Second-order volume reaction

If the forward reaction is pseudo first order and the reverse reaction second order, then, as discussed in Sections 1.4.4 and 1.4.5 in Volume 3, the rate equation may be written as ... 

Formula (5.3.5) guarantees an exact asymptotic result in both limit cases fcv - 0 and kv - oo for any function /v(c). For a first-order volume reaction (/v = c), the approximate formula (5.3.5) is reduced to the exact result (5.3.4). The maximum error of formula (5.3.5) for a chemical volume reaction of the order n = 1 /2 (/v = fc) in the entire range of the dimensionless reaction rate constant fcv is 5% for a second-order volume reaction (/v = c2), the error of (5.3.5) is 7% [360], The mean Sherwood number decreases with the increase of the rate order n and increases with kv. 

Herein Pa and Pb are the micelle - water partition coefficients of A and B, respectively, defined as ratios of the concentrations in the micellar and aqueous phase [S] is the concentration of surfactant V. ai,s is fhe molar volume of the micellised surfactant and k and k , are the second-order rate constants for the reaction in the micellar pseudophase and in the aqueous phase, respectively. The appearance of the molar volume of the surfactant in this equation is somewhat alarming. It is difficult to identify the volume of the micellar pseudophase that can be regarded as the potential reaction volume. Moreover, the reactants are often not homogeneously distributed throughout the micelle and... 

Herein [5.2]i is the total number of moles of 5.2 present in the reaction mixture, divided by the total reaction volume V is the observed pseudo-first-order rate constant Vmrji,s is an estimate of the molar volume of micellised surfactant S 1 and k , are the second-order rate constants in the aqueous phase and in the micellar pseudophase, respectively (see Figure 5.2) V is the volume of the aqueous phase and Psj is the partition coefficient of 5.2 over the micellar pseudophase and water, expressed as a ratio of concentrations. From the dependence of [5.2]j/lq,fe on the concentration of surfactant, Pj... 

This development has been generalized. Results for zero- and second-order irreversible reactions are shown in Figure 10. Results are given elsewhere (48) for more complex kinetics, nonisothermal reactions, and particle shapes other than spheres. For nonspherical particles, the equivalent spherical radius, three times the particle volume/surface area, can be used for R to a good approximation. 

The minimum total volume of a CSTR battery for first-order reaction, and near-minimum for second-order, is obtained when all vessels are the same size. 

Tubular flow reactors—minimum volume for second-order reversible reactions, maximum yield of consecutive reactions, minimum cost with and without recycle, and maximum profit with recycle... 

FIG. 23 14 Comp arison of maximiim mixed, segregated, and ping flows, (a) Relative volumes as functions of variance or n, for several reaction orders, (h) Second-order reaction with n = 2 or 3. (c) Second-order, n = 2. (d) Second-order, n = 5. 

FIG. 23-15 Chemical conversion by the dispersion model, (a) First-order reaction, volume relative to plug flow against residual concentration ratio, (h) Second-order reaction, residual concentration ratio against kC t. 

Assuming that the reaction is second order in a constant volume batch system, the rate equation is... 

Since the reaction is carried out in a batch system of constant volume, the rate expression for a second order rate law is... 

A certain reaction is first-order in A and second-order in B. In the box shown below, which is assumed to have a volume of one liter, a mole of A is represented by a mole... 

This form assumes that the effect of pressure on the molar volume of the solvent, which accelerates reactions of order > 1 by increasing the concentrations when they are expressed on the molar scale, has been allowed for. This effect is usually small, ignored but in the most precise work. Equation (7-41) shows that In k will vary linearly with pressure. We shall refer to this graph as the pressure profile. The value of A V is easily calculated from its slope. The values of A V may be nearly zero, positive, or negative. In the first case, the reaction rate shows little if any pressure dependence in the second and third, the applied hydrostatic pressure will cause k to decrease or increase, respectively. A positive value of the volume of activation means that the molar volume of the transition state is larger than the combined molar volume of the reactant(s), and vice versa. 

Figures 1.6 and 1.7 display the conversion behavior for flrst-and second-order reactions in a CSTR and contrast the behavior to that of a piston flow reactor. It is apparent that piston flow is substantially better than the CSTR for obtaining high conversions. The comparison is even more dramatic when made in terms of the volume needed to achieve a given conversion see Figure 1.8. The generalization that...

On this basis AF should be more positive for an inner-sphere than for an outer-sphere reaction since a water molecule occupies a greater volume in the liquid phase than if it is coordinated. Second-order rate coefficients were determined at various pressures in the range 0.001 to 3.5 kbars, the rate decreasing with increase in pressure. The apparatus used was a modification of that first described by Osborn and Whalley . Values of AF were calculated from the slopes of plots of log k versus pressure, since... 

A constant volume batch reactor is used to convert reactant. A, to product, B, via an endothermic reaction, with simple stoichiometry, A —> B. The reaction kinetics are second-order with respect to A, thus... 

The kinetics formation of [Ni([9]aneN3)2]3+ have been studied in great detail. Inter alia, the volume of activation for peroxodisulfate oxidation of [Ni([9]aneN3)2]2+ has been determined (—25.8 2.3 cm3 mol 1),105 and the kinetics of this reaction have been determined as a function of peroxodisulfate concentration and temperature.106 The reaction is first-order in both reagents (second-order rate constant 1.13 mol dm 3 s 1 at 298 K), and the activation energy is 38 1.8 kJ mol-1. In mixed solvents, the rate is slower. 

Second-Order Reactions inConstant Volume Systems. There are two primary types of second-order reactions for the first the rate is proportional to the square of the concentration of a single reacting species for the second the... 

For Class II second-order rate expressions of the form of equation 3.1.10, the rate can be expressed in terms of the extent of reaction per unit volume as... 

These investigators report that the second-order rate constant for reaction B is equal to 1.15 x 10 3 m3/mole-ksec at 20 °C. Determine the volume of plug flow reactor that would be necessary to achieve 40% conversion of the input butadiene assuming isothermal operating conditions and a liquid feed rate of 0.500 m3/ksec. The feed composition is as follows. 

From the units on the reaction rate constant, the reaction is second order. There is a volume change on reaction and S = —1/2. Thermal expansion will also occur, so equations 3.1.44 and 3.1.46 must be combined to get the reactant concentrations. Since equimolar concentrations of reactants are used, the design equation becomes... 

The first-order rate constants are written in (3) and (4) as second-order rate constants, kw and A M, for reaction of a reagent, Y, where the mole ratio, my = [Ym]/([D] — cmc). (Here, and elsewhere, the quantities in square brackets denote molarity in terms of total solution volume, which is approximately that of the aqueous pseudophase.)... 

Some examples of reactions of inorganic ions with organic substrates are given in Tables 3-5. It is important to note that different values are assumed for the volume element of reaction, i.e. the molar volume of the micellar pseudophase (VM) and it should be noted that the second-order rate constant, k in the micellar pseudophase varies proportionately (13). 

In a functional micelle in which the reactive group is fully deprotonated there is a 1 1 relationship between the concentrations of reactive nucleophile and micellar head group in the micellar pseudophase. If under these conditions the substrate is fully micellar bound, (5) or (6) take the very simple form (19). This rate constant, kM, can then be converted into the second-order rate constant, k in M 1s 1, estimating the volume element of reaction, VM, which can be assumed to be that of the micelle or of its Stem layer, and these second-order rate constants can be compared with reaction in water of a chemically similar, non-micellized, nucleophile. 

There is also a problem in defining the volume element of reaction in a microemulsion droplet, but despite these uncertainties second-order rate constants in the droplet are similar to those in cationic micelles for reactions of anionic nucleophiles in alcohol-swollen droplets (Bunton et al., 1983b). Thus, the rate enhancements seem to be due to concentration of reactants in the droplet. 

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