Second-order rate expression

Preequilibria and rate law. The rate of isotopic exchange between U(IV) and U(VI) follows a second-order rate expression at constant [H+] 24... 

Greater success in extending kinetic measurements to higher degrees of polymerization has been achieved with polyesterifications catalyzed by a small amount of a strong acid catalyst. The catalyst concentration being constant throughout the process, the second-order rate expression... 

For Class I second-order rate expressions, combining equations 3.0.7, 3.1.2, and 3.1.9 gives... 

For Class II second-order rate expressions of the form of equation 3.1.10, the rate can be expressed in terms of the extent of reaction per unit volume as... 

Class II second-order rate expressions are one of the most common forms one encounters in the laboratory. They include the gas phase reaction of molecular hydrogen and iodine (H2 + I2 -> 2HI), the reactions of free radicals with molecules (e.g., H -f Br2 -> HBr -f Br), and the hydrolysis of organic esters in nonaqueous media. 

Slight modifications of the above procedure are often employed to reduce the numerical calculations required. For example, the value of corresponding to a Class II second-order rate expression is... 

In some cases one attempts to cause a simplification of the rate expression by using stoichiometric ratios of reactants. For example, a mixed second-order rate expression (r = kCACB) becomes a Class I second-order rate expression (r = kCA2) if a stoichiometric mixture of A and B is used. If, by mistake, a non-stoichiometric mixture is used, positive or negative deviations can be observed, depending on which species is present in excess. 

Since an elementary reaction occurs on a molecular level exactly as it is written, its rate expression can be determined by inspection. A unimolecular reaction is first-order process, bimolecular reactions are second-order, and termolecular processes are third-order. However, the converse statement is not true. Second-order rate expressions are not necessarily the result of an elementary bimolecular reaction. While a... 

Loukidou et al. (2005) fitted the data for the equilibrium sorption of Cd from aqueous solutions by Aeromonas caviae to the Langmuir and Freundlich isotherms. They also conducted, a detailed analysis of sorption rates to validate several kinetic models. A suitable kinetic equation was derived, assuming that biosorption is chemically controlled. The so-called pseudo second-order rate expression could satisfactorily describe the experimental data. The adsorption data of Zn on soil bacterium Pseudomonas putida were fit with the van Bemmelen-Freundlich model (Toner et al. 2005). 

This problem may be solved by linear regression using equations 3.4-11 (n = 1) and 3.4-9 (with n = 2), which correspond to the relationships developed for first-order and second-order kinetics, respectively. However, here we illustrate the use of nonlinear regression applied directly to the differential equation 3.4-8 so as to avoid use of particular linearized integrated forms. The method employs user-defined functions within the E-Z Solve software. The rate constants estimated for the first-order and second-order cases are 0.0441 and 0.0504 (in appropriate units), respectively (file ex3-8.msp shows how this is done in E-Z Solve). As indicated in Figure 3.9, there is little difference between the experimental data and the predictions from either the first- or second-order rate expression. This lack of sensitivity to reaction order is common when fA < 0.5 (here, /A = 0.28). 

For the complex reaction with stoichiometry A + 3B 2R + S and with second-order rate expression... 

Example 17.4 gives the final rate expression for film mass transfer followed by a second-order rate expression for reaction on a plane surface. Please derive this expression and show that it is correct. 

For bimolecular reactions involving two molecular reactants, e.g., 03 + alkenes, concentration-time profiles of both reactants can often be recorded simultaneously, and the data can be treated using the integrated second-order rate expressions to derive the corresponding rate constants ... 

The second model was discarded because the value of the rate constant k2 found by non-linear regression of the data was approximately zero which was not considered reasonable on a physical basis. Of the other two models, the second order rate expression was found to fit the data better. For denitrogenation only the first and second order models were examined, and in this case also, the latter model was found to represent the data better. 

Equation (2) is known as second order rate expression. 

Although Diels-Alder polymerizations involve a reaction of an unsaturated molecule, and the polymer does not have a structure typical of a condensation polymer, the characteristics of the polymerization reaction are those of a condensation polymerization. The kinetics of a Diels-Alder polymerization should be those of a typical condensation polymerization and involve a series of individual reactions, and not a chain type mechanism. A typical second-order rate expression, -dC/dt = kC2 should hold for a good portion of the polymerization reaction, where C is the concentration of both diene and dienophile ends. Typically, this rate would break down only when the molecular mobility is extremely low or when shielding of functional groups occurs in dilute solutions. 

Both interchange (I) and dissociative (D) pathways represented by schemes (7.1), and (7.2) frequently lead to second-order rate expressions first-order in complex and first-order in incoming or attacking ligand. In a dilute solution, the observed second-order rate constant for the interchange path (7.1) is... 

Product oil or syncrude is collected in a chilled receiver with light gas collected in a glass receiver usually by the displacement of water. Hie syncrude and gas are analyzed chrcmatographically and the percent carbon on the spent catalyst is determined instrumentally. Conversion (weight percent) is defined as 100 minus (weight percent light cycle oil plus heavy cycle oil) on a weight % of feed basis. A kinetic term called activity is expressed as a simple second order rate expression defined as conversion/ (100-conversion). ... 

Fig. 8.17 Competition kinetic scheme of the reaction of hydroxyl radicals with a substrate M in the presence of scavengers S/ and the corresponding second order rate expressions.

Char combustion kinetics have been previously reported for Antrim shale by Rostam-Abadi and Mickelson (9). In that study the authors reported that the rate was second order with respect to the char remaining and that there was noticeable chemisorption of (>2 Attempts to fit our data for the Antrim shale to a second order rate expression were unsuccessful and, in all cases, the data appeared to follow first order kinetics. Although we did not have the precision to measure O2 chemisorption, this phenomenon is consistent with our previous observations (6 ) of catalytic activity in those shales containing decomposed mineral carbonates. That is, the catalytic activity of CaO was attributed to its ability to chemisorb 02 As will be discussed in more detail below, the Antrim shale sample did not contain such carbonates and no catalytic behavior was observed. However, the magnitude of the rate constants reported by Rostam-Abadi and Mickelson (9) are very similar to those measured here. 

Scheme 4. Second order rate expressions for the interchange of aryls between two different triarylphosphines or between molecules of a non-symmetrically substituted triarylphosphine.

The second-order rate expression in terms of conversion is... 

In other words, the second-order rate expression, which is first order in A and B, may appear to be first order in A only if the experiments were done with a large excess of B present and its change in concentration went imdetected Let us see how this works out in the next section, by considering second-order kinetics for the same reaction. 

Look carefully at this equation because it is misleading. Our goal was to nondimensionalize it. Therefore we expect the accumulation term, that is, the LHS, to be dimensionless. But how can it be The RHS clearly is not dimensionless, and it has units of concentration This is the key to seeing where we went wrong our error was in assuming that the rate constant k for this second-order rate expression had the same units as those used for the first-order system. It does not. The dimensions for this second-order rate are vol/mol/tim in other words, inverse time and inverse concentration. Why Because the accumulation term on the LHS must have the same dimensions of mol/volume/time regardless of the order or complexity of the rate expression... 

In reacting ammonia with organic compounds it is usually the practice to use a substantial excess of ammonia. Under this condition there is no appreciable change in concentration of ammonia compared to the other reactant, and the rate expression for the process becomes pseudo first order. Actually, the reaction is bimolecular in nature and involves the reaction of one molecule of ammonia with one molecule of the other reactant so tliat the second-order rate expression is more indicative of the actual mechanism. 

While the second-order rate expression was applied to these reactions, there was much better agreement in the values of the rate constant through... 

Green and blue-green algae are capable of efficiently catalyzing the removal of trace hydrogen peroxide from water. A second-order rate expression describes the removal. The median second-order... 

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