Reaction term

The acetoacetic ester condensation (involving the acylation of an ester by an ester) is a special case of a more general reaction term the Claisen condensation. The latter is the condensation between a carboxylic ester and an ester (or ketone or nitrile) containing an a-hydrogen atom in the presence of a base (sodium, sodium alkoxide, sodamide, sodium triphenylmethide, etc.). If R—H is the compound containing the a- or active hydrogen atom, the Claisen condensation may be written ... 

The generalized transport equation, equation 17, can be dissected into terms describing bulk flow (term 2), turbulent diffusion (term 3) and other processes, eg, sources or chemical reactions (term 4), each having an impact on the time evolution of the transported property. In many systems, such as urban smog, the processes have very different time scales and can be viewed as being relatively independent over a short time period, allowing the equation to be "spht" into separate operators. This greatly shortens solution times (74). The solution sequence is... 

Here R=r is the parameter for radiative heat transfer in K units, p is a heat of reaction term, in K/atm units tj is the fluid temperature in the j-th axial position e is the particle emissivity 1 is the celt dimension in m 6 is the clock time in minutes... 

It will be easier to see the importance of the various terms if this equation is rearranged as Hulbert ° recommended it. For this the reaction term will be put on the right hand side and terms representing physical changes to the left hand side. 

Saturation kinetics are also called zero-order kinetics or Michaelis-Menten kinetics. The Michaelis-Menten equation is mainly used to characterize the interactions of enzymes and substrates, but it is also widely applied to characterize the elimination of chemical compounds from the body. The substrate concentration that produces half-maximal velocity of an enzymatic reaction, termed value or Michaelis constant, can be determined experimentally by graphing r/, as a function of substrate concentration, [S]. 

It must be kept in mind that the reaction term will not occur in the overall mass balance equations of reactive systems because S(R- = 0, i.e., there is no net mass gain or loss as a result of chemical reactions. 

Heuristically, the dynamics proceed as follows the reaction term makes all active ((7 = 1) and refractory a = 2) sites cycle to their respective successor states. The diffusion term defines the manner in which activity (defined by sites with value (7 = 1) diffuses through the lattice. 

The reaction terms are evaluated at the outlet conditions since the entire reactor inventory is at these conditions. The set of component balances can be summarized as... 

Example 6.6 Suppose the reactions in Equation (6.1) are exothermic rather than endothermic. Specihcally, reverse the sign on the heat of reaction terms... 

Solution A transformation to dimensionless temperatures can be useful to generalize results when physical properties are constant, and particularly when the reaction term is missing. The problem at hand is the classic Graetz problem and lends itself perfectly to the use of a dimensionless temperature. Equation (8.52) becomes... 

Note that the accumulation and reaction terms are based on the volume of the liquid phase but that the mass transfer term is based on the working volume, V=Vi + Vg. The gas-phase balance is... 

Solution The initial liquid-phase concentration of oxygen is 0.219mol/m as in Example 11.1. The final oxygen concentration will be 1.05 mol/m. The phase balances. Equations (11.11) and (11.12), govern the dynamic response. The flow and reaction terms are dropped from the liquid phase balance to give... 

The pseudohomogeneous reaction term in Equation (11.42) is analogous to that in Equation (9.1). We have explicitly included the effectiveness factor rj to emphasis the heterogeneous nature of the catalytic reaction. The discussion in Section 10.5 on the measurement of intrinsic kinetics remains applicable, but it is now necessary to ensure that the liquid phase is saturated with the gas when the measurements are made. The void fraction s is based on relative areas occupied by the liquid and soUd phases. Thus,... 

Dynamic differential equation balances were written to calculate the molar concentration of each species in the reactor. These equations consist of inflow, outflow, accumulation, and reaction terms for a CSTR. If there are no outflow terms, the equations reduce to semibatch... 

While many enzymes have a single substrate, many others have two—and sometimes more than two—substrates and products. The fundamental principles discussed above, while illustrated for single-substrate enzymes, apply also to multisubstrate enzymes. The mathematical expressions used to evaluate multisubstrate reactions are, however, complex. While detailed kinetic analysis of multisubstrate reactions exceeds the scope of this chapter, two-substrate, two-product reactions (termed Bi-Bi reactions) are considered below. 

The effect of chemical kinetics on mass transport in incompressible flows is summarized by the reaction term r in Eq. (89). Applied to a chemical species a, it describes the rate of disappearance of this species per unit volume ... 

The Kronecker delta, 5, means that the reaction term is non-zero only when reactant is present, i.e. 

A quasi homogeneous form for the reaction term is assumed. 

The above model was solved numerically by writing finite difference approximations for each term. The equations were decoupled by writing the reaction terms on the previous time steps where the concentrations are known. Similarly the equations were linearized by writing the diffusivities on the previous time step also. The model was solved numerically using a linear matrix inversion routine, updating the solution matrix between iterations to include the proper concentration dependent diffusivities and reactions. 

The sum of these reactions yields a reaction termed disproportionation ... 

The mass balance of a continuous flow stirred-tank reactor (CSTR) with a first-order chemical reaction is very similar to the problem in Section 2.8.1 (p. 2-20). We just need to add the chemical reaction term. The balance written for the reactant A will appear as ... 

The negative sign before the reaction term k C,n indicates that the drug is consumed rather than being produced as in the derivation of the generalized mass balance equation. 

This combined equation represents a differential total material balance of a component, whether present as HA or the reaction product A-, within the reacting phase. The reader is referred to Olander s original paper for a more complete rationale for generating these differential component material balances for systems of reacting species near equilibrium. By using Olander s technique, the system of four differential equations with reaction terms can be simplified significantly to two differential equations with no reaction terms. 

Each differential equation contains a flow term identified by Q/V (flow rate/reactor volume) and also a reaction term which can be identified by a rate of reaction or equilibrium constant (k, K, k ). These reaction and equilibrium constants are functions of temperature which, in this study, was fixed. The viscosity dependence of the equilibrium constant (relating reactive species to total polymer) shown in Equations 6 and 7 was observed experimentally and is known as the Trommsdorf effect (6). Table I lists values and units of all parameters in Equations 1-7. 

Since the spontaneous reaction term dominates in the pH range 4 to 6, studies of the reaction in this range are particularly suitable for measuring the small catalytic effects of weak acids and weak bases. If one employs the more general expression for k involving contributions from the undissociated acid and the anion resulting from dissociation, determine the coefficients of these terms from the data below. 

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