Chemistry
The Change of Concentration with Time
"The Change of Concentration with Time" refers to the rate at which the concentration of a substance changes over a specific period. In chemical reactions, this concept is crucial for understanding reaction kinetics and determining the order of a reaction. By measuring how the concentration of reactants or products varies with time, scientists can gain insights into the underlying mechanisms of chemical reactions.
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11 Key excerpts on "The Change of Concentration with Time"
- Frank R. Foulkes(Author)
- 2012(Publication Date)
- CRC Press(Publisher)
1. In general, the concentrations initially change quite rapidly with time, and then level off as they 1 Unless the reaction started 75,000 years ago! 2 Sometimes, for gas phase reactions, we plot pressure vs. time instead of concentration vs. time; but plotting concentration is more common. products reactants Concn time c o 0 eqm Fig. 1 26-2 CHEMICAL REACTION KINETICS approach equilibrium. 3 Usually, as products begin to form, there is the possibility that some of them will start reacting back to reactants again: C + D A A + B In general the process is a dynamic one in which the net rate is given by: Reaction Rate Net = Forward Rate – Reverse Rate . . . [1] At equilibrium, the net rate equals zero; i.e., we have a dynamic equilibrium for which Forward Rate = Reverse Rate . . . [2] The reverse rate generally starts to become significant as equilibrium is approached. 26.3 EXPRESSION OF REACTION RATES When the reactants are first brought together (i.e., before any significant amount of product has formed), the net rate is approximately just the forward rate. Consider the reaction (in a constant volume reactor) of hydrogen with oxygen to form water: 2H 2(g) + O 2(g) A 2H 2 O (g) The rates of change with time of the concentrations of the various reactants and products are related according to < 1 2 d [H ] dt 2 = < d [O ] dt 2 = + 1 2 d [H O] dt 2 . . . [3] That is, H 2 is used up twice as fast as O 2 , and H 2 O is produced at the same rate at which H 2 is consumed. In general, for the reaction aA + bB A cC + dD Rate (mol L –1 s –1 ) = r = < 1 a d[A] dt = < 1 b d[B] dt = + 1 c d[C] dt = + 1 d d[D] dt . . . [4] Example 26-1 The rate of the reaction A + 2B A 3C + D was reported as 2.0 mol L –1 s –1 . State the rates of consumption or formation of each participant. 3 The introductory treatment of chemical kinetics given in this chapter deals mainly with constant volume batch processes in closed systems.
eBook - PDFProblems in Metallurgical Thermodynamics and Kinetics
International Series on Materials Science and Technology
- G. S. Upadhyaya, R. K. Dube, D. W. Hopkins(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
CHAPTER 9 KINETICS 9.1 Introduction A chemical or metallurgical reaction is thermodynamically possible only when there is a decrease in free energy. All the equations which we use in the thermodynamic treatment of a metallurgical reaction refer to equilibrium conditions. A reaction may be thermodynamically possible, but in practice the reaction may not proceed to completion in a measurable period of time. In other words, the thermodynamic treatment does not provide information on the rate of reaction. For this reason, another theoretical approach -'kinetics 1 - has been used to study the rate of reaction. The rate or velocity of a reaction may be defined as the rate of decrease of the concentration of a reactant or as the rate of increase of a product of the reaction. If a reactant of initial concentration C has a concentra-tion C at any time t , the rate is expressed as (-dC/dt). If the con-centration of the product is x at any time t , the rate is expressed as (dx/dt). 9.2 Effect of Concentration on the Reaction Rate The rate of a chemical reaction is proportional to the concentration of the reacting substances. The sum of the powers to which the concentration of the reacting atoms or molecules must be raised to determine the rate of reaction, is known as the 'order of reaction'. The order of reaction does not bear any relation to the molecularity of the reaction. The expressions for the rates of reactions of different orders can be evaluated as follows. 203 204 PROBLEMS IN METALLURGICAL THERMODYNAMICS AND KINETICS 9.2.1 First-Order Reaction In a first-order reaction, for example, A = X + Y, the rate of reaction is given by -£ = kC, (9.1) at where C is the concentration of A at any time t , and k is a constant known as the velocity constant, rate constant, or specific reaction rate. On integrating Eq.(9.1) within the limits C = C at t = 0, and C = C at t = t, k = 1^91 log A . , (9.2) t C - x o where x is the amount of A reacted in time t.
eBook - PDFEngineering Chemistry
Fundamentals and Applications
- Shikha Agarwal(Author)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
The rate of the reaction may be expressed as Rate of reaction = Decrease in concentration of A Time taken Rate of reaction = Increase in concentration of B Time taken Figure 11.1 Variation of concentration of reactants /products with time 530 Engineering Chemistry: Fundamentals and Applications If D[A] is the decrease in the concentration of reactant A in time D[t]and D[B] is the increase in the concentration of B in the same time duration then the change in the concentration of reactants and products during a time interval D[t] can also be expressed as Rate of reaction = [A] t D − D or, Rate of reaction = + [B] t D D The square brackets express the molar concentration (moles/L) of the reactants or products. The value of D[A] will be negative as the concentration of reactants is decreasing with time (D[A] = Final conc of A – initial conc of A). This would make the rate of reaction [A] t D D negative. However, the rate of a reaction cannot be a negative quantity; therefore, a minus sign is put before D[A] / Dt so that the rate becomes positive. The minus sign simply indicates that as time increases the concentration of the reactant A decreases. Thus, Rate of reaction [A] [B] t t D D = − = + D D In the above example the stoichiometric coefficients of A and B are same. Therefore, the rate of decrease in concentration of A is equal to the rate of increase in concentration of B. Consider another reaction A + B → C + D The rate of the reaction may be expressed by any one of the following expressions [A] [B] [C] [D] t t t t D D D D − = − = + = + D D D D Rate of reaction when the stoichiometric coefficients of reactant and products are different Consider the reaction X + Y → 2Z In the above example, one mole of X reacts with one mole of Y to give two moles of Z. This means that the rate of disappearance of X is equal to the rate of disappearance of Y, but the rate of appearance of Z is twice the rate of disappearance of either X or Y.
- Nils O. Petersen(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
Recall that we describe reactions in terms of the order of reactions, which reflects the number of species involved in the rate limiting step of the reaction. We are generally interested in measuring the rate of a chemical reaction because it can be a clue to the mechanism of the reaction. In order to measure the rate, we would initiate the reaction with only the reactant(s) present and then measure its (their) disappearance or the appearance of the product(s). Measuring concentration changes ■ 207 First order kinetics For a first order reaction A k1 → B 1 the rate of disap- pearance of the reactant is − d[A] dt = k 1 [A], (13.13) which leads to [A] = [A] 0 e −k1t , (13.14) where [A] 0 is the initial concentration of the reactant. Correspondingly, the rate of appearance of the product (see problems as- signment) is [B] = [A] 0 ( (1 − e −k1t ) . (13.15) When the reaction is reversible, the reverse reaction B k2 → A will become important as the product concentration increases and the reactant concentra- tion decreases. At some point in time, the concentrations reach the equilibrium concentrations where the rate of the forward reaction equals that of the reverse reaction, and hence − d[A] dt = k 1 [A] eq = d[B] dt = k 2 [B] eq , (13.16) which tells us that the equilibrium constant K = [B]eq [A]eq = k1 k2 . Parallel first order kinetics We shall later encounter parallel first order reactions, in which a single reactant can yield two or more products: A kB → B and A kC → C. This leads to a faster rate of disappearance of A such that − d[A] dt = (k B + k C )[A], (13.17) which leads to [A] = [A] 0 e −(kB+kC)t . (13.18) Correspondingly, d[B] dt = k B [A] = k B [A] 0 e −(kB+kC)t d[C] dt = k C [A] = k C [A] 0 e −(kB+kC)t (13.19) 1 We will use the symbol → for the forward reaction, ← for the reverse reaction, ⇋ for a reversible reaction, and = for the reaction at equilibrium.
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 - ePub- Nils O. Petersen(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
This is possible in many systems and is the classical way to measure transport phenomena. We shall see later that it is also possible to measure the rates of diffusion or flow of molecules by measuring fluctuations of numbers of molecules in small volumes. 13.1.2 Reaction kinetics Chemical reactions will change the concentrations of the species involved in the reaction until an equilibrium is reached, but even at equilibrium, the concentrations will fluctuate. Recall that we describe reactions in terms of the order of reactions, which reflects the number of species involved in the rate limiting step of the reaction. We are generally interested in measuring the rate of a chemical reaction because it can be a clue to the mechanism of the reaction. In order to measure the rate, we would initiate the reaction with only the reactant(s) present and then measure its (their) disappearance or the appearance of the product(s). First order kinetics For a first order reaction A → k 1 B 1 the rate of disappearance of the reactant is (13.13) - d [ A ] d t = k 1 [ A ], which leads to (13.14) [ A ] = [ A ] 0 e - k 1 t, where [ A ] 0 is the initial concentration of the reactant. Correspondingly, the rate of appearance of the product (see problems assignment). is (13.15) [ B ] = [ A ] 0 ((1 - e - k 1 t). When the reaction is reversible, the reverse reaction B → k 2 A will become important as the product concentration increases and the reactant concentration decreases. At some point in time, the concentrations reach the equilibrium concentrations where the rate of the forward reaction equals that of the reverse reaction, and hence (13.16) - d [ A ] d t = k 1 [ A ] e q = d [ B ] d t = k 2 [ B ] e q, which tells us that the equilibrium constant K = [ B ] e q [ A ] e q = k 1 k 2. Parallel first order kinetics We shall later encounter parallel first order reactions, in which a single reactant can yield two or more products: A → k B B and A → k C C
- Charles G. Hill, Thatcher W. Root(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
A may be determined), it is possible to use equation (3.1.47) or (3.1.50) to determine the fraction conversion of the limiting reagent at this time. Equation (3.1.51) may then be used to determine the reaction rate corresponding to this conversion:3.1.51Thus, for both variable and constant volume batch reactors, one can manipulate concentration versus time data to obtain values of the reaction rate as a function of time or as a function of the concentrations of the various species present in the reaction mixture. The task then becomes one of fitting these data to a reaction rate expression of the form of equation (3.0.13).Since data are almost invariably taken under isothermal conditions to eliminate the temperature dependence of reaction rate constants, one is concerned primarily with determining the concentration dependence of the rate expression φ(Ci) and the rate constant at the temperature in question. Next we consider two differential methods that can be used in the analysis of rate data.3.3.1.1 Differentiation of Data Obtained in the Course of a Single Experimental Run
If one has experimental results in the form of concentration versus time data, the following general differential procedure may be used to determine φ(Ci) and the rate constant at the temperature in question:- Set forth a hypothesis as to the form of the concentration dependent portion of the rate function, φ(Ci).
- From the experimental concentration versus time data, determine reaction rates corresponding to various times.
- At the selected times, prepare a table listing the reaction rate and the concentrations of the various species present in the reaction mixture. Calculate φ(Ci) at each of these points.
- Prepare a plot of the reaction rate versus φ(Ci). If the plot is linear and passes through the origin, the form of φ(Ci
eBook - PDFChemistry
The Molecular Nature of Matter
- 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 - ePubWater Quality Engineering
Physical / Chemical Treatment Processes
- Mark M. Benjamin, Desmond F. Lawler(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
P are expressed on a molar basis (e.g., mol/L min), they must be related as follows:(3-1a)(3-1b)Thus, by characterizing r A as a function of system composition and temperature, we are implicitly characterizing r B and r P as well.The chapter begins with an introduction to fundamental concepts and terminology that are used to describe reaction kinetics, followed by a discussion of the approaches that are commonly used to evaluate reaction kinetics empirically. The presentation focuses on the conversion of raw experimental data into mathematical expressions that can be used to model reaction kinetics, thereby allowing the rate of reaction to be predicted for conditions that have not been studied directly. The types of reactions that are explored in this section gradually increase in complexity, from single, irreversible reactions to overall reactions that might include multiple and/or reversible steps. In the process of describing these reactions, the idea of a characteristic reaction time is introduced, and the linkage between the kinetics of reversible reactions and the equilibrium constant for those reactions is explained. In the final section, the dependence of reaction rates on temperature is described.Many textbooks have been written devoted to reaction kinetics and reactor engineering. Excellent texts that consider reaction kinetics primarily in the context of basic chemical and chemical engineering systems include those by Levenspiel (1999) and Hill (1977). Applications of kinetics to environmental systems are introduced in many water chemistry texts (e.g., Stumm and Morgan, 1996; Morel and Hering, 1993; Benjamin, 2010), and are covered at a more advanced level in texts devoted specifically to that topic (e.g., Brezonik, 1994; Stumm, 1990).
eBook - PDFAn Introduction to Chemical Metallurgy
International Series on Materials Science and Technology
- R. H. Parker, D. W. Hopkins(Authors)
- 2016(Publication Date)
- Pergamon(Publisher)
Continuous tem-perature measurement will also be necessary as temperature influences reaction rates considerably, and it is encouraging to hear that techniques of continuous pyrometry are being de-veloped at the same time as continuous analysis for reactions in liquid phases at temperatures as high as 1600°C. There are two types of reaction which will be considered: Homogeneous reactions take place entirely within one phase, such as reactions between gas molecules to produce gaseous products, or reactions in an aqueous solution where the reactants and products all remain dissolved in water. Heterogeneous reactions involve more than one phase, such as the transfer of a substance from liquid slag to a liquid metal in a smelting process, or the reaction between gaseous oxygen and a solid metal to form an oxide film on the metal surface. Homogeneous reaction mechanisms tend to be less complex than those of heterogeneous reactions, so that the preliminary theoretical treatment will be mainly that for homogeneous REACTION KINETICS 123 reactions. Metallurgical processes usually involve heterogen-eous reactions, and consideration is given later in this chapter and Chapter 6 to the kinetics of these reactions. 4.2. Effect of Concentration of Reacting Substances It was first demonstrated by L. Wilhelmy (1850) that the rate of a chemical reaction is proportional to the concentration of the reacting substances. This led to the formulation of the Law of Mass Action and the concept of the Equilibrium Con-stant which were discussed in Chapter 2. For the moment we will assume that the temperature, which has an important effect on reaction rates, remains constant. We can define the order of the reaction as the sum of the powers to which the concentrations of the reacting atoms or molecules must be raised to determine the rate of reaction.
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