Rate expression

The reactions I and III are endothermic while the water-gas shift reaction II is weakly exothermic. The above rate equations cannot be used when the concentration of hydrogen is zero because then the rate expressions become infinite. Hence the presence of hydrogen is necessary for these equations to be applicable. From a technical point of view, the feed cannot be hydrogen free in order to reduce any Ni oxide with the packed catalyst and therefore the division by the partial pressure of hydrogen pn2 in equation (7.124) does not cause any practical problems. 

12Martin Hans Christian Knudsen, Danish physicist, 1871-1949 13Anders Jonas Angstrom, Swedish physicist, 1814-1874 

The viscosity correlations and specific heats are readily available in the literature. 

The overall heat-transfer coefficient U between the catalyst tubes and their surroundings is given by the equation 

Here Xg denotes the process gas conductivity in kJ/(m h C), Rep is the Reynolds14 number based on the equivalent particle diameter. It is equal to Rep = g Dp/p, where Pr is the Prandtl15 number Pr = Cp p/Xg, Xer is the effective conductivity of the catalyst bed measured in kJ/ (m-h - C), and A°r is the static contribution of the effective conductivity of the catalyst bed with the value 

With a further assumption, we can define two useful kinetic regions. If k dif kel, then kq = Kdifkel, where Kdif is the equilibrium constant for formation of the encounter complex, i.e., Kdif = kdif/k dif. In this region, kcl values can be estimated from Stem-Volmer procedures, which measure kq values. In the second region, k dif ke and kq = Kdif. In this case, the reaction is dominated by diffusion dynamics and is said to be diffusion-controlled. 

The activated rate constant of electron transfer, k , is given by 

For an electron transfer between two stationary molecules — perhaps those in the solid state or in an organized matrix — Eq. (24) can be written  

With kq values obtained by Stern-Volmer experiments, ka can be calculated from Eqs. (23) or (24). For determining k, in Eq. (23), kdif can be estimated by various modifications of classical Smoluchowski theory [35] or by statistical nonequilibrium thermodynamic theory [36]. 

In the classical theory of Marcus, the rate determining factors involve nuclear reorganization. We write the first-order rate constant, ke, as [37] 

So far we have only looked at irreversible reactions but as mentioned earlier all reactions are in principle reversible meaning that the reactions run in both directions. In the prior example with the decomposition of nitrogen dioxide the following reaction may also take place if there is excess O2 available  

Such contra dictionaiy reactions have naturally importance regarding the writing of the rate expressions. Typically, complications are avoided but neglecting the reversible reaction and thereby assume that the rate of reaction only depends on the concentration of the reactants. For the decomposition of nitrogen dioxide the rate expressions is written as  

order reactions, where n equals 1, the general rate expressions for the use of reactant A may be expressed as  

This differential equation may be solve analytically and it may be shown that the concentration of reactant A depends on the initial concentration of A (symbolised by [A]o), rate constant k and time t in the following manner  

order reactions where the parameter is 2 a similar expression for the use of reactant A is  

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The kinetics of reactions in which a new phase is formed may be complicated by the interference of that phase with the ease of access of the reactants to each other. This is the situation in corrosion and tarnishing reactions. Thus in the corrosion of a metal by oxygen the increasingly thick coating of oxide that builds up may offer more and more impedance to the reaction. Typical rate expressions are the logarithmic law,... 

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... 

C(ads) + H2 CH2(ads) followed by fast steps. The corresponding rate expression proposed is... 

In many instances tire adiabatic ET rate expression overestimates tire rate by a considerable amount. In some circumstances simply fonning tire tire activated state geometry in tire encounter complex does not lead to ET. This situation arises when tire donor and acceptor groups are very weakly coupled electronically, and tire reaction is said to be nonadiabatic. As tire geometry of tire system fluctuates, tire species do not move on tire lowest potential energy surface from reactants to products. That is, fluctuations into activated complex geometries can occur millions of times prior to a productive electron transfer event. 

Wlretlrer adiabatic or nonadiabatic, it is tire case tlrat botlr solvent aird intramolecular degrees of freedom respond to ET events. As such, tire two rate expressions given above cair be generalized such tlrat... 

Rate fonrrulations that treat tire iinrer-sphere nrode(s) quairtum nrechairically aird tire outer sphere modes classically are used ratlrer widely. The rate expression for a single hanrronic quairtum mode is... 

The algebraic form of the expression (9.24) for the enhancement factor is specific to the particular reaction rate expression we have considered, and corresponding results can easily be obtained for other reactions in binary mixtures, for example the irreversible cracking A—2B. ... 

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. 

Applying the first-order eleetrie dipole transition rate expressions... 

The first-order El "golden-rule" expression for the rates of photon-induced transitions can be recast into a form in which certain specific physical models are easily introduced and insights are easily gained. Moreover, by using so-called equilibrium averaged time correlation functions, it is possible to obtain rate expressions appropriate to a... 

The rate of a reaction r is dependent on the reactant concentrations. For example, a bimolecular reaction between the reactants B and C could have a rate expression, such as... 

Retention Behavior. On a chromatogram the distance on the time axis from the point of sample injection to the peak of an eluted component is called the uncorrected retention time The corresponding retention volume is the product of retention time and flow rate, expressed as volume of mobile phase per unit time ... 

To deal with the case of termination by combination, it is convenient to write some reactions by which an n-mer might be formed. Table 6.5 lists several specific chemical reactions and the corresponding rate expressions as well as the general form for the combination of an (n - m)-mer and an m-mer. On the assumption that all kj values are the same, we can write the total rate of change of [M -] ... 

External Fluid Film Resistance. A particle immersed ia a fluid is always surrounded by a laminar fluid film or boundary layer through which an adsorbiag or desorbiag molecule must diffuse. The thickness of this layer, and therefore the mass transfer resistance, depends on the hydrodynamic conditions. Mass transfer ia packed beds and other common contacting devices has been widely studied. The rate data are normally expressed ia terms of a simple linear rate expression of the form... 

The term dqljdt represents the overall rate of mass transfer for component / (at time t and distance averaged over a particle. This is governed by a mass transfer rate expression which may be thought of as a general functional relationship of the form... 

Also shown are the corresponding curves calculated for the same system assuming a diffusion model in place of the linear rate expression. For intracrystalline diffusion k = 15Dq/v, whereas for macropore diffusion k = 15e /R ) Cq/q ), in accordance with the Glueckauf approximation (21). 

The enhanced rate expressions for regimes 3 and 4 have been presented (48) and can be appHed (49,50) when one phase consists of a pure reactant, for example in the saponification of an ester. However, it should be noted that in the more general case where component C in equation 19 is transferred from one inert solvent (A) to another (B), an enhancement of the mass-transfer coefficient in the B-rich phase has the effect of moving the controlling mass-transfer resistance to the A-rich phase, in accordance with equation 17. Resistance in both Hquid phases is taken into account in a detailed model (51) which is apphcable to the reversible reactions involved in metal extraction. This model, which can accommodate the case of interfacial reaction, has been successfully compared with rate data from the Hterature (51). 

Low temperatures strongly favor the formation of nitrogen dioxide. Below 150°C equiUbrium is almost totally in favor of NO2 formation. This is a slow reaction, but the rate constant for NO2 formation rapidly increases with reductions in temperature. Process temperatures are typically low enough to neglect the reverse reaction and determine changes in NO partial pressure by the rate expression (40—42) (eq. 13). The rate of reaction, and therefore the... 

Combining equation 6 with the heat- and mass-transfer rate expressions gives... 

The energy tiansfeiied on both sides of the interface in equation 28 can also be written in terms of the appropriate rate expressions. For the hquid phase, it is... 

Division of equation 109 by the time interval, 8t, transforms it into a rate expression ... 

The proposed rate expression for the ammonia /H O process is as follows ... 

The two dashed lines in the upper left hand corner of the Evans diagram represent the electrochemical potential vs electrochemical reaction rate (expressed as current density) for the oxidation and the reduction form of the hydrogen reaction. At point A the two are equal, ie, at equiUbrium, and the potential is therefore the equiUbrium potential, for the specific conditions involved. Note that the reaction kinetics are linear on these axes. The change in potential for each decade of log current density is referred to as the Tafel slope (12). Electrochemical reactions often exhibit this behavior and a common Tafel slope for the analysis of corrosion problems is 100 millivolts per decade of log current (1). A more detailed treatment of Tafel slopes can be found elsewhere (4,13,14). 

FIG. 5-24 Flowchart iUnstrating problem solving approach using mass-transfer rate expressions in the context of mass conservation. 

Use of Mass-Transfer-Rate Expression Figure 14-3 shows a section of a packed absorption tower together with the nomenclature that will be used in developing the equations which follow. In a differential section dh, we can equate the rate at which solute is lost from the gas phase to the rate at which it is transferred through the gas phase to the interface as follows ... 

For this derivation we use the gas-phase rate expression Na = J -ciy yi) and integrate over the tower to obtain... 

Although the right-hand side of Eq. (14-60) remains valid even when chemical reactions are extremely slow, the mass-transfer driving force may become increasingly small, until finally c — Cj. For extremely slow first-order irreversible reactions, the following rate expression can be derived from Eq. (14-60) ... 

At the other extreme, when the ratio ki /mkc is much smaller than unity, the interfacial concentration of reactant A may be approximated by the equihbrium relation Xi = y/m, and the specific absorption rate expression is... 

Mass Transfer Relationships for calculating rates of mass transfer between gas and liquid in packed absorbers, strippers, and distillation columns may be found in Sec. 5 and are summarized in Table, 5-28. The two-resistance approach is used, with rates expressed as transfer units ... 

Dry Solids or Filtrate Rate Filtration rate, expressed either in terms of diy solids or filtrate volume, may be plotted as a function of time on log-log paper. However, it is more convenient to delavthe rate calculation until the complete cycle of operations has been defined. 

This rate expression is known as the parabolic law. It is obeyed by oxidation of Ni, Ti, Cu, and Cr and by halogenation of silver. The product coat retards both diffusion and heat transfer. 

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