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#### Fundamentals of Chemical Reaction Engineering

**By Mark E. Davis, Robert J. Davis**

**Dover Publications, Inc.**

**Copyright © 2003 Mark E. Davis and Robert J. Davis**

All rights reserved.

ISBN: 978-0-486-29131-4

All rights reserved.

ISBN: 978-0-486-29131-4

CHAPTER 1

**The Basics of Reaction Kinetics for Chemical Reaction Engineering**

**1.1 | The Scope of Chemical Reaction Engineering**

The subject of chemical reaction engineering initiated and evolved primarily to accomplish the task of describing how to choose, size, and determine the optimal operating conditions for a reactor whose purpose is to produce a given set of chemicals in a petrochemical application. However, the principles developed for chemical reactors can be applied to most if not all chemically reacting systems (e.g., atmospheric chemistry, metabolic processes in living organisms, etc.). In this text, the principles of chemical reaction engineering are presented in such rigor to make possible a comprehensive understanding of the subject. Mastery of these concepts will allow for generalizations to reacting systems independent of their origin and will furnish strategies for attacking such problems.

The two questions that must be answered for a chemically reacting system are: (1) what changes are expected to occur and (2) how fast will they occur? The initial task in approaching the description of a chemically reacting system is to understand the answer to the first question by elucidating the thermodynamics of the process. For example, dinitrogen (N2) and dihydrogen (H2) are reacted over an iron catalyst to produce ammonia (NH3):

N2 + 3H2 = 2NH3, -ΔHr = 109 kJ/mol (at 773 K)

where Δ*Hr* is the enthalpy of the reaction (normally referred to as the heat of reaction). This reaction proceeds in an industrial ammonia synthesis reactor such that at the reactor exit approximately 50 percent of the dinitrogen is converted to ammonia. At first glance, one might expect to make dramatic improvements on the production of ammonia if, for example, a new catalyst (a substance that increases the rate of reaction without being consumed) could be developed. However, a quick inspection of the thermodynamics of this process reveals that significant enhancements in the production of ammonia are not possible unless the temperature and pressure of the reaction are altered. Thus, the constraints placed on a reacting system by thermodynamics should always be identified first.

**EXAMPLE 1.1.1**

In order to obtain a reasonable level of conversion at a commercially acceptable rate, ammonia synthesis reactors operate at pressures of 150 to 300 atm and temperatures of 700 to 750 K. Calculate the equilibrium mole fraction of dinitrogen at 300 atm and 723 K starting from an initial composition of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (*Xi* is the mole fraction of species *i*). At 300 atm and 723 K, the equilibrium constant, *Ka,* is 6.6 × 10-3. (K. Denbigh, *The Principles of Chemical Equilibrium,* Cambridge Press, 1971, p. 153).

**Answer**

(See **Appendix A** for a brief overview of equilibria involving chemical reactions):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The definition of the activity of species *i* is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] fugacity at the standard state, that is, 1 atm for gases and thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Use of the Lewis and Randall rule gives:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] fugacity coefficient of pure component *i* at *T* and *P* of system then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Upon obtaining each [bar.φ]*i* from correlations or tables of data (available in numerous references that contain thermodynamic information):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If a basis of 100 mol is used ([xi] is the number of moles of N2 reacted):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, [xi] = 13.1 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. At 300 atm, the equilibrium mole fraction of ammonia is 0.36 while at 100 atm it falls to approximately 0.16. Thus, the equilibrium amount of ammonia increases with the total pressure of the system at a constant temperature.

The next task in describing a chemically reacting system is the identification of the reactions and their arrangement in a *network.* The kinetic analysis of the network is then necessary for obtaining information on the rates of individual *reactions* and answering the question of how fast the chemical conversions occur. Each reaction of the network is stoichiometrically simple in the sense that it can be described by the single parameter called the *extent of reaction* (see **Section 1.2**). Here, a stoichiometrically simple reaction will just be called a reaction for short. The expression "simple reaction" should be avoided since a stoichiometrically simple reaction does not occur in a simple manner. In fact, most chemical reactions proceed through complicated sequences of *steps* involving reactive intermediates that do not appear in the stoichiometries of the reactions. The identification of these intermediates and the sequence of steps are the core problems of the kinetic analysis.

If a step of the sequence can be written as it proceeds at the molecular level, it is denoted as an *elementary step* (or an *elementary reaction*), and it represents an irreducible molecular event. Here, elementary steps will be called *steps* for short. The hydrogenation of dibromine is an example of a stoichiometrically simple reaction:

H2 + Br2 [??] 2HBr

If this reaction would occur by H2 interacting directly with Br2 to yield two molecules of HBr, the step would be elementary. However, it does not proceed as written. It is known that the hydrogenation of dibromine takes place in a sequence of two steps involving hydrogen and bromine atoms that do not appear in the stoichiometry of the reaction but exist in the reacting system in very small concentrations as shown below (an initiator is necessary to start the reaction, for example, a photon: Br2 + light -> 2Br, and the reaction is terminated by Br + Br + *TB ->* Br2 where *TB* is a third body that is involved in the recombination process—see below for further examples):

Br + H2 -> HBr + H

H + Br2 -> HBr + Br

In this text, stoichiometric reactions and elementary steps are distinguished by the notation provided in **Table 1.1.1**.

In discussions on chemical kinetics, the terms *mechanism* or *model* frequently appear and are used to mean an assumed reaction network or a plausible sequence of steps for a given reaction. Since the levels of detail in investigating reaction networks, sequences and steps are so different, the words *mechanism* and *model* have to date largely acquired bad connotations because they have been associated with much speculation. Thus, they will be used carefully in this text.

As a chemically reacting system proceeds from reactants to products, a number of species called *intermediates* appear, reach a certain concentration, and ultimately vanish. Three different types of intermediates can be identified that correspond to the distinction among networks, reactions, and steps. The first type of intermediates has reactivity, concentration, and lifetime comparable to those of stable reactants and products. These intermediates are the ones that appear in the reactions of the network. For example, consider the following proposal for how the oxidation of methane at conditions near 700 K and atmospheric pressure may proceed (see Scheme 1.1.1). The reacting system may evolve from two stable reactants, CH4 and O2, to two stable products, CO2 and H2O, through a network of four reactions. The intermediates are formaldehyde, CH2O; hydrogen peroxide, H2O2; and carbon monoxide, CO. The second type of intermediate appears in the sequence of steps for an individual reaction of the network. These species (e.g., free radicals in the gas phase) are usually present in very small concentrations and have short lifetimes when compared to those of reactants and products. These intermediates will be called *reactive intermediates* to distinguish them from the more stable species that are the ones that appear in the reactions of the network. Referring to Scheme 1.1.1, for the oxidation of CH2O to give CO and H2O2, the reaction may proceed through a postulated sequence of two steps that involve two reactive intermediates, CHO and HO2. The third type of intermediate is called a *transition state,* which by definition cannot be isolated and is considered a species in transit. Each elementary step proceeds from reactants to products through a transition state. Thus, for each of the two elementary steps in the oxidation of CH2O, there is a transition state. Although the nature of the transition state for the elementary step involving CHO, O2, CO, and HO2 is unknown, other elementary steps have transition states that have been elucidated in greater detail. For example, the configuration shown in **Fig. 1.1.1** is reached for an instant in the transition state of the step:

OH- + C2H5Br -> HOC2H5 + Br-

The study of elementary steps focuses on transition states, and the kinetics of these steps represent the foundation of chemical kinetics and the highest level of understanding of chemical reactivity. In fact, the use of lasers that can generate femtosecond pulses has now allowed for the "viewing" of the real-time transition from reactants through the transition-state to products (A. Zewail, *The Chemical Bond: Structure and Dynamics,* Academic Press, 1992). However, in the vast majority of cases, chemically reacting systems are investigated in much less detail. The level of sophistication that is conducted is normally dictated by the purpose of the work and the state of development of the system.

The transition state (trigonal bipyramid) of the elementary step:

OH- + C2H5Br -> HOC2H5 + Br-

The nucleophilic substituent OH- displaces the leaving group Br-.

**1.2 | The Extent of Reaction**

The changes in a chemically reacting system can frequently, but not always (e.g., complex fermentation reactions), be characterized by a stoichiometric equation. The stoichiometric equation for a simple reaction can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.1)

where *NCOMP* is the number of components, *Ai,* of the system. The stoichiometric coefficients, *vi* are positive for products, negative for reactants, and zero for inert components that do not participate in the reaction. For example, many gas-phase oxidation reactions use air as the oxidant and the dinitrogen in the air does not participate in the reaction (serves only as a diluent). In the case of ammonia synthesis the stoichiometric relationship is:

N2 + 3H2 = 2NH3

Application of Equation (1.2.1) to the ammonia synthesis, stoichiometric relationship gives:

0 = 2NH3 - N2 - 3H2

For stoichiometric relationships, the coefficients can be ratioed differently, e.g., the relationship:

0 = 2NH3 - N2 - 3H2

can be written also as:

0 = NH3 - 1/2N2 - 3/2H2

since they are just mole balances. However, for an elementary reaction, the stoichiometry is written as the reaction should proceed. Therefore, an elementary reaction such as:

2NO + O2 -> 2NO2 (correct)

CANNOT be written as:

NO + 1/2O2 -> NO2 (not correct)

**EXAMPLE 1.2.1**

If there are several simultaneous reactions taking place, generalize Equation (1.2.1) to a system of *NRXN* different reactions. For the methane oxidation network shown in Scheme 1.1.1, write out the relationships from the generalized equation.

**Answer**

If there are *NRXN* reactions and *NCOMP* species in the system, the generalized form of Equation (1.2.1) is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.2)

For the methane oxidation network shown in Scheme 1.1.1:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or in matrix form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that the sum of the coefficients of a column in the matrix is zero if the component is an intermediate.

Consider a closed system, that is, a system that exchanges no mass with its surroundings. Initially, there are n0i moles of component *Ai* present in the system. If a single reaction takes place that can be described by a relationship defined by Equation (1.2.1), then the number of moles of component *Ai* at any time *t* will be given by the equation:

ni(t) = n0i + viΦ(t) (1.2.3)

that is an expression of the *Law of Definitive Proportions* (or more simply, a mole balance) and defines the parameter, Φ, called the *extent of reaction.* The extent of reaction is a function of time and is a natural reaction variable.

Equation (1.2.3) can be written as:

Φ(t) = ni(t) - n0i/vi (1.2.4)

Since there is only one Φ for each reaction:

ni(t) - n0i/vi = nj(t) - n0j/vj (1.2.5)

or

nj(t) = n0j + (vj/vi) [ni(t) - n0i] (1.2.6)

Thus, if *ni* is known or measured as a function of time, then the number of moles of all of the other reacting components can be calculated using Equation (1.2.6).

**EXAMPLE 1.2.2**

If there are numerous, simultaneous reactions occurring in a closed system, each one has an extent of reaction. Generalize Equation (1.2.3) to a system with *NRXN* reactions.

**Answer**

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.7)

**EXAMPLE 1.2.3**

Carbon monoxide is oxidized with the stoichiometric amount of air. Because of the high temperature, the equilibrium:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

has to be taken into account in addition to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The total pressure is one atmosphere and the equilibrium constants of reactions (1) and (2) are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and *Xi* is the mole fraction of species *i* (assuming ideal gas behavior). Calculate the equilibrium composition.

**Answer**

Assume a basis of 1 mol of CO with a stoichiometric amount of air ([xi]1 and [xi]2 are the number of moles of N2 and CO reacted, respectively):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The simultaneous solution of these two equations gives:

[xi]1 = 0.037, [xi]2 = 0.190

Therefore,

[TABLE OMITTED]

*(Continues...)*

Excerpted fromFundamentals of Chemical Reaction EngineeringbyMark E. Davis, Robert J. Davis. Copyright © 2003 Mark E. Davis and Robert J. Davis. Excerpted by permission of Dover Publications, Inc..

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