Chemistry
Rate Law
Rate law in chemistry refers to the mathematical expression that relates the rate of a chemical reaction to the concentration of reactants. It is determined experimentally and provides insight into the reaction mechanism. The rate law equation is typically of the form rate = k[A]^m[B]^n, where k is the rate constant and m and n are the reaction orders with respect to the reactants A and B, respectively.
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12 Key excerpts on "Rate Law"
eBook - PDF- Paul Flowers, Klaus Theopold, Richard Langley, William R. Robinson(Authors)
- 2019(Publication Date)
- Openstax(Publisher)
12.3 Rate Laws LEARNING OBJECTIVES By the end of this section, you will be able to: • Explain the form and function of a Rate Law • Use Rate Laws to calculate reaction rates • Use rate and concentration data to identify reaction orders and derive Rate Laws As described in the previous module, the rate of a reaction is often affected by the concentrations of reactants. Rate Laws (sometimes called differential Rate Laws) or rate equations are mathematical expressions that describe the relationship between the rate of a chemical reaction and the concentration of its reactants. As an example, consider the reaction described by the chemical equation 12.3 • Rate Laws 607 where a and b are stoichiometric coefficients. The Rate Law for this reaction is written as: in which [A] and [B] represent the molar concentrations of reactants, and k is the rate constant, which is specific for a particular reaction at a particular temperature. The exponents m and n are the reaction orders and are typically positive integers, though they can be fractions, negative, or zero. The rate constant k and the reaction orders m and n must be determined experimentally by observing how the rate of a reaction changes as the concentrations of the reactants are changed. The rate constant k is independent of the reactant concentrations, but it does vary with temperature. The reaction orders in a Rate Law describe the mathematical dependence of the rate on reactant concentrations. Referring to the generic Rate Law above, the reaction is m order with respect to A and n order with respect to B. For example, if m = 1 and n = 2, the reaction is first order in A and second order in B. The overall reaction order is simply the sum of orders for each reactant. For the example Rate Law here, the reaction is third order overall (1 + 2 = 3). A few specific examples are shown below to further illustrate this concept.
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 - PDF- David Ball(Author)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Finally, we will discuss a little bit of theoretical kinetics, to leave you with the idea that not all kinetics is phenomenological. More and more, basic physical chemical principles are applied at the molecular level in attempts to describe adequate models for chemical reactions—which are, after all, of fundamental interest to chemists. 20.2 Rates and Rate Laws One of the most basic descriptions of a chemical reaction is how fast it goes. But when we speak of how fast a reaction goes, we are not thinking “fast” as in a velocity in meters per second. Rather, we are thinking about how quickly amounts (that is, moles) of reactants are converted into amounts (moles) of products. The “quick-ness” implies that time (in units of seconds, minutes, hours, days, and so on) will be a concern also. The rate of a reaction is an indication of how many moles of a reactant or product are reacted or produced over a period of time. Rates of reactions are a central issue in kinetics. Understand that it is difficult to predict before the fact how fast a reaction will be (although we will explore some of the factors that influence the rate of reactions). A lot of information about kinetics of reactions is experimentally determined. Reaction rates also pro-vide the fundamental information needed to deduce the individual actions that reactant species take in order to make products. (We will consider this near the end of this chapter.) Furthermore, in a closed system, the rates of most reactions change over time. Typically, amounts of reactants decrease over time. When discussing rates of reactions, it is important to indicate at what point along the extent of the reaction we are. (Extents of reaction, j , were discussed previously in Chapter 5.) It is con-ventional to define rates of reactions as they would be at the very beginning of a chemical process, in which only reactants are present, no products.
eBook - PDF- Paul Flowers, Klaus Theopold, Richard Langley, William R. Robinson(Authors)
- 2015(Publication Date)
- Openstax(Publisher)
Catalysis will be discussed in greater detail later in this chapter as it relates to mechanisms of reactions. Link to Learning Chapter 12 | Kinetics 659 Figure 12.8 The presence of a catalyst increases the rate of a reaction by lowering its activation energy. Chemical reactions occur when molecules collide with each other and undergo a chemical transformation. Before physically performing a reaction in a laboratory, scientists can use molecular modeling simulations to predict how the parameters discussed earlier will influence the rate of a reaction. Use the PhET Reactions & Rates interactive (http://openstaxcollege.org/l/16PHETreaction) to explore how temperature, concentration, and the nature of the reactants affect reaction rates. 12.3 Rate Laws By the end of this section, you will be able to: • Explain the form and function of a Rate Law • Use Rate Laws to calculate reaction rates • Use rate and concentration data to identify reaction orders and derive Rate Laws As described in the previous module, the rate of a reaction is affected by the concentrations of reactants. Rate Laws or rate equations are mathematical expressions that describe the relationship between the rate of a chemical reaction and the concentration of its reactants. In general, a Rate Law (or differential Rate Law, as it is sometimes called) takes this form: rate = k[ A] m [B] n [C] p … in which [A], [B], and [C] represent the molar concentrations of reactants, and k is the rate constant, which is specific for a particular reaction at a particular temperature. The exponents m, n, and p are usually positive integers (although it is possible for them to be fractions or negative numbers). The rate constant k and the exponents m, n, and p must be determined experimentally by observing how the rate of a reaction changes as the concentrations of the reactants are Link to Learning 660 Chapter 12 | Kinetics This OpenStax book is available for free at http://cnx.org/content/col11760/1.9
eBook - PDFChemistry
The Molecular Nature of Matter
- James E. Brady, Neil D. Jespersen, Alison Hyslop(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
Experiments yielded the following results: Initial Concentrations (mol L -1 ) Initial Rate of Formation of C (mol L -1 s -1 ) [A] [B] 0.40 0.30 1.00 Ž 10 -4 0.60 0.30 2.25 Ž 10 -4 0.80 0.60 1.60 Ž 10 -3 (a) What is the Rate Law for the reaction? (b) What is the value of the rate constant? (c) What are the units for the rate constant? (d) What is the overall order of this reaction? (Hint: Solve for the exponent of [A], then use it to solve for the exponent of [B].) 13.4 | Integrated Rate Laws The Rate Law tells us how the speed of a reaction varies with the concentrations of the reac- tants. Often, however, we are more interested in how the concentrations change over time. For instance, if we were preparing some compound, we might want to know how long it will take for the reactant concentrations to drop to some particular value, so we can decide when to isolate the products. The relationship between the concentration of a reactant and time can be derived from a Rate Law using calculus. By summing or “integrating” the instantaneous rates of a reaction from the start of the reaction until some specified time, t, we can obtain integrated Rate Laws that quantitatively give concentration as a function of time. The form of the integrated Rate Law depends on the order of the reaction. The mathematical expressions that relate concentra- tion and time in complex reactions can be complicated, so we will concentrate on using integrated Rate Laws for a few simple first- and second-order reactions with only one reactant. First-Order Reactions A first-order reaction is a reaction that has a Rate Law of the type rate = k 3 A 4 Practice Exercise 13.13 Practice Exercise 13.14 644 Chapter 13 | Chemical Kinetics Using calculus, 2 the following equation can be derived that relates the concentration of A and time: ln 3 A 4 0 3 A 4 t = kt (13.5) The symbol “ln” means natural logarithm.
eBook - PDFChemistry
The Molecular Nature of Matter
- Neil D. Jespersen, Alison Hyslop(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Experiments yielded the following results: Initial Concentrations (mol L −1 ) Initial Rate of Formation of C (mol L −1 s −1 ) [A] [B] 0.40 0.30 1.00 × 10 −4 0.60 0.30 2.25 × 10 −4 0.80 0.60 1.60 × 10 −3 (a) What is the Rate Law for the reaction? (b) What is the value of the rate constant? (c) What are the units for the rate constant? (d) What is the overall order of this reaction? (Hint: Solve for the exponent of [A], then use it to solve for the exponent of [B].) 13.4 Integrated Rate Laws 665 13.4 Integrated Rate Laws The Rate Law tells us how the speed of a reaction varies with the concentrations of the reac- tants. Often, however, we are more interested in how the concentrations change over time. For instance, if we were preparing some compound, we might want to know how long it will take for the reactant concentrations to drop to some particular value, so we can decide when to isolate the products. The relationship between the concentration of a reactant and time can be derived from a Rate Law using calculus. By summing or “integrating” the instantaneous rates of a reaction from the start of the reaction until some specified time, t, we can obtain integrated Rate Laws that quantitatively give concentration as a function of time. The form of the integrated Rate Law depends on the order of the reaction. The mathematical expressions that relate concen- tration and time in complex reactions can be complicated, so we will concentrate on using integrated Rate Laws for a few simple first- and second-order reactions with only one reactant. First-Order Reactions A first-order reaction is a reaction that has a Rate Law of the type rate = k[A] Using calculus, 2 the following equation can be derived that relates the concentration of A and time: ln [A] 0 ____ [A] t = kt (13.5) The symbol “ln” means natural logarithm.
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 - 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.
eBook - PDF- Chung K. Law(Author)
- 2010(Publication Date)
- Cambridge University Press(Publisher)
A prominent example is the unimolecular reaction to be studied in Section 2.3, for which the reaction order is 1 at high pressures but becomes 2 at low pressures. The measured reaction order is sometimes called the pseudo- molecularity of the reaction. This potential loss of sensitivity at large concentrations also forms the basis of the isolation method in determining the reaction orders of individual components of a complex reaction scheme. That is, by keeping all but one of the reactants in high concentrations, the reaction order of the lean component can be approximately identified as the apparent overall reaction order, which can be easily measured. This concept is also useful in modeling premixed combustion phenomena because the concentration of the stoichiometrically abundant reactant can be assumed to be constant during the course of reaction. Since reaction order describes both elementary and global reactions, it will be used from now on in specifying all reactions. 2.2. THEORIES OF REACTION RATES: BASIC CONCEPTS 2.2.1. The Arrhenius Law The specific reaction rate constant k(T) gives the functional dependence of the re- action rate on temperature. For an elementary reaction the Arrhenius law states that d ln k(T) dT = E a R o T 2 , (2.2.1) where E a is called the activation energy of the reaction, having the unit of cal/mole or joule/mole. If E a is a constant with respect to temperature, integrating Eq. (2.2.1) yields k(T) = Ae −E a / R o T , (2.2.2) where A is called the frequency factor or the preexponential factor. We note that since R o is a constant, it is convenient to define a new quantity, T a = E a R o , and call it the activation temperature of the reaction. Furthermore, following tradi- tion, we shall also use the uppercase letter to designate molar quantities. For constant values of A and E a , a plot of ln k(T) versus 1/ T exhibits a linear relationship, with A and E a respectively determined from the intercept and slope of such a plot.
eBook - PDFChemistry
An Atoms First Approach
- Steven Zumdahl, Susan Zumdahl, Donald J. DeCoste, , Steven Zumdahl, Steven Zumdahl, Susan Zumdahl, Donald J. DeCoste(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
b. The integrated Rate Law shows how concentration depends on time. The integrated Rate Laws corresponding to zero-order, first-order, and second-order kinetics of one-reactant reactions are given in Table 11.6. 3. Whether we determine the differential Rate Law or the integrated Rate Law depends on the type of data that can be collected conveniently and accurately. Once we have experimentally determined either type of Rate Law, we can write the other for a given reaction. 4. The most common method for experimentally determining the differential Rate Law is the method of initial rates. In this method several experiments are run at different initial concentrations and the instantaneous rates are determined for each at the same value of t (as close to t 5 0 as possible). The point is to evaluate the rate before the concentrations change significantly from the initial values. From a comparison of the initial rates and the initial concentrations, the dependence of the rate on the concen- trations of various reactants can be obtained—that is, the order in each reactant can be determined. 5. To experimentally determine the integrated Rate Law for a reaction, concentrations are measured at various values of t as the reaction proceeds. Then the job is to see which integrated Rate Law correctly fits the data. Typically this is done visually by ascertaining which type of plot gives a straight line. A summary for one-reactant reactions is given in Table 11.6. Once the correct straight-line plot is found, the correct integrated Rate Law can be chosen and the value of k obtained from the slope. Also, the (differential) Rate Law for the reaction can then be written. (Box continues on the following page) 466 CHAPTER 11 Chemical Kinetics Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
- Charles G. Hill, Thatcher W. Root(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
In addition to spectrophotometric or spectroscopic measurements, there are a number of other optical measurements that can be used to monitor the course of various reactions. Among the optical properties that can be used for these studies are optical rotation, refractive indices, fluorescence, and colorimetry. There also are several electrical measurements that may be used for analysis of solutions under in situ conditions. Among the properties that may be measured are dielectric constants, electrical conductivity or resistivity, and the redox potential of solutions. These properties are easily measured with instrumentation that is readily interfaced to a computer for data acquisition and manipulation for analysis. However, most of these techniques should be used only after careful calibration and do not give better than 1% accuracy without unusual care in the experimental work.3.3 Techniques for the Interpretation of Kinetic Data
In Section 3.1 the mathematical expressions that result from integration of various reaction rate functions were discussed in some detail. Our present problem is the converse of that considered earlier (i.e., given data on the concentration of a reactant or product as a function of time, how does one proceed to determine the reaction rate expression?).Determination of the rate expression normally involves a two-step procedure. First, the concentration dependence is determined at a fixed temperature. Then the temperature dependence of the rate constants is evaluated to obtain a complete reaction rate expression. The form of this temperature dependence is given by equation (3.0.14), so our present problem reduces to that of determining the form of the concentration dependence and the value of the rate constant at the temperature of the experiment.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.
eBook - PDF- Young, William Vining, Roberta Day, Beatrice Botch(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
The Rate Laws for each step in this mechanism are different, so how is it possible to determine the Rate Law for the net reaction? The rate of a reaction can be no greater than the rate of the slowest step in the mechanism, the rate-determining step (sometimes called the rate-limiting step ). Thus, it makes sense that a reaction’s Rate Law directly Table 15.6.1 Rate Law and Elementary Steps Molecularity Order of Elementary Step Example Rate Law Unimolecular First order A S products rate 5 k 3 A 4 Bimolecular Second order A 1 B S products 2A S products rate 5 k 3 A 43 B 4 rate 5 k 3 A 4 2 Termolecular Third order A 1 B 1 C S products 2 A 1 B S products 3A S products rate 5 k 3 A 43 B 43 C 4 r ate 5 k 3 A 4 2 3 B 4 r ate 5 k 3 A 4 3 Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 Unit 15 Chemical Kinetics 487 relates to the stoichiometry of the rate-determining step in the mechanism. For the H 2 O 2 decomposition reaction shown previously, the experimentally determined Rate Law is rate 5 k 3 H 2 O 2 43 I 2 4 . This Rate Law matches the Rate Law for the first step in the mechanism. This suggests that the first step in the decomposition reaction is the rate-determining step in the mechanism. The experimentally determined Rate Law is one of the most useful pieces of information to have when trying to determine a reaction mechanism. The process of proving or disprov-ing a mechanism usually involves three different steps: 1. Proposing a mechanism, including identifying the rate-determining step. 2. Predicting the Rate Law using the stoichiometry of the rate-determining step. 3. Determining the experimental Rate Law for the reaction using one of the methods cov-ered earlier in this unit. If the predicted and experimental Rate Laws do not match, then the proposed mechanism is incorrect. If they match, then the proposed mechanism might be correct.
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