtained here the bacteria are dying more quickly than predicted by first-order kinetics and the rate constant diminishes in value continuously during the disinfection process. 

The thermal degradation of anthocyanins, both in extracts and model systems, was reported to follow first-order reaction kinetics in all studies. The stability of anthocyanins and all pigments found in foods decreased with increases in temperature. 

Eqnation (27) shows that the eonversion for Case II obeys the first-order reaction kinetics. 

Adsorption and desorption. The user can choose to handle this using either temperature-corrected first order reaction kinetics, in which case the concentrations are always moving towards equilibrium but never quite reach it, or he can use a Freundlich isotherm, in which instantaneous equilibrium is assumed. With the Freundlich method, he can elect either to use a single-valued isotherm or a non-single-valued one. This was included in the model because there is experimental evidence which suggests that pesticides do not always follow the same curve on desorption as they do on adsorption. 

Adsorption and desorption between the solution phase and sand, silt and clay in suspension and on the bed. First order reaction kinetics are used. 

It has been found that both the anhydrous Form III and dihydrate phases of carbamazepine exhibit fluorescence in the solid state [78]. The fluorescence intensity associated with the dihydrate phase was determined to be significantly more intense than that associated with the anhydrate phase, and this difference was exploited to develop a method for study of the kinetics of the aqueous solution-mediated phase transformation between these forms. Studies were conducted at temperatures over the range of 18 40 °C, and it was found that the phase transformation was adequately characterized by first-order reaction kinetics. The temperature dependence in the calculated rate constants was used to calculate activation energy of 11.2 kCal/ mol (47.4 cal/g) for the anhydrate-to-dihydrate phase conversion. 

Other workers have used the tristimulus parameters to study the kinetics of decomposition reactions. The fading of tablet colorants was shown to follow first-order reaction kinetics, with the source of the illumination energy apparently not affecting the kinetics [49]. The effect of excipients on the discoloration of ascorbic acid in tablet formulations has also been followed through determination of color changes [50]. In this latter work, it was established that lactose and Emdex influenced color changes less than did sorbitol. 

Kinetics of the photooxidation of organic water impurities on illuminated titania surfaces has been generally regarded to be based on the Langmuir-Hinshelwood equation with first-order reaction kinetics vs. initial substrate concentration was established univocally by many authors... 

With first-order reaction kinetics, the rate of metabolism is proportional to the substrate the following relation can be expressed ... 

A continuous flow stirred reactor operates off the decomposition of gaseous ethylene oxide fuel. If the fuel injection temperature is 300 K, the volume of the reactor is 1500 cm3, and the operating pressure is 20 atm, calculate the maximum rate of heat evolution possible in the reactor. Assume that the ethylene oxide follows homogeneous first-order reaction kinetics and that values of the reaction rate constant k are... 

Thus, for an element whose removal from seawater follows first-order reaction kinetics, its MORT is the inverse of its removal rate constant. This relationship predicts that reactive elements should have short residence times. As shown in Figure 21.3, the actual data do demonstrate a linear relationship (r = 0.79, p = 0.00), although a log-log plot is required to cover the several orders of magnitude diversity of MORT and concentrations exhibited by the solutes in seawater. A similar relationship exists between the MORT and the seawater-crustal rock partition coefficient (Ay). The latter is defined as the ratio of the mean seawater concentration of an element to its mean concentration in crustal rocks. Elements with high partitioning coefficients would be expected to have low seawater concentrations. As shown in Figure 21.4, this is seen in the data and... 

As we will see in detail in chapter 11, the production of radiogenic isotopes follows perfect first-order reaction kinetics. 

Hydrogenation experiments were conducted at 400 and 20 MPa for 2 h with a number of mono and bimetallic catalysts, all on an acidic alumina support Rate constants were calculated by assuming first order reaction kinetics for the disappearance of phenanthrene. The values for the rate constants and the results of the GC analysis (phenanthrene and its hydroderivatives) are summarised in Table III. 

First-order reaction kinetics is frequently observed in organic chemistry in the form of an SnI reaction, indicating it is a first-order nucleophilic substitution type. An example is the solvolysis of tcrt-butylbromide at alkaline pH to form t rt-butanol and bromide ion. The reaction probably proceeds in two steps ... 

CHEMICAL KINETICS First-order rate behavior, AUTOPHOSPHORYLATION FIRST-ORDER REACTION KINETICS ORDER OF REACTION HALF-LIFE... 

As mentioned in Section 4, the analysis of rate data resulting from unimolecular reactions is considerably easier than the analysis of such data for bimolecular reactions, and the same is true for pseudounimolecular reactions. Kinetic probes currently used to study the micellar pseudophase showing first-order reaction kinetics are almost exclusively compounds undergoing hydrolysis reactions showing in fact pseudofirst-order kinetics. In these cases, water is the second reactant and it is therefore anticipated that these kinetic probes report at least the reduced water concentration (or better water activity in the micellar pseudophase. As for solvatochromic probes, the sensitivity to different aspects of the micellar pseudophase can be different for different hydrolytic probes and as a result, different probes may report different characteristics. Hence, as for solvatochromic probes, the use of a series of hydrolytic probes may provide additional insight. 

From this work, it is apparent that the inactivation of DPP IV by u and / nitrile does not follow pseudo-first-order reaction kinetics. The inactivation process is dependent principally on inhibitor concentration, and only slightly changes with incubation time. Secondly, the inhibitory potency of inhibitor u and / nitrile is nearly equivalent (for u K, = 6.03 pM and for /K = 7.69 pM), that is, the inhibitors interact with DPP IV relatively little difference in potency. K of both u and / are four to five times lower than Ala-Pro-NHO-Bz(4-N02). Both u and / nitrile exhibit superior inhibitory activity to the previously prepared Ala-Pro-NHO-Bz(4-N02) compound. Surprisingly both the u and / nitrile have superior activity to mechanism based Alai/ [CF=C]-Pro-NHO-Bz inhibitor. 

Assuming first order reaction kinetics, the sorption rate that was determined for adsorption and desorption was 0.187 sec . A reaction rate of 0.187 sec implies a half time of reaction of 3.7 seconds. 

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