Rate expressions initiation

Acceleration period Monomer exponent in rate expression Initiator exponent in rate expression Energy of activation... 

Rate expressions. Initially, to determine how rate equations are developed from a proposed reaction mechanism, we will consider simple reaction mechanisms consisting of only one elementary reaction For example, the differential rate equation for the hypothetical reaction in Equation 1.15... 

A rather different method from the preceding is that based on the rate of dissolving of a soluble material. At any given temperature, one expects the initial dissolving rate to be proportional to the surface area, and an experimental verification of this expectation has been made in the case of rock salt (see Refs. 26,27). Here, both forward and reverse rates are important, and the rate expressions are... 

Before elosing this ehapter, it is important to emphasize the eontext in whieh the transition rate expressions obtained here are most eommonly used. The perturbative approaeh used in the above development gives rise to various eontributions to the overall rate eoeffieient for transitions from an initial state i to a final state f, these eontributions inelude the eleetrie dipole, magnetie dipole, and eleetrie quadrupole first order terms as well eontributions arising from seeond (and higher) order terms in the perturbation solution. 

Any one of these expressions gives the rate of initiation Rj for the particular catalytic system employed. We shall focus attention on the homolytic decomposition of a single initiator as the mode of initiation throughout most of this chapter, since this reaction typifies the most widely used free-radical initiators. Appropriate expressions for initiation which follows Eq. (6.6) are readily derived. 

This relationship permits the measurement of the ratio k /ky. The initial concentrations [B—X]o and [B—Y]o are known from the conditions of the experiment The reaction can be stopped at some point when some of both B—X and B—Y remain unreacted, or an excess of B—X and B—Y can be used so that neither is completely consumed when A—A has completely reacted. Determination of [B—X], and [B—Y], then provides the information needed to calculate k /ky. It is clear that the reactions being compared must be of the same order. If they were not, division of the two rate expressions would leave uncanceled concentration terms. 

Table 3-11 gives the initial rate data [-d(B2Hg)/dt] reported for the gas phase reaetion of diborane and aeetone at 114°C BjHg -i-dMejCO —> 2(Me2CHO)2BH. If a rate expression is of the form Rate = PmcjCO determine n, m, and k. 

Briggs and Haldane [8] proposed a general mathematieal deseription of enzymatie kinetie reaetion. Their model is based on the assumption that after a short initial startup period, the eoneentration of the enzyme-substrate eomplex is in a pseudo-steady state (PSS). Eor a eonstant volume bateh reaetor operated at eonstant temperature T, and pH, the rate expressions and material balanees on S, E, ES, and P are... 

One way to find the order of a reaction is to measure the initial rate (Le., the rate at t = 0) as a function of the concentration of reactant Suppose, for example, that we make up two different reaction mixtures differing only in the concentration of reactant A. We now measure the rates at the beginning of reaction, before the concentration of A has decreased appreciably. This gives two different initial rates (rate , rate2) corresponding to two different starting concentrations of A, [A]x and [A]2. From the rate expression,... 

Our initial experimental results indicated that the kinetic model— first order in liquid phase CO concentration—was the leading candidate. We designed an experimental program specifically for this reaction model. The integrated rate expression (see Appendix for nomenclature) can be written as ... 

In a gas-liquid contactor, a pure gas is absorbed in a solvent and the Penetration Theory provides a reasonable model by which to describe the transfer mechanism. As fresh solvent is exposed to the gas, the transfer rate is initially limited by the rate at which the gas molecules can reach the surface. If at 293 K and a pressure of 1 bar the maximum possible rate of transfer of gas is 50 m3/m2s, express this as an equivalent resistance, when the gas solubility is 0.04 kmol/m3. 

However, the mechanisms by which the initiation and propagation reactions occur are far more complex. Dimeric association of polystyryllithium is reported by Morton, al. ( ) and it is generally accepted that the reactions are first order with respect to monomer concentration. Unfortunately, the existence of associated complexes of initiator and polystyryllithium as well as possible cross association between the two species have negated the determination of the exact polymerization mechanisms (, 10, 11, 12, 13). It is this high degree of complexity which necessitates the use of empirical rate equations. One such empirical rate expression for the auto-catalytic initiation reaction for the anionic polymerization of styrene in benzene solvent as reported by Tanlak (14) is given by ... 

In this work, the characteristic "living" polymer phenomenon was utilized by preparing a seed polymer in a batch reactor. The seed polymer and styrene were then fed to a constant flow stirred tank reactor. This procedure allowed use of the lumped parameter rate expression given by Equations (5) through (8) to describe the polymerization reaction, and eliminated complications involved in describing simultaneous initiation and propagation reactions. 

As given above, the statements that adjust the exponents m and n have been commented out and the initial values for these exponents are zero. This means that the program will fit the data to. = k. This is the form for a zero-order reaction, but the real purpose of running this case is to calculate the standard deviation of the experimental rate data. The object of the fitting procedure is to add functionality to the rate expression to reduce the standard deviation in a manner that is consistent with physical insight. Results for the zero-order fit are shown as Case 1 in the following data ... 

Use these initial rate data to estimate the constants in the rate expression... 

The appearance of this heterogeneous form for the rate expression reflects the presence of a mass transfer step in series with the reaction step. If the parameter values are known, this ODE for bi i) can be integrated subject to the initial condition that bi=(bi)o at t = 0. The result can then be used to find a (f). 

When testing, the permeation rate is initially measured as change in pressure (E2) in the low-pressure reservoir versus time, to give a plot of the same form as shown in Figure 23.10, but with pressure rather than volume being plotted. To convert to volume, the following expression (referring strictly to an ideal gas) is applied to obtain rate Aq/At from the slope of the plots ... 

The model was verified for initial temperatures ranging from 20 to 40 C and for initial amine to epoxide molar ratios from 0.9 to 1.2. The rate expression shown in Equation 9 was the model ultimately used in the finite element program. This was done to... 

Interestingly, one can easily deduce an expression for the relative rate of coke formation as compared to that of methanation. The rate of initial coke formation depends on the combination probability of carbon atoms and hence is given by... 

LDPE polymerization reaction consists of various elementary reactions such as initiation, propagation, termination, chain transfer to polymer and monomer, p-scission and so forth [1-3], By using the rate expression of each elementary reaction in our previous work [4], we can construct the equations for the rate of formation of each component. 

If the radiation is absorbed by a photosensitizer which decomposes into radicals on photoactivation, or which brings about initiation in some other manner, then Iahs = cCsIo for sufficiently small concentrations Cs of the sensitizer. The rate expression resembles Eq. (12) Rp is proportional to and to... 

Initial Rate Measurements. Another differential method useful in the determination of reaction rate expressions is the initial rate approach. It involves a series of rate measurements at different initial reactant concentrations but restricted to very small conversions of the limiting reagent (5 to 10% or less). This technique differs from those discussed previ-... 

Because the steady-state assumption leads to the equilibrium relation for the bromine atom concentration (4.2.20), it does not matter what mechanism is assumed to be responsible for establishing this equilibrium. Alternative elementary reactions for the initiation and termination processes, which give rise to the same equilibrium relationship, would also be consistent with the observed rate expression for HBr formation. For example, the following reactions give rise to the same equilibrium ... 

In order for the overall rate expression to be 3/2 order in reactant for a first-order initiation process, the chain terminating step must involve a second-order reaction between two of the radicals responsible for the second-order propagation reactions. In terms of our generalized Rice-Herzfeld mechanistic equations, this means that reaction (4a) is the dominant chain breaking process. One may proceed as above to show that the mechanism leads to a 3/2 order rate expression. 

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