General expression for the rate

Two-process models nucleation and growth 10.3.1. General expression for the rate 

We consider now that at any time there are simultaneous processes of nucleation on the free surface and growth of the nuclei formed earlier. 

Consider a nucleus G bom at r time. At t time later than t, its speed of growth will be (r, t). But between times x and x + 5x, not only a nucleus was bom but r(x) dt (if r(x) indicates the frequency of nucleation or number of nuclei manufactm-ed per unit of time). At time t, the reaction speed due to the growth of all these nuclei will thus be  

Numerous nuclei were bom at any time between 0 and f therefore, to express the rate of the reaction, it is necessary to integrate the preceding expression, that is. 

In this expression, rg(r, f) is the contribution of a nucleus bom at the x time to the speed of growth. 

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The reaction order is equal to the exponent of the concentration in the rate law. A general expression for the rate law can be written using the standard reaction A + B —> products, where A and B represent reactants. The general rate law for this expression can be written as... 

Obtain a general expression for the rate at which entropy is produced the reactions are taking place in a container maintained at constant temperature and pressure. 

Therefore, in the general expression for the rate of reaction on heterogenous surfaces... 

At low temperatures (/3ho)c>l) the contributions from one-phonon, two-phonon processes, etc., can be systematically extracted from the general expression for the rate constant, and the type of the dominant process is determined by the bath spectrum and temperature. The results of Leggett et al. [1987] show that quantum dynamics of a TLS crucially depends on the spectrum of the bath. For sufficiently strong coupling, the bath may dramatically slow the tunneling rate, even to the point of localizing the particle in one of the wells. This strong dependence on the bath spectrum is inherent to the quantum dynamics and does not show up in classical transitions. 

In the preceding sections we derived an approximate expression for the thermal rate constant k(T). These derivations were not based on the general expressions for the rate constant that were derived in the first chapters. We consider here a derivation of the TST result that is based on an exact expression for a bimolecular rate constant. 

It will therefore be natural to first derive a general expression for the rate constant that includes the possibility that the structure of the activated complex is different from the structure in the gas phase, and then specialize this result to the case where it is assumed that the structure will not be perturbed. We will see that the general result does not allow us to extract a simple physical picture of the solvent effect on the rate constant, whereas more insight may be obtained with the simplifying assumption that the structure of the complex is not perturbed. 

Having established that the general expression for the rate constant in Eq. (10.40) simplifies to the well-known expression from the gas phase in the absence of a solvent, let us then write down a general expression for the relation between the two rate constants, when the activated complex is unperturbed by the solvent. We find, using Eqs (10.40) and (10.51),... 

The general expression for the rate constant (70) transforms to expression (75) at AE — 0, if the following limitation on the tunneling distance rDA is realized ... 

X A + B — 60 Kcal, orX— C-f-D —50 Kcal. (a) Show that the existence of the low-energy path, reaction 2, makes the high-energy reaction 1 less likely to happen. (6) By using the RRK model, derive a general expression for the rates of the two reactions. 

As we discussed in Sect. 3.1.1, Hansen et al. [15] made significant improvements to the concept of the radical capture efficiency proposed by Nomura et al. [ 14]. Taking this concept into consideration, they examined the effect of radical desorption on micellar particle formation in emulsion polymerization [ 65 ]. Assuming that radical entry is proportional to the x power of the micelle radius and the polymer particle radius, they proposed the following general expression for the rate of particle formation ... 

The relative rates of the reactions leading to formation of either ground-state or excited-state products can be evaluated in terms of formalisms developed by Marcus [26], Hopfield [27], Jortner [28], and others [29]. The development of the semiclassical and quantum-mechanical expressions for electron transfer are discussed in Chapters. 3-5 (Volume I, Part 1). A general expression for the rate constant of a non-adiabatic electron-transfer process is given below. 

Usually, however, Z, is dispersed as droplets (phase h) and the rate of transport from the droplets to the bulk of the continuous phase (Kn,) must he taken into account. Setting = Kk, the following general expression for the rate of swelling is obtained (Ugelstad et a ., 1980a). 

This is a general expression for the rate law. The exponent, n, is called the order of the reaction. It is usually a whole number, often 1 or 2, but it could be a fraction. Occasionally, n equals 0, which means that the reaction rate is independent of the reactant concentration. The term k is the rate constant, a proportionality constant that varies with temperature. 

Since estimation of the rate of reaction from the experimental data involves amplification of experimental error, an improved estimate of the rate constant is usually sought using an integrated version of the general expression for the rate of reaction once it is known. For a first-order reaction (n = 1), integration of equation (7.2.2) gives the result... 

Martinov and MuUer reported a general expression for the rate constants of the reverse process ... 

The role of hindered rotation in the breakdown of the BO approximation can, in fact, be analyzed in considerable detail. It can be shown that the molecular mode whose equilibrium positions differ greatly in the initial and final electronic states (the strongly coupled degree of freedom) may be separated out from the other vibrations in the general expression for the rate of reaction. It is then found that, under conditions appropriate to the... 

This creates a generalized expression for the rate.of reaction carried out at any pressure P but still at a constant volume. The expression can be used- as- it stands for constant volume reactions. Since volume contraction occurs in this reaction, we next have to account for it. In this connection we are not so much interested in the change in the output mol fraction of CO (yco) as we are in the fraction of moles at the input that has been converted. We introduce this by rewriting the rate in terms of Xco> the fraction of the original CO converted (see equation 7.60). This, after some rearrangement, leads to the equation ... 

In the case of vinyl chloride, the kinetics of miniemulsion polymerization is complicated by the fact that almost from the beginning of the reaction, two reaction zones inside the droplets need to be considered (1) a practically pure monomer phase of decreasing volume, and (2) a precipitated, monomer-swollen polymer phase which increases in volume with conversioa The general expression for the rate of polymerization (moles dm H2O) is... 

In a system which displays yield stress behaviour, the integral in the general expression for the rate of shear need not be evaluated from the bob all the way to the cup. This is due to the fact that, for such a system, no shearing takes place where t is less than the yield value, tq. Thus the integral need only be evaluated from the bob to the critical radius, the radius at which T = To. This gives... 

One can notice that the general expressions for the rate coefficients depend on the cross-section of corresponding reactions as well as on the distribution functions, and, consequently, on the... 

In the past it has been a standard practice to derive a simple but general expression for the rate of polsmierization, Rp. This expression correlates the rate of polymerization with the initiator and monomer concentrations on one side and the... 

If we consider more carefully the definition of the A-1 process (equation 1 and 2) and its general expression for the rate equation (equation 5"), we may easily realize (see equation 9, where S g stands for the stoicheiometric concentration of the substrate) that... 

Let us start by considering the general expression for the rate of relaxation for a quadrupolar nucleus under extreme narrowing conditions (eq. 4). We have discussed already the... 

This equation can be substituted into Equation (2.29) to give a general expression for the rate of monomer consumption which more commonly is called the rate of polymerization and signified by Rp... 

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