First-order reaction, definition

According to the definition given, this is a first-order reaction. Now let us multiply the rate law by a factor that is exactly unity ... 

The importance of dilfusion in a tubular reactor is determined by a dimensionless parameter, SiAt/S = QIaLKuB ), which is the molecular diffusivity of component A scaled by the tube size and flow rate. If SiAtlB is small, then the elfects of dilfusion will be small, although the definition of small will depend on the specific reaction mechanism. Merrill and Hamrin studied the elfects of dilfusion on first-order reactions and concluded that molecular diffusion can be ignored in reactor design calculations if... 

This interpretation of Ha may be deduced from its definition. For example, for a first-order reaction, from equation 9.2-40,... 

For a first-order reaction (equation 3.4-10), and from the definition of /a>... 

It should be noted, and will be further discussed later, that the definition of kuni in Equation 14.13 does not mean that the reaction is a first order reaction. A first order reaction would imply that kunj is independent of concentration. As indicated by Equation 14.13, kunj depends on the concentration of the third body M with which A collides for activation and deactivation. For k2 [M] much larger than ka, which means high pressure, kunj does indeed become independent of concentration. At low pressure, however, kunj depends on [M] and the overall rate of making products becomes second order, depending linearly on both [M] and [A],... 

Since all of the above-mentioned interconversion reactions are reversible, any kinetic analysis is difficult. In particular, this holds for the reaction Sg - Sy since the backward reaction Sy -+ Sg is much faster and, therefore, cannot be neglected even in the early stages of the forward reaction. The observation that the equilibrium is reached by first order kinetics (the half-life is independent of the initial Sg concentration) does not necessarily indicate that the single steps Sg Sy and Sg Sg are first order reactions. In fact, no definite conclusions about the reaction order of these elementary steps are possible at the present time. The reaction order of 1.5 of the Sy decomposition supports this view. Furthermore, the measured overall activation energy of 95 kJ/mol, obtained with the assumption of first order kinetics, must be a function of the true activation energies of the forward and backward reactions. The value found should therefore be interpreted with caution. 

By definition, the laminar-flow reactor is segregated. Each radial element of fluid is assumed to slide past its adjacent elements with no mixing. Thus, eqn. (34) may be used to predict reactor conversion. The case of a first-order reaction has been analysed by Cleland and Wilhelm [43]. As we have seen previously... 

A simple way to characterize the rate of a reaction is the time it takes for the concentration to change from the initial value to halfway between the initial and final (equilibrium). This time is called the half-life of the reaction. The half-life is often denoted as ti/z. The longer the half-life, the slower the reaction. The half-life is best applied to a first-order reaction (especially radioactive decay), for which the half-life is independent of the initial concentration. For example, using the decay of " Sm as an example, [ Sm] = [ Sm]o exp( kt) (derived above). Now, by definition,... 

Fractional Change Method From the equations of half life tor reactions of various orders except first order reaction, time required to complete a definite fraction of the reaction is inversely proportional to af- where n is the order of reaction and a is initial concentration. 

Two important ways in which heterogeneously catalyzed reactions differ from homogeneous counterparts are the definition of the rate constant k and the form of its dependence on temperature T. The heterogeneous rate equation relates the rate of decline of the concentration (or partial pressure) c of a reactant to the fraction / of the catalytic surface area that it covers when adsorbed. Thus, for a first-order reaction,... 

Riesenfeld and Bohnholtzer and Riesenfeld and Schumacher used ozone concentrated by liquefaction and distillation. From their kinetic measurements they conclude that a reaction of the second order and one of the first order take place simultaneously at quite low pressures, 6-60 mm. Hg the first order reaction predominates. The velocity constants of the second order reaction are not influenced by the total pressure, while those of the first order reaction appear to be inversely proportional to the total pressure. The figures given show that the first order reaction at the lower pressures is considerably influenced by the surface, and is quite probably a heterogeneous reaction, though the authors themselves do not consider this to be definitely shown. The decomposition appears to be rather sensitive to catalysts such as dust particles. 

For uniform poisoning, the effectiveness is obtained by simply replacing k with kv(l-/3) in the definition of For pore mouth poisoning the equation for if is in P7.06.07. These lesuLts are for first order reaction in slab geometry. 

After the absence of film diffusion effects has been verified and if the reaction order n is known, the expression for the rate equation r = r) kcat[E][S]buik/KM (first-order reaction assumed) can be inserted into the definition for 7j and the unknown rate constant k can be eliminated (Weisz, 1954) [Eq. (5.66)]. 

It is prudent at this point to note that dilatometry does not fulfil the initial requirement of a reaction at constant volume (see Sections 3.2.1 and 3.5.2.1), so the definition of reaction rate needs to be revised to take account of the possibility that the volume may change as the reaction proceeds. Thus, for a first-order reaction, for example,... 

In Table 5.4 the kj values show a definite trend with time while the k2 values show small variations with no definite trend. Therefore, the first-order reaction mechanism does not adequately describe the experimental kinetic data. 

In a first order reaction the fraction of the molecules decomposing per unit of time is constant, irrespective of concentration. In other words the rate of decomposition is independent of collision. Consider, for example, molecules of a gas in a cylinder decomposing at a definite rate. Now imagine that a piston is raised and the volume doubled. The number of collisions is one-fourth as large as before and yet experiments show that if the reaction is first order the decomposition will nevertheless take place at the same rate as before. 

It follows from the above definition that for an irreversible first-order reaction, the utilization factor is given by ... 

In order to use equation (7.6) we need a definite expression for rate r considering a first-order reaction... 

In the past, a number of attempts have been made to generalize the definition of the Thiele modulus. Aris [6] noticed that all the Thiele moduli for first-order reactions were of the form ... 

The diffusion modulus diffusion effect are entirely determined by the magnitude of the modulus v>, involving size (72), diffusivity (7)en), and intrinsic activity (fc,) of the catalyst. In many practical cases of experimentation the intrinsic activity constant fc, will not be directly known, but instead it will be desirable to estimate tp from the diffusivity and size of the solid, and the actvxilly observed reaction rate dnjdt. Use is made in such cases (Wagner, 16) of the definition of the modulus p and the basic activity equation (first-order reaction being used here) ... 

The simplest kinetics to assume would be a one-step reaction from an organometallic compound to metallic deposit, i.e. the A to B first order reaction. This is essentially the approach used by Sato et al [1], which led to a straight forward definition of the Thiele modulus and the pattern of metal deposition as the fantiliar... 

For a first-order reaction, their light-off criterion (deemed unsafe by the classical definition, but desired in this application) accounting for washcoat diffusion is given by... 

Solution (a) In Example 11-6 the proper definition of 0 for a reversible first-order reaction was shown to be given by Eq. (11-57). From this definition, the criterion for j 1 is... 

For the single-reaction cases, we performed dimensional analysis and found a dimensionless number, the Thiele modulus, which measures the rate of production divided by the rate of diffusion of some component. A complete analysis of the first-order reaction in a sphere suggested a general approach to calculate the production rate in a pellet in terms of the rate evaluated at the pellet exterior surface conditions. This motivated the definition of the pellet effectiveness factor, which is a function of the Thiele modulus. 

The reactor stability decreases with increasing values of a, since the fraction converted at the peak temperature is lower when ATa j is higher. One study showed that the allowable value of 9 for a first-order reaction ranged from 2.4 to 1.1 as a increased from about 7 to 70 [11,12]. There have been many other studies of the stability of tubular reactors and batch reactors, and some complex correlations for the stability limit allowing for changes in coolant temperature with length and the thermal capacity of the reactor wall [13]. However, it is generally not necessary to get the exact stability limit. The conservative criterion that 6> < 1 is often used unless calculations for different conditions show that even with 9 > the reactor is definitely stable to all likely disturbances. 

For a first-order reaction, the time for a definite fraction of the reactant to be consumed is independent of the original concentration. The units of k, the rate constant, are s". ... 

The above integral is easiest done by recognizing that Eq. 12.4-3 is really the definition of the Laplace transform with respect to 9 of (0), but with k substituted for s. Thus, the exit concentration fora first-order reaction is found from the transforms of any E(9) by merely substituting k for s. 

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