Lecture notes on Chemical Reaction Engineering

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Fundamentals of Chemical Reaction Engineering___1 The Basics of Reaction Kinetics for Chemical Reaction Engineering 1.1 I 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 chem­ icals in a petrochemical application. However, the principles developed for chemi­ cal reactors can be applied to most if not all chemically reacting systems (e.g., at­ mospheric 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 (N ) and dihydrogen (H ) are reacted over an iron catalyst 2 2 to produce ammonia (NH ): 3 N + 3H = 2NH , - b.H, = 109 kllmol (at 773 K) 3 2 2 where b.H, is the enthalpy of the reaction (normally referred to as the heat of reac­ tion). This reaction proceeds in an industrial ammonia synthesis reactor such that at the reactor exit approximately 50 percent of the dinitrogen is converted to ammo­ nia. At first glance, one might expect to make dramatic improvements on the production of ammonia if, for example, a new catalyst (a substance that increases2 CHAPTER 1 The Basics of Reaction Kinetics for Chemical Reaction Engineering the rate of reaction without being consumed) could be developed. However, a quick inspection of the thermodynamics of this process reveals that significant enhance­ ments 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 sys­ tem by thermodynamics should always be identified first. EXAMPLE 1.1.1 I In order to obtain a reasonable level of conversion at a commercially acceptable rate, am­ monia 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 X = 0.25, X = 0.75 (Xi is the mole fraction of species i). Hz N2 3 At 300 atm and 723 K, the equilibrium constant, K , is 6.6 X 10- . (K. Denbigh, The Prin­ a ciples of Chemical Equilibrium, Cambridge Press, 1971, p. 153). • Answer (See Appendix A for a brief overview of equilibria involving chemical reactions):CHAPTER 1 The Basics of Rear.tion Kinetics for Chemical Reaction Engineering 3 The definition of the activity of species i is: fugacity at the standard state, that is, 1 atm for gases and thus K = _lN3 (,)I/2(Y/2 fNH; J;2J;2 I atm a fI/2 f3/2 (t'O ) N, H, JNH Use of the Lewis and Randall rule gives: /; = Xj cPj P, cPj = fugacity coefficient of pure component i at T and P of system then XNH; cPNH; I K = K K-K = - P- 1 atm a X (p P XlI2 X /2 -:1,1(2-:1,3/2 3 N, H, 'VN, 'VH, Upon obtaining each cPj from correlations or tables of data (available in numerous ref­ erences that contain thermodynamic information): If a basis of 100 mol is used (g is the number of moles of N reacted): 2 N 25 2 75 Hz NH o 3 total 100 then (2g)(100 - 2g) - = 2.64 2 (25 - g)l/2(75 - 3g)3/ Thus, g = 13.1 and X , (25 - 13.1)/(100 26.2) = 0.16. At 300 atm, the equilibrium N 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.4 CHAPTER 1 The Basics of Reaction Kinetics for Chemical Reaction Engineering The next task in describing a chemically reacting system is the identifica­ tion 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 individ­ ual 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 Sec­ tion 1.2). Here, a stoichiometrically simple reaction will just be called a reaction for short. The expression "simple reaction" should be avoided since a stoichio­ metrically simple reaction does not occur in a simple manner. In fact, most chem­ ical reactions proceed through complicated sequences of steps involving reactive intermediates that do not appear in the stoichiometries of the reactions. The iden­ tification 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 ir­ reducible molecular event. Here, elementary steps will be called steps for short. The hydrogenation of dibromine is an example of a stoichiometrically simple reaction: If this reaction would occur by Hz interacting directly with Brz to yield two mole­ cules of HBr, the step would be elementary. However, it does not proceed as writ­ ten. 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 stoi­ chiometry of the reaction but exist in the reacting system in very small concentra­ tions as shown below (an initiator is necessary to start the reaction, for example, a Brz + light -+ 2Br, and the reaction is terminated by Br + Br + TB -+ Brz photon: where TB is a third body that is involved in the recombination process-see below for further examples): Br + Hz -+ HBr + H H + Brz -+ HBr + Br In this text, stoichiometric reactions and elementary steps are distinguished by the notation provided in Table 1.1.1. Table 1.1.1 I Notation used for stoichiometric reactions and elementary steps. Irreversible (one-way) Reversible (two-way) Equilibrated Rate-determiningCHAPTER 1 The Basics of Reaction Kinetics for Chemical Reaction EnginAering 5 In discussions on chemical kinetics, the terms mechanism or model fre­ quently appear and are used to mean an assumed reaction network or a plausi­ ble sequence of steps for a given reaction. Since the levels of detail in investi­ gating 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 care­ fully 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 compara­ ble to those of stable reactants and products. These intermediates are the ones that appear in the reactions of the network. For example, consider the follow­ ing proposal for how the oxidation of methane at conditions near 700 K and atmospheric pressure may proceed (see Scheme l.l.l). The reacting system may evolve from two stable reactants, CH and ° , to two stable products, CO and 4 2 2 H 0, through a network of four reactions. The intermediates are formaldehyde, 2 CH 0; hydrogen peroxide, H 0 ; and carbon monoxide, CO. The second type 2 2 2 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 pres­ ent in very small concentrations and have short lifetimes when compared to those of reactants and products. These intermediates will be called reactive in­ termediates 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 CH 0 to give CO and H 0 , the reaction may proceed through a 2 2 2 postulated sequence of two steps that involve two reactive intermediates, CHO and H0 . The third type of intermediate is called a transition state, which by 2 definition cannot be isolated and is considered a species in transit. Each ele­ mentary step proceeds from reactants to products through a transition state. Thus, for each of the two elementary steps in the oxidation of CH 0, there is 2 a transition state. Although the nature of the transition state for the elementary step involving CHO, 02' CO, and H0 is unknown, other elementary steps have 2 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: 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 gen­ erate femtosecond pulses has now allowed for the "viewing" of the real-time transition from reactants through the transition-state to products (A. Zewail, TheCHAPTER 1 The Basics of Reaction Kinetics for Chemical Reaction Engineering 6CHAPTER 1 The Basics of Reaction Kinetics for Chemical Reaction Engineering 7 Br Br- Br I ) C H/ I "'CH H H 3 I H 3 H '"C/ CH I OW OH OH • J ) .. Figure 1.1.1 I The transition state (trigonal bipyramid) of the elementary step: OH- + C H Br HOC H + Br- 2 s 2 s The nucleophilic substituent OH- displaces the leaving group Br-.8 CHAPTER 1 The Basics of Reaction Kinetics for Chemical Reaction Engineering 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. 1.2 I 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: NCOMP 0= L: viA; (1.2.1) i=1 where NCOMP is the number of components, A;, of the system. The stoichiomet­ ric 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 par­ ticipate in the reaction (serves only as a diluent). In the case of ammonia synthesis the stoichiometric relationship is: Application of Equation (1.2.1) to the ammonia synthesis, stoichiometric relation­ ship gives: For stoichiometric relationships, the coefficients can be ratioed differently, e.g., the relationship: can be written also as: since they are just mole balances. However, for an elementary reaction, the stoi­ chiometry is written as the reaction should proceed. Therefore, an elementary re­ action such as: 2NO + O -+ 2N0 (correct) 2 2 CANNOT be written as: (not correct)CHAPTER 1 The Basics of Reaction Kinetics for Chemical Reaction Engineering 9 EXAMPLE 1.2.1 I If there are several simultaneous reactions taking place, generalize Equation (1.2.1) to a sys­ tem 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 Equa­ tion (1.2.1) is: NCOMP o = 2: vi,jA , j 1; ",NRXN (1.2.2) i i For the methane oxidation network shown in Scheme 1.1.1: 0= OCO + IHzO lO + OCO + OHzO + lCH O - ICH z z z z 4 0= OCO + OHp - lO + lCO + 1HzO - lCH O + OCH z z z z 4 o = ICO + ORzO Oz - ICO + OHzO + OCHzO + OCH z z 4 0= OCO + IH O + Oz + OCO - I HzO + OCHp + OCH z z z 4 or in matrix form: 1 o o I o 1 -I 1 -1 o o o -ol HzO z -1 I o o CHp CH 4 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 sur­ roundings. Initially, there are n? moles of component Ai present in the system. If a single reaction takes place that can be described by a relationship defined by Equa­ tion (1.2.1), then the number of moles of component Ai at any time t will be given by the equation: ni (t) = n? + Vi p(t) (1.2.3) that is an expression of the Law of Definitive Proportions (or more simply, a mole balance) and defines the parameter, P, 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: p(t) = ni (t) - n? (1.2.4) Vi10 CHAPTER 1 The Basics of Reaction Kinetics for Chemical Reaction Engineering Since there is only one P for each reaction: (1.2.5) or (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 I 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 NRXN (1.2.7) nj = n? + 2: Vj,jP j jl EXAMPLE 1.2.3 I Carbon monoxide is oxidized with the stoichiometric amount of air. Because of the high temperature, the equilibrium: N + Oz =:: 2NO (1) z has to be taken into account in addition to: (2) The total pressure is one atmosphere and the equilibrium constants of reactions (l) and (2) are: (XNO)Z K , x (XN,)(X ,)' o whereK , = 8.26 X 1O-3,K , = 0.7, andXjis the mole fraction of species i (assuming ideal x x gas behavior). Calculate the equilibrium composition. • Answer Assume a basis of 1 mol of CO with a stoichiometric amount of air (; I and tz are the num­ ber of moles of N and CO reacted, respectively): zCHAPTER 1 The Basics of Reaction Kinetics for Chemical Reaction Engineering 11 gj N 1.88 1.88 z gj Oz 0.5 0.5 g2 CO 1 1 gz gz COZ 0 NO 0 2gj total 3.38 3.38 gz The simultaneous solution of these two equations gives: gj = 0.037, gz = 0.190 Therefore, N 0.561 z Oz 0.112 CO 0.247 COz 0.058 NO 0.022 1.000 EXAMPLE 1.2.4 I Using the results from Example 1.2.3, calculate the two equilibrium extents of reaction. • Answer q q P1 g1 = 0.037 Q p Q = 0.190 2 VIGNETTE 1.2.112 CHAPTER 1 The Basics of Reaction Kinetics for Chemical Reaction EngineeringCHAPTER 1 The Basics of Reaction Kineticshemical Reaction Engineering 13 significantly contributed to pollution reduction and are one of the major success stories for chemical reaction ensim,ering. Insulation cover The drawback of I is that it is an extensive variable, that is, it is dependent upon the mass of the system. The fractional conversion, f, does not suffer from this problem and can be related to 1. In general, reactants are not initially present in stoichiometric amounts and the reactant in the least amount determines the maxi­ mum value for the extent of reaction, lmax. This component, called the limiting component (subscript f) can be totally consumed when I reaches lmax. Thus, (1.2.8) The fractional conversion is defined as: f(t) = I(t) (1.2.9) P max and can be calculated from Equations (1.2.3) and (1.2.8): (1.2.10) Equation (1.2.10) can be rearranged to give: (1.2.11) where 0 :::; fi :::; 1. When the thermodynamics of the system limit I such that it can­ not reach lmax (where n/ 0), I will approach its equilibrium value leg (n/ =1= 0 value of n/ determined by the equilibrium constant). When a reaction is limited by thermodynamic equilibrium in this fashion, the reaction has historically been called14 CHAPTER 1 The Basics of Reaction Kinetics for Chemical Rear,tion Engineering reversible. Alternatively, the reaction can be denoted as two-way. When peg is equal to P for all practical purposes, the reaction has been denoted irreversible or one­ max way. Thus, when writing the fractional conversion for the limiting reactant, (1.2.12) where f14 is the fractional conversion at equilibrium conditions. Consider the following reaction: aA + bB + ... = sS + wW + ... (1.2.13) Expressions for the change in the number of moles of each species can be written in terms of the fractional conversion and they are assume A is the limiting reac­ tant, lump all inert species together as component I and refer to Equations (1.2.6) and (1.2.11): nTO = nOTAL + n TAL or n s + w + ... a b ... j (1.2.14) + nOTAL nOTAL a By defining SA as the molar expansion factor, Equation (1.2.14) can be written as: (1.2.15) where (1.2.16)CHAPTER 1 The Basics of Reaction Kinetics for Chemical Reaction Engineering 15 Notice that SA contains two tenus and they involve stoichiometry and the initial mole fraction of the limiting reactant. The parameter SA becomes important if the density of the reacting system is changing as the reaction proceeds. EXAMPLE 1.2.5 I Calculate SA for the following reactions: (i) n-butane = isobutane (isomerization) (ii) n-hexane =? benzene + dihydrogen (aromatization) (iii) reaction (ii) where 50 percent of the feed is dinitrogen. • Answer (i) CH CH CH CH = CH CH(CH h, pure n-butane feed 3 2 2 3 3 3 noTAL o nTOTAL 1-11 _ nOTAL 4 + 1- 1 SA - 0 = 4 nTOTAL 1-11 (iii) CH3CH2CH2CH2CH2CH3 ==0 + 4H , 50 percent of feed is n-hexane 2 SA = 0.5;oTAL 4 + 1 - 1 = 2 nTOTAL 1-11 EXAMPLE 1.2.6 I If the decomposition of N 0 into N 0 and O were to proceed to completion in a closed 2 S 2 4 2 volume of size V, what would the pressure rise be if the starting composition is 50 percent N 0 and 50 percent N ? 2 S 2 • Answer The ideal gas law is: PV = nTOTALRgT (R : universal gas constant) g At fixed T and V, the ideal gas law gives:CHAPTER 1 The Basics of Reaction Kinetics for Chemical Reaction Engineering 16 The reaction proceeds to completion so fA = 1 at the end of the reaction. Thus, with A _- 0.50nOTAL 1 + 0.5 1 o = 0.25 nTOTAL 1-11 Therefore, p pO = 1.25 1.3 I The Rate of Reaction For a homogeneous, closed system at uniform pressure, temperature, and composition in which a single chemical reaction occurs and is represented by the stoichiometric Equation (1.2.1), the extent of reaction as given in Equation (1.2.3) increases with time, t. For this situation, the reaction rate is defmed in the most general way by: dp (mOl) (1.3.1) dt time This expression is either positive or equal to zero if the system has reached equi­ librium. The reaction rate, like P, is an extensive property of the system. A specific rate called the volumic rate is obtained by dividing the reaction rate by the volume of the system, V: 1 dp mol ) r= (1.3.2) ( V dt time-volume Differentiation of Equation (1.2.3) gives: (1.3.3) dni = VidP Substitution of Equation (1.3.3) into Equation (1.3.2) yields: 1 dni r= (1.3.4) v;V dt since Vi is not a function of time. Note that the volumic rate as defined is an exten­ sive variable and that the definition is not dependent on a particular reactant or prod­ uct. If the volumic rate is defined for an individual species, ri, then: 1 dn; r· = v·r= (1.3.5) I , V dtCHAPTER 1 The Basics of Reaction Kinetics for Chemical Reaction Engineering 17 Since Vi is positive for products and negative for reactants and the reaction rate, dp/ dt, is always positive or zero, the various ri will have the same sign as the Vi (dnJdt has the same sign as ri since r is always positive). Often the use of molar concentrations, C , is desired. Since C = nJV, Equation (1.3.4) can be written as: i i i r = _1_ d (CV) = I dC + dV (1.3.6) v;V dt I Vi dt ViV dt Note that only when the volume of the system is constant that the volumic rate can be written as: I dC i r = - , constant V (1.3.7) Vi dt When it is not possible to write a stoichiometric equation for the reaction, the rate is normally expressed as: r = (COEF) dni (COEF) = , reactant (1.3.8) V dt ' +, product For example, with certain polymerization reactions for which no unique stoichio­ metric equation can be written, the rate can be expressed by: I dn r=- V dt where n is the number of moles of the monomer. Thus far, the discussion of reaction rate has been confined to homogeneous reactions taking place in a closed system of uniform composition, temperature, and pressure. However, many reactions are heterogeneous; they occur at the in­ terface between phases, for example, the interface between two fluid phases (gas­ liquid, liquid-liquid), the interface between a fluid and solid phase, and the inter­ face between two solid phases. In order to obtain a convenient, specific rate of reaction it is necessary to normalize the reaction rate by the interfacial surface area available for the reaction. The interfacial area must be of uniform composi­ tion, temperature, and pressure. Frequently, the interfacial area is not known and alternative definitions of the specific rate are useful. Some examples of these types of rates are: I dp mol ) r=- (specific rate) ( gm dt mass-time 1 dp mol ) r= (areal rate) ( SA dt area-time where gm and SA are the mass and surface area of a solid phase (catalyst), respec­ tively. Of course, alternative definitions for specific rates of both homogeneous and heterogeneous reactions are conceivable. For example, numerous rates can be defined18 C H APT E R 1 The Basics of Reaction Kinetics for Chemical Reaction Engineering for enzymatic reactions, and the choice of the definition of the specific rate is usu­ ally adapted to the particular situation. For heterogeneous reactions involving fluid and solid phases, the areal rate is a good choice. However, the catalysts (solid phase) can have the same surface area but different concentrations of active sites (atomic configuration on the catalyst capable of catalyzing the reaction). Thus, a definition of the rate based on the num­ ber of active sites appears to be the best choice. The turnover frequency or rate of turnover is the number of times the catalytic cycle is completed (or turned-over) per catalytic site (active site) per time for a reaction at a given temperature, pressure, reactant ratio, and extent of reaction. Thus, the turnover frequency is: 1 dn r = (1.3.9) t S dt where S is the number of active sites on the catalyst. The problem of the use of r t is how to count the number of active sites. With metal catalysts, the number of metal atoms exposed to the reaction environment can be determined via techniques such as chemisorption. However, how many of the surface atoms that are grouped into an active site remains difficult to ascertain. Additionally, different types of active sites probably always exist on a real working catalyst; each has a different reaction rate. Thus, r is likely to be an average value of the catalytic activity and a lower t bound for the true activity since only a fraction of surface atoms may contribute to the activity. Additionally, r is a rate and not a rate constant so it is always neces­ t sary to specify all conditions of the reaction when reporting values of r • t The number of turnovers a catalyst makes before it is no longer useful (e.g., due to an induction period or poisoning) is the best definition of the life of the catalyst. In prac­ 6 tice, the turnovers can be very high, 10 or more. The turnover frequency on the other hand is commonly in the range of r = 1 S-1 to r = 0.01 s-1 for practical applications. t t Values much smaller than these give rates too slow for practical use while higher values give rates so large that they become influenced by transport phenomena (see Chapter 6). EXAMPLE 1.3.1 I Gonzo and Boudart 1. Catal., 52 (1978) 462 studied the hydrogenation of cyclohexene over Pd supported on A1 0 and Si0 at 308 K, atmospheric pressure of dihydrogen and 0.24M 2 3 2 cyclohexene in cyclohexane in a stirred flask: Pd 0 + H === 2 O 4 The specific rates for 4.88 wt. % Pd on Al 0 and 3.75 wt. % Pd on Si0 were 7.66 X 10- 2 3 2 3 and 1.26 X 10- mo/(gcat . s), respectively. Using a technique called titration. the percent­ age of Pd metal atoms on the surface of the Pd metal particles on the Al 0 and Si0 was 2 3 2 21 percent and 55 percent, respectively. Since the specific rates for Pd on Al 0 and Si0 2 3 2 are different, does the metal oxide really influence the reaction rate?CHAPTER 1 The Basics of Reac1lollKLoetics for Chemical Reaction Engineering 19 Titration is a technique that can be used to measure the number of surface metal atoms. The procedure involves first chemisorbing (chemical bonds formed between adsorbing species and surface atoms) molecules onto the metal atoms exposed to the reaction environment. Second, the chemisorbed species are reacted with a second component in order to recover and count the number of atoms chemisorbed. By knowing the stoichiometry of these two steps, the number of surface atoms can be calculated from the amount of the recovered chemisorbed atoms. The technique is illustrated for the problem at hand: Pd metal particle Oxygen atoms chemisorbed on surface Pd atoms \ (Step I) G2J + °z • Metal oxide Metal oxide Step I (only surface Pd) 2Pd, + Oz -+ 2PdP Step II (not illustrated) PdP + Hz =? Pd,H + HzO By counting the number of Hz molecules consumed in Step II, the number of surface Pd atoms (PdJ can be ascertained. Thus, the percentage of Pd atoms on the surface can be cal­ culated since the total number of Pd atoms is known from the mass of Pd. • Answer The best way to determine if the reaction rates are really different for these two catalysts is to compare their values of the turnover frequency. Assume that each surface Pd atom is an active site. Thus, to convert a specific rate to a turnover frequency: _ I ( mOl) ( gcat ) (mOleCUlar weight of metal) r (s ) = r . . t gcat. s mass metal fraction of surface atoms 4 = (7.66 X 10- ). (oo88)(106.4)(o.21)-1 1 8.0s- Likewise for Pd on SiO , z Since the turnover frequencies are approximately the same for these two catalysts, the metal oxide support does not appear to influence the reaction rate. 1.4 I General Properties of the Rate Function for a Single Reaction The rate of reaction is generally a function of temperature and composition, and the development of mathematical models to describe the form of the reaction rate is a central problem of applied chemical kinetics. Once the reaction rate is known,

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