Intrinsic rates

Reactor Configuration. The horizontal cross-sectional area of a reactor is a critical parameter with respect to oxygen mass-transfer effects in LPO since it influences the degree of interaction of the two types of zones. Reactions with high intrinsic rates, such as aldehyde oxidations, are largely mass-transfer rate-limited under common operating conditions. Such reactions can be conducted effectively in reactors with small horizontal cross sections. Slower reactions, however, may require larger horizontal cross sections for stable operation. 

In the design of synthesis facilities, the rate at which the ammonia is formed has to be considered. The most widely used equation for the intrinsic rate of reaction is the Temkia-Pyzher equation (11) ... 

When a relatively slow catalytic reaction takes place in a stirred solution, the reactants are suppHed to the catalyst from the immediately neighboring solution so readily that virtually no concentration gradients exist. The intrinsic chemical kinetics determines the rate of the reaction. However, when the intrinsic rate of the reaction is very high and/or the transport of the reactant slow, as in a viscous polymer solution, the concentration gradients become significant, and the transport of reactants to the catalyst cannot keep the catalyst suppHed sufficientiy for the rate of the reaction to be that corresponding to the intrinsic chemical kinetics. Assume that the transport of the reactant in solution is described by Fick s law of diffusion with a diffusion coefficient D, and the intrinsic chemical kinetics is of the foUowing form... 

The effect of physical processes on reactor performance is more complex than for two-phase systems because both gas-liquid and liquid-solid interphase transport effects may be coupled with the intrinsic rate. The most common types of three-phase reactors are the slurry and trickle-bed reactors. These have found wide applications in the petroleum industry. A slurry reactor is a multi-phase flow reactor in which the reactant gas is bubbled through a solution containing solid catalyst particles. The reactor may operate continuously as a steady flow system with respect to both gas and liquid phases. Alternatively, a fixed charge of liquid is initially added to the stirred vessel, and the gas is continuously added such that the reactor is batch with respect to the liquid phase. This method is used in some hydrogenation reactions such as hydrogenation of oils in a slurry of nickel catalyst particles. Figure 4-15 shows a slurry-type reactor used for polymerization of ethylene in a sluiTy of solid catalyst particles in a solvent of cyclohexane. 

The rate of evaporation is given by the intrinsic rate of dissociation of bound molecules from the surface... 

As the pre-equilibria in Schemes 5-10 and 5-11 are not identical and their equilibrium constants are therefore likely to be different from one another, the rate constants k - and are not intrinsic rate constants of the corresponding slow dissociation steps, but are dependent in addition on the constants of these pre-equilibria. 

Assuming that reactions 1 and 2 are reversible and are fast in comparison with reaction 5, the measured rate constants ( obs) at pH 8.2-9.8 are related to the intrinsic rate constant k5 according to Scheme 5-16, as shown by Ritchie and Wright (1971a). ... 

Broxton and Roper measured the rate of dissociation (A 3) of the (ii)-diazo ether, A 2, and the rate of the protection reaction (A p), i.e., the transformation of the (Z)-into the (ii)-ether ( protection because the diazo ether is protected against dediazoniation almost completely if present as the ( >isomer). Rate constants kx and k are known from Ritchie and Virtanen s work (1972). The results demonstrate firstly that the initial reaction of the diazonium ion takes place in such a way that almost exclusively the (Z)-ether is formed directly (ki/k3 = 120). The protection rate constant kp is a simple function of the intrinsic rate constants as shown in Scheme 6-4. 

In addition to the influence of the complexation equilibrium constant K, the observed reaction rate of arenediazonium salts in the presence of guest complexing reagents is influenced by the intrinsic reaction rate of the complexed arenediazonium ion. This system of reactions can be rationalized as in Scheme 11-1. Here we are specifically interested in the numerical value of the intrinsic rate constant k3 of the complexed diazonium ion relative to the rate constant k2 of the free diazonium ion. 

Does this model give us a practical solution for the synthesis of monosubstitution products in high yields The model teaches us that reactions are not disguised by micromixing if the intrinsic rate constant (in Scheme 12-84 k2o and k2v>) is significantly less than 1 m-1s-1. As discussed in Section 12.7, the intrinsic rate constant refers to unit concentrations of the acid-base equilibrium species involved in the substitution proper, not to analytical concentrations. Therefore, if the azo coupling reaction mentioned above is not carried out within the range of maximal measured rates (i.e., with the equilibria not on the side of the 1-naphthoxide ion and... 

Intrinsic Rate Expressions from Equality of Rates... 

Suppose a gradientless reactor is used to obtain intrinsic rate data for a catalytic reaction. Gas-phase concentrations are measured, and the data are fit to a rate expression using the methods of Chapter 7. The rate expression can be arbitrary ... 

This result is experimentally indistinguishable from the general form, Equation (10.12), derived in Example 10.1 using the equality of rates method. Thus, assuming a particular step to be rate-controlling may not lead to any simplification of the intrinsic rate expression. Furthermore, when a simplified form such as Equation (10.15) is experimentally determined, it does not necessarily justify the assumptions used to derive the simplified form. Other models may lead to the same form. 

The ratio of actual rate to intrinsic rate is the effectiveness factor ... 

It depends only on J sJkj A, which is a dimensionless group known as the Thiele modulus. The Thiele modulus can be measured experimentally by comparing actual rates to intrinsic rates. It can also be predicted from first principles given an estimate of the pore length =2 . Note that the pore radius does not enter the calculations (although the effective diffusivity will be affected by the pore radius when dpore is less than about 100 run). 

Actual rate] = r (6) [Intrinsic rate of fresh catalyst]... 

According to the method as described by Hiromi et al [5,6] the measured parameters Km and Umax were used for the calculation of the subsite affinities A - and the intrinsic rate constant. The calculated subsite affinities are summarized in Table 3. 

The material balance was calculated for EtPy, ethyl lactates (EtLa) and CD by solving the set of differential equation derived form the reaction scheme Adam s method was used for the solution of the set of differential equations. The rate constants for the hydrogenation reactions are of pseudo first order. Their value depends on the intrinsic rate constant of the catalytic reaction, the hydrogen pressure, and the adsorption equilibrium constants of all components involved in the hydrogenation. It was assumed that the hydrogen pressure is constant during... 

Figure 6.3 Dependence of the apparent rate constants, determined by fitting the experimental transients with (6.5), on the step fraction. The final potentials are 0.73 V (triangles), 0.755 V (diamonds), 0.78 V (squares), and 0.805 V (circles). The value of the step fraction for Pt(lll) was estimated using a procedure described in [Lebedeva et al., 2002c]. The inset shows the independence of the apparent intrinsic rate constant per step.

Figure 6.4 Dependence of the apparent rate constants and the apparent intrinsic rate constant on the potential ( Tafel plots ), determined by fitting the experimental transients with (6.5).

Chemical kinetics is an area that received perhaps most of the attention of chemical engineers from a parameter estimation point of view. Chemical engineers need mathematical expressions for the intrinsic rate of chemical reactions... 

The first equation gives the rate of gas consumption as moles of gas (n) versus time. This is the only state variable that is measured. The initial number of moles, nO is known. The intrinsic rate constant, K is the only unknown model parameter and it enters the first model equation through the Hatta number y. The Hatta number is given by the following equation... 

In electrochemical literature the standard rate constant fe is often designated as fes h or fe9, called the specific heterogeneous rate constant or the intrinsic rate constant. According to eqns. 3.5 and 3.6, we have... 

The material factor is a measure of the intrinsic rate of energy release from the burning, explosion, or other chemical reaction of the material. Values for the MF for over 300 of... 

For reversible reactions one normally assumes that the observed rate can be expressed as a difference of two terms, one pertaining to the forward reaction and the other to the reverse reaction. Thermodynamics does not require that the rate expression be restricted to two terms or that one associate individual terms with intrinsic rates for forward and reverse reactions. This section is devoted to a discussion of the limitations that thermodynamics places on reaction rate expressions. The analysis is based on the idea that at equilibrium the net rate of reaction becomes zero, a concept that dates back to the historic studies of Guldberg and Waage (2) on the law of mass action. We will consider only cases where the net rate expression consists of two terms, one for the forward direction and one for the reverse direction. Cases where the net rate expression consists of a summation of several terms are usually viewed as corresponding to reactions with two or more parallel paths linking reactants and products. One may associate a pair of terms with each parallel path and use the technique outlined below to determine the thermodynamic restrictions on the form of the concentration dependence within each pair. This type of analysis is based on the principle of detailed balancing discussed in Section 4.1.5.4. 

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