Irreversible Reactions of First Order

3 INTEGRATED RATE EQUATIONS 8.3.1 Irreversible Reactions of First Order 

By irreversibility, we mean, of course, that the back reaction can be neglected. In Equation 8.8, = 1 while all other a s are equal to zero. The reaction is thus 

x stands for the decrease of the concentration of A. This equation may be integrated to 

k may be obtained by measuring the concentration of reactants in time intervals and plotting the logarithm as a function of time. A linear plot is obtained where k is the slope. 

The time it takes for the concentration to decrease to half of the original concentration is called the half life time and is denoted tyi- By substituting [A] = [A]q/2 in Equation 8.13, we find that 

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As an example, the four simplest one-plus rate equations for an irreversible reaction of first order in A and orders between zero and plus one in B and C are ... 

For irreversible reactions of first order, the rate constant k may easily be obtained from Bassett and Habgood s equation, often used in the literature ... 

Let us consider the case in which a component (A) undergoes two consecutive, irreversible reactions of first order ... 

Here cb disappears, indicating that the ratio dcR/dcA does not depend on the concentration level of B. It is of interest to note that for consecutive reactions, A R S, an equation similar to Equation 3.266 is obtained. Therefore, the product distribution should become similar to the consecutive, irreversible reactions of first order. This can be verified by solving Equation 3.266. Consequently, we obtain the following equations for cr and dcR/dcA ... 

More re( ent work by Hawkins and coworkers [R. E. Fuguitt and J. E. Hawkins, J. Am. Chem. Soc., 69, 319 (1947) 67, 242 (1945) H. G. Hunt and J. E. Hawkins, ihid.y 72, 5618 (1950)] on the licjuid-phase isomerization. shows that there are three simultaneous reactions of first order occurring, the racemization to /3-pinenc with log A = 14.0, E = 44.1 Kcal/mole, and the two listed above. In addition, the allocimcne undergoes reversible dimerization and irreversible first-order decomjwsition to pyrenones. The values of Smith in parentheses cannot therefore be taken too seriously. 

The example of this reaction demonstrates another important facet of kinetics. Figure 5.4 shows side by side the experimental data plotted as a first order-first order reversible reaction and as an irreversible reaction of order 1.5. Over a limited conversion range (here about two thirds of the way to equilibrium) the second plot is linear within the scatter of the data points. Although evaluation of the full conversion range leaves no doubt that the reaction is indeed reversible and first order-first order, its rate up to a rather high conversion is approximated surprisingly well by the equation for an irreversible reaction of higher order, in this instance of order 1.5 ... 

The ester of an organic base is hydrolyzed in a CSTR. The rate of this irreversible reaction is first-order in each reactant. The liquid volume in the vessel is 6500 L. A jacket with coolant at 18°C maintains the reactant mixture at 30°C. Additional data ... 

The principle underlying the method is that the ratio of the times at which two specific conversions occur depends solely on the order of the reaction. The method is applicable to single irreversible reactions of any order—positive or negative, integral or fractional—as well as to reversible first-order reactions. 

In first order-first order reversible reactions, the rate of approach to equilibrium is proportional to the fractional distance from equilibrium, measured in terms of concentrations or any other quantity that is a linear function of the concentrations. Unless carried to high conversion, the rate of a reversible reaction may be indistinguishable from that of an irreversible reaction of higher order. 

ERA The reaction was performed in a batch reactor and in the gas phase. 10% N2 was introduced in the reactor at 2 atm and 450°C. After 50 min, the pressure was 3.3 atm. The reaction is irreversible and of first order. Calculate the specific reaction rate constant. If the same reaction would be performed in a piston system with variable volume, how volume changes keeping pressure constant at 2 atm and considering the same conversion as before Calculate the initial concentration ... 

The first step in heterogenous catalytic processes is the transfer of the reactant from the bulk phase to the external surface of the catalyst pellet. If a nonporous catalyst is used, only external mass and heat transfer can influence the effective rate of transformation. The same situation will occur for very fast reactions, where the reactants are completely consumed at the external catalyst surface. As no internal mass and heat transfer resistances are considered, the overall catalyst effectiveness factor corresponds to the external effectiveness factor, For a simple irreversible reaction of nth order, the following relation results ... 

The microchannel reactor described in Example 5.1 is previewed for a type B reaction with a characteristic reaction time of l,. = 5s at 323 K. The reaction is irreversible and of first order. The reaction enthalpy... 

Figure 5-38 shows plots of the dynamic response to changes in the inlet concentration of component A. The figure represents possible responses to an abrupt change in inlet concentration of an isothermal CFSTR with first order irreversible reaction. The first plot illustrates the situation where the reactor initially contains reactant at and... 

An interesting method, which also makes use of the concentration data of reaction components measured in the course of a complex reaction and which yields the values of relative rate constants, was worked out by Wei and Prater (28). It is an elegant procedure for solving the kinetics of systems with an arbitrary number of reversible first-order reactions the cases with some irreversible steps can be solved as well (28-30). Despite its sophisticated mathematical procedure, it does not require excessive experimental measurements. The use of this method in heterogeneous catalysis is restricted to the cases which can be transformed to a system of first-order reactions, e.g. when from the rate equations it is possible to factor out a function which is common to all the equations, so that first-order kinetics results. 

Example 4.3 represents the simplest possible example of a variable-density CSTR. The reaction is isothermal, first-order, irreversible, and the density is a linear function of reactant concentration. This simplest system is about the most complicated one for which an analytical solution is possible. Realistic variable-density problems, whether in liquid or gas systems, require numerical solutions. These numerical solutions use the method of false transients and involve sets of first-order ODEs with various auxiliary functions. The solution methodology is similar to but simpler than that used for piston flow reactors in Chapter 3. Temperature is known and constant in the reactors described in this chapter. An ODE for temperature wiU be added in Chapter 5. Its addition does not change the basic methodology. 

Example 4.13 treated the case of a piston flow reactor inside a recycle loop. Replace the PER with two equal-volume stirred tanks in series. The reaction remains first order, irreversible, and at constant density. 

The overall reaction between CO2 and GMA was assumed to consist of two elementary reactions such as a reversible reaction of GMA and catalyst to form an intermediate and an irreversible reaction of this intermediate and carbon dioxide to form five-membered cyclic carbonate. Absorption data for CO2 in the solution at 101.3 N/m were interpreted to obtain pseudo-first-order reaction rate constant, which was used to obtain the elementary reaction rate constants. The effects of the solubility parameter of solvent on lc2/k and IC3 were explained using the solvent polarity. 

The reaction may be considered as irreversible. The progress of the reaction was monitored by observing the absorbance of the solution at 723 nm where Np(IV) is the principal absorbing species. From the data below determine the reaction rate constant for this reaction if the reaction is first order in both Np(IV) and V(II). 

If the first reaction is regarded as zero-order irreversible (i.e., the enzyme is saturated with substrate), and the second reaction is first-order in the product B, determine the time-dependent behavior of the concentration of species B if no B is present initially. How long does it take to reach 98% of the steady-state value if kx = 0.833 mole/m3-ksec and k2 = 0.767 sec 1 What is this steady-state value ... 

It is readily apparent that equation 8.3.21 reduces to the basic design equation (equation 8.3.7) when steady-state conditions prevail. Under the presumptions that CA in undergoes a step change at time zero and that the system is isothermal, equation 8.3.21 has been solved for various reaction rate expressions. In the case of first-order reactions, solutions are available for both multiple identical CSTR s in series and individual CSTR s (12). In the case of a first-order irreversible reaction in a single CSTR, equation 8.3.21 becomes... 

For the case where all of the series reactions obey first-order irreversible kinetics, equations 5.3.4, 5.3.6, 5.3.9, and 5.3.10 describe the variations of the species concentrations with time in an isothermal well-mixed batch reactor. For series reactions where the kinetics do not obey simple first-order or pseudo first-order kinetics, the rate expressions can seldom be solved in closed form, and it is necessary to resort to numerical methods to determine the time dependence of various species concentrations. Irrespective of the particular reaction rate expressions involved, there will be a specific time... 

A - B was taking place. The reaction is first-order, irreversible with k = 0.0433 sec L Pure A enters the reactor. The exit stream consists of 10% A and 90% B. 

Consider the gas-phase decomposition of ethane (A) to ethylene at 750°C and 101 kPa (assume both constant) in a PFR. If the reaction is first-order with kA = 0.534s-1(Fro-ment and Bischoff, 1990, p. 351), and r is 1 s, calculate /a- For comparison, repeat the calculation on the assumption that density is constant. (In both cases, assume the reaction is irreversible.)... 

Two stirred tanks are available, one 100 m3 in volume, the other 30 m3 in volume. It is suggested that these tanks be used as a two-stage CSTR for carrying out a liquid phase reaction A + B product. The two reactants will be present in the feed stream in equimolar proportions, the concentration of each being 1.5 kmol/m3. The volumetric flowrate of the feed stream will be 0.3 x 10-3 m3/s. The reaction is irreversible and is of first order with respect to each of the reactants A and B, i.e. second order overall, with a rate constant 1.8 x 10-4 m3/kmols. 

A series of experiments were performed using various sizes of catalyst spheres. The reaction was first order irreversible. The first two columns of the table record the diameter dp in cm and the rate in mol/(h)(cc). The surface concentration was Cs = 0.0002 mol/cc. Find the true specific rate and the effective diffusivity. 

The reesterification model reaction kinetics of methylbenzoate by heptanole-1 in catalyst (TBT) presence and without it was studied at 443 K on the gas chromatograph Biokhrom using diphenyloxide according to the earlier described method [5] as an internal standard. The rate constant k was calculated according to the equation of irreversible reaction of the first order. 

First-order chemical reaction. Among first-order chemical complications following electron transfers, the most convenient case to study by chronoamperometry is that of a first-order irreversible chemical reaction (ErQ), which constitutes a frequently encountered case ... 

A semibatch reactor is run at constant temperature by varying the rate of addition of one of the reactants, A. The irreversible, exothermic reaction is first order in reactants A and B. 

For this situation an irreversible chemical reaction with first-order kinetics with respect to A and B has been used, where a very high value of the reaction rate constant has been taken to simulate an instantaneous... 

Example 4.1 We wish to produce 5 in the reaction A B in d continuous reactor at V = 4 liter/min with C o = 2 moles/liter. However, we find that there is a second reaction B C that can also occur. We find that both reactions are first order and irreversible with ki = 0.5 min and 2 = 0.1 min h Find r, V, Cg, Sg, and Yb for 90% conversion of A. 

Ethanol is to be oxidized to aeetaldehyde in aqueous solution. The aldehyde can further oxidize to aeetic acid, which can decarboxylate to methane. There is an excess of O2 in the solution so all reactions are first order in the organics and irreversible. 

We assume that the right-hand side of this equation is constant because C s usually does not change much as the reaction proceeds. The rate expression could be more complicated, but we will assume a first-order irreversible reaction of the reactant gas at the surface, r" = k"CAs, throughout this chapter. 

The heat sterilization of microorganisms and heat inactivation of enzymes are examples of first-order reactions. In the case of an enzyme being irreversibly heat-inactivated as follows ... 

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