First-Order Rate Equations

Activation Parameters. Thermal processes are commonly used to break labile initiator bonds in order to form radicals. The amount of thermal energy necessary varies with the environment, but absolute temperature, T, is usually the dominant factor. The energy barrier, the minimum amount of energy that must be suppHed, is called the activation energy, E. A third important factor, known as the frequency factor, is a measure of bond motion freedom (translational, rotational, and vibrational) in the activated complex or transition state. The relationships of yi, E and T to the initiator decomposition rate (kJ) are expressed by the Arrhenius first-order rate equation (eq. 16) where R is the gas constant, and and E are known as the activation parameters. 

Sets of first-order rate equations are solvable by Laplace transform (Rodiguin and Rodiguina, Consecutive Chemical Reactions, Van Nostrand, 1964). The methods of linear algebra are applied to large sets of coupled first-order reactions by Wei and Prater Adv. Catal., 1.3, 203 [1962]). Reactions of petroleum fractions are examples of this type. 

Separating the variables and integrating between the limits shown below yields Eqs. (2-6), (2-7), and (2-8) as equivalent forms of the integrated first-order rate equation. 

It frequently happens that we plot or analyze data in terms of quantities that are transformed from the raw experimental variables. The discussion of the propagation of error leads us to ask about the distribution of error in the transformed variables. Consider the first-order rate equation as an important example ... 

The reactant A follows a first-order rate equation, hence. 

We now consider the solution of differential equations by means of Laplaee transforms. We have already solved one equation, namely, the first-order rate equation, but the technique is capable of more than this. It allows us to solve simultaneous differential equations. 

To develop the ideas we take the simple first-order rate equation as an example, namely, dddt = —kc. Let c = y, t = x, so... 

The reversible hydration of nitrogen-containing heterocyclic bases and their hydroxy and mercapto derivatives is acid-base catalyzed and, at constant pH, the reactions obey first-order rate equations. 2 26.27-33... 

For Sa << Km, the rate model is reduced to a first-order rate equation The Michaelis-Menten rate equation is ... 

The reaction rate for simple fermentation systems is normally given by the Monod equation. This model indicates that the specific conversion rate is constant when applied to an immobilised cell system (Table 8.7). If a first-order rate equation for sugar consumption is used, (8.7.4.2) yields ... 

It is worthwhile to first review several elementary concepts of reaction rates and transition state theory, since deviations from such classical behavior often signal tunneling in reactions. For a simple unimolecular reaction. A—>B, the rate of decrease of reactant concentration (equal to rate of product formation) can be described by the first-order rate equation (Eq. 10.1). 

If one were to mix fixed concentrations E and / at time zero and then measure the concentration of El complex as a function of time after mixing, the data would appear to be described by the pseudo-first-order rate equation ... 

Thus for hydrolysis in 50% aqueous acetone, a mixed second and first order rate equation is observed for phenylchloromethane (benzyl chloride, 10)—moving over almost completely to the SV1 mode in water alone. Diphenylchloromethane (11) is found to follow a first order rate equation, with a very large increase in total rate, while with triphenylchloromethane (trityl chloride, 12) the ionisation is so pronounced that the compound exhibits electrical conductivity when dissolved in liquid S02. The main reason for the greater promotion of ionisation—with consequent earlier changeover to the SW1 pathway in this series—is the considerable stabilisation of the carbocation, by delocalisation of its positive charge, that is now possible ... 

The integrated first order rate equation lnrI = akt... 

Assume the initial amount of carbon-11 is 100%. Calculate the time required for decay, t, using the first order rate equation. 

In the simplest case a 1 1 complex is formed between the host in solution and the guest immobilized on the surface. The response of the SPR sensor, R, is proportional to the concentration of the complex formed, and thus pseudo-first-order rate equations can be used to analyze the data.73 If no host is initially bound the function for R... 

For systems where the mineralization rate is dependent on the substrate concentration, first-order kinetics can be used to describe carbon turnover (Paul and Clark 1989 Six and Jastrow 2002). The first-order rate equation is... 

For first-order rate equations, the MRT (MRT = —) and half-fives can be calculated... 

The concentrations of the reactants are proportional to (11.08-K). Accordingly the first order rate equation... 

Reactor volume was 0.276 liters. Data are presented of P in torr, na0 mols/sec of entering reactant, y = nd/na0 mols carrier gas per mol of reactant, and x fractional conversion. Check a first order rate equation. 

Deduce the first order rate equation. Cb) How much acetochloranilid would be expected after 8 hours The amount actually found was 4.8 gms. What could be the reason for the discrepancy ... 

Bethea Karchmer, Ind Eng Chem 48 370, 1956). Composition of the charge was 58% propylene, 41% propane, 0.5% butylene, 0.5% butane. On an average, 1 mol of polymer required 3.4 mols of propylene. The results at 360 F are tabulated. Check a first order rate equation. 

Equation (8.24) is the integrated first-order rate equation. Being a logarithm, the left-hand side of Equation (8.24) is dimensionless, so the right-hand side must also be dimensionless. Accordingly, the rate constant k will have the units of s-1 when the time is expressed in terms of the SI unit of time, the second. 

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