Kinetic rate law, first-order

Reduction of these PHMs took place in experiments containing both Fe(ll) and iron oxide minerals, under anoxic conditions. The transformation of PHMs by surface-bound Fe(ll) generally follows a pseudo-first-order kinetic rate law, expressed by... 

Initiator decomposition reactions usually follow a first order kinetic rate law, which allows the following formal kinetic description of radical formation ... 

As we will see in a moment, such a reaction will always obey a first order kinetic rate law and, therefore, it is convenient to define a first order rate constant for it as... 

Interestingly, in the experiments devoted solely to computational chemistry, molecular dynamics calculations had the highest representation (96-98). The method was used in simulations of simple liquids, (96), in simulations of chemical reactions (97), and in studies of molecular clusters (98). One experiment was devoted to the use of Monte Carlo methods to distinguish between first and second-order kinetic rate laws (99). One experiment used DFT theory to study two isomerization reactions (100). 

If a reaction occurs by this first mechanism, it is commonly termed an SN2 reaction (i.e., substitution, nucleophilic, bimolecular). It represents an example of a simple elementary bimolecular reaction, as we discussed in Section 12.3, and it therefore follows a second-order kinetic rate law ... 

Figure 4.17 Frequency shift as a function of time measured for a gold-coated SAW device exposed to an Na gas stream containing CH3(CH2)isSH at tproximately 25% of its saturation vapor pressure. Leveling of the frequency shift at approximately 1.15 monolayers indicates that the polycrystalline gold Him has a roughness factor of 1.15. The kinetic data fit a simple, first-order Langmuir rate law. (Reprinted with permission. See Ref. [143J. 1991 American Chemical Society.)...

Example. Determine the best value of the pseudo-first-order kinetic rate constant that provides the closest match between the actual rate law and the first-order irreversible rate law. 

The adsorption/desorption equilibrium constant for each component is Kf = 0.25 atm and forward is the kinetic rate constant for the forward chemical reaction on the catalytic surface with units of moles per area per time. The reason that forward has the same units as Ehw is because rate laws for heterogeneous catalysis are written in terms of fractional surface coverage by the adsorbed species that participate in the reaction. Langmuir isotherms are subsequently used to express fractional surface coverage of the reacting species in terms of their partial pressures. The best value for the pseudo-first-order kinetic rate constant is calculated from... 

Figure 15-1 Total pressure dependence of the best pseudo-first-order kinetic rate constant when a first-order rate law approximates a Hougen-Watson model for dissociative adsorption of diatomic A2 on active catalytic sites. Irreversible triple-site chemical reaction between atomic A and reactant B (i.e., 2Acr - - Bcr -> products) on the catalytic surface is the rate-limiting step. The adsorption/desorption equilibrium constant for each adsorbed species is 0.25 atm. ...

It is advantageous to linearize the rate law, given by equations (22-38), because analytical solutions are available for diffusion and chemical reaction within porous catalysts of all geometries when the kinetics are first-order. Consequently, one calculates the effectiveness factor in spherical pellets rather easily after linearization is performed. The best value of the pseudo-first-order kinetic rate constant for irreversible reactions that achieve 100% conversion is... 

Sometimes, even under pseudo-first-order conditions, the kinetic observations do not obey the first-order integrated rate law. This may indicate a number of chemical problems, such as impurities, a nonlinear analytical method or precipitate formation. However, it is also possible that the system is more complex, with parallel and/or successive reactions, as shown in the following system ... 

Kinetic Anaiysis Kinetic studies were conducted by measuring the rate of TCE degraded over time and is modeled based on the pseudo-first-order reaction rate law ... 

The integrated form of the rate law for equation 13.4, however, is still too complicated to be analytically useful. We can simplify the kinetics, however, by carefully adjusting the reaction conditions. For example, pseudo-first-order kinetics can be achieved by using a large excess of R (i.e. [R]o >> [A]o), such that its concentration remains essentially constant. Under these conditions... 

If (A i[X ]/A 2[Y ]) is not much smaller than unity, then as the substitution reaction proceeds, the increase in [X ] will increase the denominator of Eq. (8-65), slowing the reaction and causing deviation from simple first-order kinetics. This mass-law or common-ion effect is characteristic of an S l process, although, as already seen, it is not a necessary condition. The common-ion effect (also called external return) occurs only with the common ion and must be distinguished from a general kinetic salt effect, which will operate with any ion. An example is provided by the hydrolysis of triphenylmethyl chloride (trityl chloride) the addition of 0.01 M NaCl decreased the rate by fourfold. The solvolysis rate of diphenylmethyl chloride in 80% aqueous acetone was decreased by LiCl but increased by LiBr. ° The 5 2 mechanism will also yield first-order kinetics in a solvolysis reaction, but it should not be susceptible to a common-ion rate inhibition. 

Now we shall apply this to different rate laws. For first-order kinetics either of two forms can be used ... 

Second-order kinetics, (a) Derive expressions for the half-time and lifetime of A if the rate law for its disappearance is v = fc[A]2 (b) calculate t]/i and t for the data presented in Section 2.2 (c) calculate the second half-life, t /i(2), i.e., the time elapsed between 50 percent and 75 percent completion, for the same reaction (d) compare fj/2(l) and fi/>(2), and contrast this result with that from first-order kinetics. 

In experiments with [sulfone]o = 3.15 x 10 5 M and excess N2H4, the reaction follows pseudo-first-order kinetics. Values of k vary with [N2H4]. Formulate the rate law and evaluate the constants therein ... 

The reaction follows first-order kinetics when studied at constant [Melm] and [CO]. Formulate the rate law and calculate any constants from the following data obtained at 23 °C in benzene with [CO] = 3.17 X 10 3 M ... 

Note that the first term in the rate law could be written ArisflH2o but if water is the solvent, as we shall assume, its activity is unity. Were one to write the term as ki,[H20], this would be tantamount to adopting a nonconventional standard state for water, which is usually not advisable. With [OH- ] [(CH3)2CHBr], the reaction follows first-order kinetics with... 

Since the slow step involves only the substrate, the rate should be dependent only on the concentration of that. Although the solvent is necessary to assist in the process of ionization, it does not enter the rate expression, because it is present in large excess. However, the simple rate law given in Eq. (10.3) is not sufficient to account for all the data. Many cases are known where pure first-order kinetics are followed, but in many other cases more complicated kinetics are found. We can explain this by taking into account the reversibility of the first step. The X formed in this step competes with Y for the cation and the rate law must be modified as follows (see Chapter 6) ... 

A, obtained if the disinfection process obeyed the first-order kinetic law. B, sigmoid curve. This shows a slow initial rate of kill, a steady rate and finally a slower rate of kill. This is the form of curve most usually encountered. C, obtained if bacteria are dying more quickly than first-order kinetics would predict. The constant, K, diminishes in value continuously during the process. 

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