Rate equation order

RATE EQUATION, ORDER OF REACTION AND RATE CONSTANT... 

Second-order rate coefficients for nitration in sulphuric acid at 25 °C fall by a factor of about 10 for every 10 % decrease in the concentration of the sulphuric acid ( 2.4.2). Since in sulphuric acid of about 90% concentration nitric acid is completely ionised to nitronium ions, in 68 % sulphuric acid [NO2+] io [HNO3]. The rate equation can be written in two ways, as follows ... 

In a curve-fitting method the concentration of a reactant or product is monitored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. Eor example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentration of a product is followed as a function of time, can be determined by fitting a rearranged form of equation 13.12... 

Activation Parameters. Thermal processes are commonly used to break labile initiator bonds in order to form radicals. The amount of thermal energy necessary varies with the environment, but absolute temperature, T, is usually the dominant factor. The energy barrier, the minimum amount of energy that must be suppHed, is called the activation energy, E. A third important factor, known as the frequency factor, is a measure of bond motion freedom (translational, rotational, and vibrational) in the activated complex or transition state. The relationships of yi, E and T to the initiator decomposition rate (kJ) are expressed by the Arrhenius first-order rate equation (eq. 16) where R is the gas constant, and and E are known as the activation parameters. 

As with the case of energy input, detergency generally reaches a plateau after a certain wash time as would be expected from a kinetic analysis. In a practical system, each of its numerous components has a different rate constant, hence its rate behavior generally does not exhibit any simple pattern. Many attempts have been made to fit soil removal (50) rates in practical systems to the usual rate equations of physical chemistry. The rate of soil removal in the Launder-Ometer could be reasonably well described by the equation of a first-order chemical reaction, ie, the rate was proportional to the amount of removable soil remaining on the fabric (51,52). In a study of soil removal rates from artificially soiled fabrics in the Terg-O-Tometer, the percent soil removal increased linearly with the log of cumulative wash time. 

Related to the preceding is the classification with respect to oidei. In the power law rate equation / = /cC C, the exponent to which any particular reactant concentration is raised is called the order p or q with respect to that substance, and the sum of the exponents p + q is the order of the reaction. At times the order is identical with the molecularity, but there are many reactions with experimental orders of zero or fractions or negative numbers. Complex reactions may not conform to any power law. Thus, there are reactions of ... 

Sets of first-order rate equations are solvable by Laplace transform (Rodiguin and Rodiguina, Consecutive Chemical Reactions, Van Nostrand, 1964). The methods of linear algebra are applied to large sets of coupled first-order reactions by Wei and Prater Adv. Catal., 1.3, 203 [1962]). Reactions of petroleum fractions are examples of this type. 

This is the equation for a plug flow reactor. It can be derived directly from the rate equations with the aid of Laplace transforms. The sequences of second-order reactions of Figs. 7-5n and 7-5c required numerical integrations. 

Mathematically, multiplicities become evident when heat and material balances are combined. Both are functions of temperature, the latter through the rate equation which depends on temperature by way of the Arrhenius law. The curves representing these b ances may intersect in several points. For first order in a CSTR, the material balance in terms of the fraction converted can be written... 

FIG. 23-17 Multiple steady states of CSTRs, stable and unstable, adiabatic except the last item, (a) First-order reaction, A and C stable, B unstable, A is no good for a reactor, the dashed line is of a reversible reaction, (h) One, two, or three steady states depending on the combination Cj, Ty). (c) The reactions A B C, with five steady states, points 1, 3, and 5 stable, (d) Isothermal operation with the rate equation = 0 /(1 -I- C y = (C o Cy/t. 

The latter kind of formulation is described at length in Sec. 7. The assumed mechanism is comprised of adsorption and desorption rates of the several participants and of the reaction rates of adsorbed species. In order to minimize the complexity of the resulting rate equation, one of the several rates in series may be assumed controlling. With several controlling steps the rate equation usually is not exphcit but can be used with some extra effort. 

When a unimolecular reaction occurs with an initial product partial pressure of the reactant A, to yield an amount of die product, jc, the first-order reaction rate equation reads... 

Several generalizations of the inelastic theory to large deformations are developed in Section 5.4. In one the stretching (velocity strain) tensor is substituted for the strain rate. In order to make the resulting constitutive equations objective, i.e., invariant to relative rotation between the material and the coordinate frame, the stress rate must be replaced by one of a class of indifferent (objective) stress rates, and the moduli and elastic limit functions must be isotropic. In the elastic case, the constitutive equations reduce to the equation of hypoelastidty. The corresponding inelastic equations are therefore termed hypoinelastic. 

Consider a reaetion involving a reaetant sueh dial A —> produets. The rate equations eorresponding to a zero, first, seeond, and third order reaetion together with their eorresponding units are ... 

This means that the eonversion is proportional to time. Eigure 3-4 shows plots of the zero order rate equations. Examples of zero order reaetions are the intensity of radiation within the vat for photoehemieal reaetions or the surfaee available in eertain solid eatalyzed gas reaetions. 

Figure 3-7 gives plots of Equations 3-54 and 3-58, respeetively. Consider the seeond order reaetion 2A-I-B—produets, whieh is first order with respeet to both A and B, and therefore seeond order overall. The rate equation is ... 

EMPIRICAL RATE EQUATIONS OF THE nth ORDER IRREVERSIBLE REACTIONS... 

An exeess of eomponent B is used, and its eoneentration remains eonstant, so that the rate equation reduees to a first order. That is. 

All other reversible seeond-order rate equations have the same solution with the boundary eonditions assumed in Equation 3-176. Table 3-4 gives solutions for some reversible reaetions. 

Integrating the rate equation is often diffieult for orders greater than 1 or 2. Therefore, the differential method of analysis is used to seareh the form of the rate equation. If a eomplex equation of the type below fits the data, the rate equation is ... 

The results of the types of reaetion being eonsidered show that the treatment of kinetie data heeomes rapidly more eomplex as the reaetion order inereases. In eases where the reaetion eonditions are sueh dial the eoneentrations of one or more of the speeies oeeurring in the rate equation remain eonstant, these terms may he ineluded in the rate eonstant k. The reaetions ean he attributed to lower order reaetions. These types of reaetions ean be defined as pseudo-nth order, where n is the sum of the exponents of those eoneentrations that ehange during the reaetion. An example of this type of reaetion is in eatalytie reaetion, where the eatalyst eoneentration remains eonstant during the reaetions. 

This method estimates the reaction order based on the reaction stoichiometry and assumptions concerning its mechanism. The assumed rate equation is then integrated to obtain a relation between the composition and time. The following procedures are used for determining the rate equations ... 

Table 3-6 gives - D with time t. For a first order rate law, the rate equation is expressed by... 

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

Assuming that the reaetion is first order in an isothermal bateh reaetor of eonstant volume, then the rate equation for the reaetion A... 

Assuming that the reactions are first order in a constant volume batch reactor, the rate equations for components A, B, C, and D, respectively, are ... 

Assume the rates to be seeond order in a eonstant volume bateh system and determine the rate equations as follows. 

From the reaetion seheme, it ean be assumed the rate equations are first order. The rate of disappearanee of eomponent A is ... 

Figure 13.36 is a plot of the preceding equation for the three types of hoods. The plot shows the curve for the actual and the worst hoods requiring a hood flow rate larger than the plume flow rate in order to get 99% fume capture. 

The power a is called the order of reaction with respect to reactant A, b is the order with respect to B, and the sum (a + b. ..) is the overall order of the reaction. Many rate equations are of forms different from Eq. (1-11)—for example, concentration teims may appear in the denominator—and then the concept of reaction order is not applicable. 

Improbable as a zero-order reaction may seem on the basis of what has been said thus far, let us consider the possibility of this rate equation ... 

Integration between the limits of c = c° when r = 0 and c = c when t = t gives the integrated zero-order rate equation. 

Equation (2-4) is the stoichiometric equation for an elementary first-order reaction, and Eq. (2-5) is the corresponding differential rate equation. 

Separating the variables and integrating between the limits shown below yields Eqs. (2-6), (2-7), and (2-8) as equivalent forms of the integrated first-order rate equation. 

---