Chemistry
Rate Equations
Rate equations in chemistry describe the speed at which a chemical reaction occurs. They are mathematical expressions that relate the rate of a reaction to the concentrations of reactants. Rate equations are often determined experimentally and can provide valuable insights into the mechanisms of chemical reactions.
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12 Key excerpts on "Rate Equations"
- A. Kayode Coker(Author)
- 2001(Publication Date)
- Gulf Professional Publishing(Publisher)
Reaction Rate Expression 109 109 CHAPTER THREE Reaction Rate Expression INTRODUCTION Chemical reactions are processes in which reactants are transformed into products. In some processes, the change occurs directly and the complete description of the mechanism of the reaction present can be relatively obtained. In complex processes, the reactants undergo a series of step-like changes and each step constitutes a reaction in its own right. The overall mechanism is made up of contributions from all the reactions that are sometimes too complex to determine from the knowledge of the reactants and products alone. Chemical kinetics and reaction mechanism as reviewed in Chapter 1 can often provide a reasonable approach in determining the reaction Rate Equations. Chemical kinetics is concerned with analyzing the dynamics of chemical reactions. The raw data of chemical kinetics are the measurements of the reactions rates. The end product explains these rates in terms of a complete reaction mechanism. Because a measured rate shows a statistical average state of the molecules taking part in the reaction, chemical kinetics does not provide information on the energetic state of the individual molecules. Here, we shall examine a series of processes from the viewpoint of their kinetics and develop model reactions for the appropriate Rate Equations. The equations are used to arrive at an expression that relates measurable parameters of the reactions to constants and to con-centration terms. The rate constant or other parameters can then be determined by graphical or numerical solutions from this relationship. If the kinetics of a process are found to fit closely with the model equation that is derived, then the model can be used as a basis for the description of the process. Kinetics is concerned about the quantities of the reactants and the products and their rates of change. Since reactants disappear in reactions, their rate expressions are given a
eBook - PDF- George W. Roberts(Author)
- 2015(Publication Date)
- Wiley(Publisher)
Chapter 2 Reaction Rates—Some Generalizations LEARNING OBJECTIVES After completing this chapter, you should be able to 1. use the Arrhenius relationship to calculate how reaction rate depends on temperature; 2. use the concept of reaction order to express the dependence of reaction rate on the individual species concentrations; 3. calculate the frequency of bimolecular and trimolecular collisions; 4. determine whether the Rate Equations for the forward and reverse rates of a reversible reaction are thermodynamically consistent; 5. calculate heats of reaction and equilibrium constants at various temperatures (review of thermodynamics). In order to design a new reactor, or analyze the behavior of an existing one, we need to know the rates of all the reactions that take place. In particular, we must know how the rates vary with temperature, and how they depend on the concentrations of the various species in the reactor. This is the field of chemical kinetics. This chapter presents an overview of chemical kinetics and introduces some of the molecular phenomena that provide a foundation for the field. The relationship between kinetics and chemical thermodynamics is also treated. The information in this chapter is sufficient to allow us to solve some problems in reactor design and analysis, which is the subject of Chapters 3 and 4. In Chapter 5, we will return to the subject of chemical kinetics and treat it more fundamentally and in greater depth. 2.1 Rate Equations A ‘‘rate equation’’ is used to describe the rate of a reaction quantitatively, and to express the functional dependence of the rate on temperature and on the species concentrations. In symbolic form, r A ¼ r A ðT , all C i Þ where T is the temperature. The term ‘‘all C i ’’ is present to remind us that the reaction rate can be affected by the concentrations of the reactant(s), the product(s), and any other compounds that are present, even if they do not participate in the reaction.
eBook - PDFFoundations of Chemistry
An Introductory Course for Science Students
- Philippa B. Cranwell, Elizabeth M. Page(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Chemists and chemical engineers must be able to control reaction conditions in order to obtain maximum yields by optimising factors such as temperature and pressure. Equally important is the time required to produce products. Fast reaction times reduce costs and energy requirements. Reaction times are critical when designing drugs and in drug deliv-ery. To be effective a drug must be sufficiently inert that it reaches its target before breaking down but must then react quickly and produce waste products that are readily removed from the body. The area of chemistry concerned with studying and controlling the rates of chemical reactions is known as chemical kinetics . Chemists study the rates at which chemical reactions occur so they can control them. The series of molecular processes that occur when a chemical reaction takes place is called the mechanism of the reaction, and understanding the mechanism helps chemists control the rate of reaction . 8.2 The rate of reaction 8.2.1 Defining the rate of a chemical reaction The rate of a chemical reaction is defined as the increase in concentration of one of the products of reaction divided by the time taken. Alternatively, it can be defined as the decrease in concentration of one of the reactants divided by the time: Rate of reaction = change in concentration of reactant or product time taken for the change A plot of concentration against time is given in Figure 8.1 for the hypothetical reaction of reactant A being converted to product B, as represented by the equa-tion A B. The rate can be expressed as: Rate of reaction = change in concentration of B time or Δ B Δ t 256 Chemical kinetics – the rates of chemical reactions The symbol Δ (Greek letter delta) means ‘ a change ’ , so Δ [B] represents a change in concentration of B and Δ t is the time taken for this change to occur. The units for reaction rate are therefore units of concentration divided by time: typically, mol dm -3 s -1 .
eBook - PDFChemistry
Principles and Reactions
- William Masterton, Cecile Hurley(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
▼ 274 ▼ Rate of Reaction 11 The rate at which the speeding train is traveling can be expressed in km/h. Similarly, the rate of a reaction can be expressed in M /h. Not every collision, not every punctilious trajectory by which billiard-ball complexes arrive at their calculable meeting places leads to reaction. Men (and women) are not as different from molecules as they think. —ROALD HOFFMANN Excerpt from Men and Molecules Chapter Outline 11-1 Meaning of Reaction Rate 11-2 Reaction Rate and Concentration 11-3 Reactant Concentration and Time 11-4 Models for Reaction Rate 11-5 Reaction Rate and Temperature 11-6 Catalysis 11-7 Reaction Mechanisms ▼ F or a chemical reaction to be feasible, it must occur at a reasonable rate. Conse-quently, it is important to be able to control the rate of reaction. Most often, this means making it occur more rapidly. When you carry out a reaction in the gen-eral chemistry laboratory, you want it to take place quickly. A research chemist trying to synthesize a new drug has the same objective. Sometimes, though, it is desirable to reduce the rate of reaction. The aging process, a complex series of biological oxida-tions, believed to involve “free radicals” with unpaired electrons such as O } H and O } O 2 is one we would all like to slow down. This chapter sets forth the principles of chemical kinetics , the study of reaction rates. The main emphasis is on those factors that influence rate. These include ■ ■ the concentrations of reactants (Sections 11-2 and 11-3). ■ ■ the process by which the reaction takes place (Section 11-4). ■ ■ the temperature (Section 11-5). ■ ■ the presence of a catalyst (Section 11-6). ■ ■ the reaction mechanism (Section 11-7). 11-1 Meaning of Reaction Rate To discuss reaction rate meaningfully, it must be defined precisely. The rate of reac-tion is a positive quantity that expresses how the concentration of a reactant or product changes with time.
eBook - PDFEngineering Chemistry
Fundamentals and Applications
- Shikha Agarwal(Author)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
[Ans (i) 1.2 × 10 –4 mol L –1 s –1 ,(ii) 3.6 × 10 –4 mol L –1 s –1 .] 11.4 Rate Law Expression It is the mathematical relation between the rate of the reaction and the concentration of the reacting species. Consider a general reaction aA + bB → products where A and B are reactants and a and b are the stoichiometric coefficients of the balanced equation. From the kinetic study of the reaction, the dependence of rate of the reaction on the concentration of the reactants is found to be Rate = k [A] p [B] q p and q are the powers of the reactants A and B on which the rate of the reaction depends and k is the rate constant. It may be noted that p and q are determined experimentally and may or may not be equal to the coefficients a and b in the reaction. Consider the reaction, 2NO (g) + O 2 (g) 2 NO 2 (g) The rate law for the above reaction is Rate = k [NO] 2 [O 2 ] In the above example, a and b are equal to p and q. Chemical Kinetics 535 Similarly, consider the decomposition of dinitrogen pentaoxide 2N 2 O 5 (g) → 4NO 2 (g) + O 2 (g) Experimental studies have shown that the rate of the reaction is proportional to [N 2 O 5 ] and not to [N 2 O 5 ] 2 . Therefore the rate law is Rate = k[N 2 O 5 ] In the above example, a and b are not equal to p and q. 11.5 Velocity Constant or Rate Constant Consider a reaction A + B → products. The rate of the reaction is given by Rate a [A][B] Rate = k [A][B] Here k is the proportionality constant known as the velocity constant or the specific reaction rate of a reaction at a given temperature. If [A] = [B] = 1 then in the above equation, Rate = k Hence, velocity constant of a reaction at a given temperature can be defined as the rate of the reaction when the concentration of each of the reactants is unity. Characteristics of rate constant 1. Rate constant is a measure of the reaction rate. Larger the value of k, faster is the reaction. Similarly, smaller value of k indicates slow reaction.
eBook - PDFElementary Chemical Reactor Analysis
Butterworths Series in Chemical Engineering
- Rutherford Aris(Author)
- 2013(Publication Date)
- Butterworth-Heinemann(Publisher)
Sometimes there is less than this, and the construction of a reactor becomes an exercise in the art of experienced guessing and scale-up. The health of the chemical industry 53 4 54 Reaction Rates Chap. 4 is a tribute to the skillful practice of this art, but the paucity of background information emphasizes the need for a general understanding of reaction and reactor behavior. The student can at least begin to get a feel for this from reaction rate expressions conceived in the conjunction of chemical theory and experiment and developed in the spirit of applied mathematics. But he should be warned that this will give him only the beginning of wisdom; it can never presume to displace the knowledge that comes of experience. In summary, all that is needed to develop a coherent description of reac-tor analysis is provided by a formula for r(£, T, P) that is in accord with chemical realities. What can be obtained in an industrial situation may, for reasons of time and money, fall far short of this precision; but whatever expression can be obtained must play the same role in practical design as does r in the theoretical analysis. 4.2 Homogeneous Reaction Rates Before developing the general expression that is often used for homogeneous reactions let us look at a simple and by now familiar reaction. Our standard way of writing the reaction A 2 + A 3 ^ Ai is A 1 -A 2 -A 3 = 0, (4.2.1) where the species A 2 and A 3 are the reactants and A x is the product. We know all about the stoichiometry of this reaction (Sec. 2.4) and about its equilibrium (Sec. 3.7) and wish presently to describe its kinetics. Consider first the situa-tion when the reactants are present in concentrations c 2 and c 3 but no product has yet been formed. The rate of reaction will depend on the concentrations of the reactants and on temperature.
eBook - PDF- Gordon Skinner(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
The rate law may change, and the rate constant and its temperature dependence may change, both in ways 4 Strictly speaking, equilibrium constants are dimensionless, and instead of actual concentrations or pressures, we put into the above equation ratios of actual concentra-tions or pressures to the standard state concentrations or pressures. In converting from K p to K c , the 1 /RT term is also taken to be dimensionless, being merely the ratio of the concentrations in the two standard states of 1 atm and 1 mole Z _ 1 . However, if one uses this convention, then all rate constants and rates must have the same units (inverse seconds, for example) and we obtain the rates by multiplying rate constants by ratios of actual concentrations to standard (unit) concentrations. Since attaching concentra-tion units to equilibrium and rate constants is such a useful method of keeping track of reaction orders, we will continue to do so, as have most practical kineticists. 13 2. HOW KINETIC RESULTS ARE EXPRESSED that are unpredictable in the absence of knowledge about the mechanism. This difficulty explains, in large part, why a knowledge of reaction mecha-nism has great practical as well as strictly scientific importance. 2.5 REACTION RATES AS DIFFERENTIAL QUANTITIES So far the rate of a reaction has been discussed as if it were an ob-servable quantity, but in most instances it is not. Usually a chemist will measure the concentration of a substance as a function of time, and deduce from this the rate during the time. From the definitions of rate laws we have used it is clear that since the concentrations are changing with time, the rates are also.
eBook - ePub- Patrick E. McMahon, Rosemary McMahon, Bohdan Khomtchouk(Authors)
- 2019(Publication Date)
- CRC Press(Publisher)
reaction step rate rule:Rule 1: The rate of any elementary reaction step is proportional to the concentrations of each reactant species involved in that step, raised to the power of the number of molecules of the specific reactant involved in that reaction step. This power will equal the reactant coefficient in the specific reaction step balanced chemical equation.The rule applies only to reaction steps, not to general balanced equations. Rate expressions for multi-step reactions cannot be analyzed by applying Rule 1 to the balanced equation. Multi-step reactions must be derived from elementary step rate expressions and the additional rules given in the next section.Example: Apply Rule 1 to the following proposed elementary reaction steps:A + B + C→ABC rate of the step=k[A][B][C]NO 2+NO 2→N 2O 4rate of the step=k[NO 2] 22 C +O 2→2 CO rate of the step=k[C] 2[O 2]CH 4+ Cl→CH 3+ HCl rate of the step=k [CH 4][Cl]Rule 2: If the complete reaction mechanism consists of only one (elementary) step, then the rate expression for the overall (complete) reaction is identical to the rate expression for the single step that describes the mechanism. In this case, the reaction step rate rule is used directly; nothing else is required.Rate expressions for multiple-step mechanisms cannot be predicted directly from the reaction step rate rule: the rule applies only to reaction steps, not to general balanced equations.Example: The balanced equation for the synthesis of ammonia is shown.N 2+ 3H 2→2NH 3Experiments have determined that this is a multi-step reaction; this could not be inferred simply from the equation. A multi-step reaction implies that the rule cannot be used directly: rate (overall) is not equal to k [N2 ] [H2 ]3
eBook - PDF- Bohdan Wojciechowski, Norman Rice(Authors)
- 2003(Publication Date)
- Elsevier Science(Publisher)
41 3. Using Kinetic Data in Reaction Studies Uninterpreted data is a neglected gem. Uncut and unpolished by the expertise of a craftsman, it is nothing but a stone devoid of bnlliance or significance. The Rate Expression The rate of a reaction is governed by reactant concentrations and by physical parameters such as pressure and temperature, but how it is governed by these conditions depends on the mechanism of the reaction. It is the mechanism that lies behind the kinetic rate ex- pression. Understanding the mechanism and its rate expression allows us to engineer the reaction by influencing elementary steps in the overall conversion process. We note from the beginning that the mechanism of a reaction does not usually change with reaction conditions; it is a fundamental property ofthe reaction. This is also true for catalytic reactions on a homologous series of catalysts at similar reaction condi- tions. In a given setting, the rates of the elementary steps of a mechanism depend solely on the concentrations of the reactants (and active sites in the case of a catalyst) and the reaction temperature. Of these, temperature affects the rate parameters only. The behav- iour of these is well understood from thermodynamic considerations and well described by the Arrhenius equation containing: 9 a pre-exponential, temperature-independent term A, multiplied by 9 an exponential term containing an energy term E. This dependence is so important that we will examine it more closely later. The concentration dependence of the rate expression has a more varied effect on the behaviour of the kinetics since it dictates the algebraic form of the rate expression. Identification of a rate expression that fits the experimental rate data and agrees with a plausible reaction mechanism for the reaction requires an extensive experimental inves- tigation of reaction rates, and much thought.
- Charles G. Hill, Thatcher W. Root(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
Unfortunately, there is no completely general method of determining the reaction rate expression or even of determining the order of a reaction. Usually, one employs an iterative trial-and-error procedure based on intelligent guesses and past experience with similar systems. Very often the stoichiometry of the reaction and knowledge of whether the reaction is “reversible” or “irreversible” will suggest a form of the rate equation to try first. If this initial guess (hypothesis) is incorrect, the investigator may then try other forms that are suggested either by assumptions about the mechanism of the reaction or by the nature of the discrepancies between the data and the mathematical model employed in a previous trial. Each reaction presents a unique problem, and success in fitting a reaction rate expression to the experimental data depends on the ingenuity of the individual investigator.The discussion that follows is largely confined to irreversible reactions with simple rate expressions of the form of equation (3.0.17), but the methods developed are more generally applicable. We have chosen this approach to keep the discussion as simple as possible and to present the material in a manner that avoids the introduction of more complex rate expressions. Most of the methods presented below are applicable regardless of the mathematical form of the rate expression, and they may readily be extended to cover the rate expressions that will be encountered in Chapters 6, 7 and 13.The techniques used to determine reaction rate expressions may be classified into three general categories:- Integral methods based on integration of the reaction rate expression. In these approaches one customarily analyzes the data by plotting some function of the reactant concentrations or the extent of reaction versus time using coordinates that would yield a straight line if the hypothesis concerning the mathematical form of the rate expression is correct.
- Differential methods based on differentiation of experimental concentration versus time data to obtain an approximation to the actual rate of reaction. In these approaches one analyzes the data by postulating various functional relations between the rate of reaction and the concentrations of the various species in the reaction mixture and tests these hypotheses using appropriate plots of the data.
- Methods based on simplification of the reaction rate expression. In these approaches one uses stoichiometric ratios of the reactants or a vast excess of one or more of the reactants in order to permit a partial evaluation of the form of the rate expression. These methods may be used in conjunction with either a differential or an integral analysis of the experimental data.
eBook - PDF- Anthonio C. Lasaga, James Kirkpatrick(Authors)
- 2018(Publication Date)
- De Gruyter(Publisher)
If we write a general overall reaction as aA + bB + . . . -s- pP + qQ + . . ., (1) the rate of the reaction can be written as a dt b dt " ' ρ dt q dt ' " η IL η η = k c a c ^ c p r q * L A Β ''* c P Q where the C represents some units of concentration and the n's can be any real number. In this section, the rate will refer to the forward rate of a reaction exclusively, -i.e., the rate at which the reaction proceeds from left to right as written. In later sections, both forward and reverse rates will be considered. The form of the rate law in equation (2) {i.e., product of concentration terms) has been validated in the majority of kinetic studies. The exact rate law may be more complicated than that in equation (2), but in most cases certain sim- plifications allow the rate law to reduce to the form in equation (2). Examples of the general rate law are given below. The units of the rate constant k will depend on the form of the rate law. Therefore, for the expression in equation (2), k has units of concentrations to the power (n^ + n^ + ... + n^ + per unit time. In the case that the reaction is heterogeneous, one or more of the "concentrations" in equation (2) should refer to the speaifia area {i.e., area/unit volume of solution) of the solids involved. In heterogeneous kinetics, it is very important to measure the total areas of the reactive solids, either directly by BET and related methods or indirectly by con- straining grain size and shape within narrow confines. The need to use aoneentration units in writing rate laws distinguishes kinetics 3 from thermodynamics. Whereas the "thermodynamic concentration," or activity, determines the equilibrium between thermodynamic components, 3 the spatial concentration (e.g., moles/cm ) of the colliding molecules determine the molecular collision rates, and hence the rates of reac- tions .
eBook - ePub- John R. Whitaker(Author)
- 2018(Publication Date)
- Routledge(Publisher)
(1), (4), and (10) ] are unity. Consider the case of the reversible reaction described by 2A + B ⇌ k − 1 k 1 3P (16) The rates of change of the various molecular species are described by the following rate. equations: 1 2 (− d A d t) = k 1 (A) 2 (B) − k − 1 (P) 3 (17) − d B d t = k 1 (A) 2 (B) − k − 1 (P) 3 (18) 1 3 (d P d t) = k 1 (A) 2 (B) − k − 1 (P) 3 (19) It will be noted. that 1 2 (− d A d t) = − d B d t = 1 3 (d P d t) = k 1 (A) 2 (B) − k − 1 (P) 3 (20) From the specific equations (16) and (20), one may then deduce that the corresponding general equations can be written. as a A + b B ⇌ k − 1 k 1 p P + q Q (21) and 1 a (− d A d t) = 1 b (− d B d t) = 1 p (d P d t) = 1 q (d Q d t) (22) Therefore, when the rate of a chemical reaction may be described by a simple stoichiometric equation, it is possible to define a rate of reaction for the system uniquely. The rate of change in the concentration of A for the forward reaction can then be written in the general form − d A d t = k A a B b C c ⋯ (23) where A, B, and C are concentrations of reactants at any time and the exponents, a, b, and c are coefficients of the balanced equation. This may or may not be true for reactions that go by alternative pathways. The molecular order of the reaction depicted in Eq. (23) is a with respect to A, b with respect to B, c with respect to C, and so on, while the overall (or total order) is the sum of the exponents a + b + c … C. Consecutive Reactions Equation (10) describes a case of consecutive reactions. This equation indicates that enzyme and substrate must combine to form an enzyme-substrate complex, which in turn reacts to form enzyme and product. The enzyme-substrate complex, EA, is an obligatory intermediate in the reaction as written. In the case of consecutive reactions it is valid to write the equation in parts as E + A ⇌ k − 1 k 1 EA (24) EA → k 2 E + P (25) D
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