Chemistry Chemical Kinetics - Molecular Approach PDF

Chemistry: A Molecular Approach Chapter 15 Chemical Kinetics What does rate of reaction mean? The speed of different chemical reactions varies hugely. Some reactions are very fast and others are very slow. rusting baking...

Chemistry: A Molecular Approach Chapter 15 Chemical Kinetics What does rate of reaction mean? The speed of different chemical reactions varies hugely. Some reactions are very fast and others are very slow. rusting baking explosion slow fast very fast Chemical Kinetics Chemical kinetics is the study of the factors that affect the rates of chemical reactions, such as temperature. The speed of a chemical reaction is called its reaction rate. The rate of a reaction is a measure of how fast the reaction makes products or uses reactants. The ability to control the speed of a chemical reaction is important. Defining Rate Rate is how much a quantity changes in a given period of time. The speed you drive your car is a rate—the distance your car travels (miles) in a given period of time (1 hour). – So, the rate of your car has units of mi/hr. Defining Reaction Rate The rate of a chemical reaction is generally measured in terms of how much the concentration of a reactant decreases (or product concentration increases) in a given period of time. For reactants, a negative sign is placed in front of the definition. – For the reaction H2 𝑔 + l2 𝑔 → 2 Hl 𝑔 Δ H2 H2 𝑡2 − H2 𝑡1 Rate = − =− Δ𝑡 𝑡2 − 𝑡1 Δ means “change in” [ ] means molar concentration t represents time The Rate of a Chemical Reaction Δ H2 H2 𝑡2 − H2 𝑡1 Rate = − =− Δ𝑡 𝑡2 − 𝑡1 ▪ For products, the reaction rate is a positive number, and for reactants, it is a negative number. ▪ Typically, reaction rates decrease with time because reactant concentrations decrease as reactants are converted to products. ▪ It always occurs in a rate expression for a reactant in order to indicate a decrease in concentration and to give a positive value for the rate. Reaction Rate Changes over Time As time goes on, the rate of a reaction generally slows down because the concentration of the reactants decreases. At some time, the reaction stops, either because the reactants run out or because the system has reached equilibrium. Reaction Rate and Stoichiometry Rates of reaction can be defined in terms of the change in concentration per unit of time of any reactant or any product aA + bB → cC + dD The change in the concentration of each substance is multiplied by 1/coefficient. 1Δ A 1Δ B 1Δ C 1Δ D Rate = − =− =+ =+ 𝑎 Δ𝑡 𝑏 Δ𝑡 𝑐 Δ𝑡 𝑑 Δ𝑡 5H2O2 (aq)+ 2MnO4- (aq) + 6H+(aq) → 2Mn2+ (aq) + 8H2O(l) + 5O2(g) At a certain time during the titration, the rate of appearance of O2(g) was 1.0x10-3 mol/(Lxs). What was the rate of disappearance of MnO4- at the same time? 5H2O2 (aq)+ 2MnO4- (aq) + 6H+(aq) → 2Mn2+ (aq) + 8H2O(l) + 5O2(g) At a certain time during the titration, the rate of appearance of O2(g) was 1.0x10-3 mol/(Lxs). What was the rate of disappearance of MnO4- at the same time? −1 Δ MnO4− 1 Δ O2 =+ 2 Δ𝑡 5 Δ𝑡 Δ MnO4− 𝟐 Δ O2 𝟐 = − = − (1.0x10-3)= - 4.0x10-4 mol/(Lxs) Δ𝑡 𝟓 Δ𝑡 𝟓 Factors Affecting Reaction Rate: Reactant Concentration Rate often depends on the concentration of one or more of the reactant molecules. Rate law is an equation relating concentration of reactants to rate when the reverse reaction is negligible. The Rate Law The rate of a reaction is directly proportional to the concentration of each reactant raised to a power. For the reaction A → products, the rate law would have the form given below. Rate = 𝑘 A 𝑛 – n is called the order; usually, it is an integer that determines rate dependence on reactant concentration. – k is called the rate constant. Reaction Order We determine the order of each reactant from experimental data. The resulting rate law would have the following form. Rate = k A m B n where m is the reaction order with respect to A and n is the reaction order with respect to B The reaction is said to have an overall order of (m+n) Reaction Order The exponent on each reactant in the rate law is called the order with respect to that reactant. The sum of the exponents on the reactants is called the order of the reaction. In the rate law, Rate = k[N O]2[O2], the reaction is second order with respect to [NO], first order with respect to [O2], and third order overall. Rate = k[A]n If a reaction is zero order, the rate of the reaction is always the same. Rate = 𝑘 A 0 = 𝑘 – Doubling [A] will have no effect on the reaction rate. If a reaction is first order, the rate is directly proportional to the reactant concentration. rate = 𝑘 A 1 = 𝑘 A – Doubling [A] will double the rate of the reaction. If a reaction is second order, the rate is directly proportional to the square of the reactant concentration. – Doubling [A] will quadruple the rate of the reaction. Rate = 𝑘 A 2 Conceptual Connection 15.2 For a particular reaction in which A → products, a doubling of the concentration of A causes the reaction rate to double. What is the order of the reaction? a. 0 b. 1 c. 2 Conceptual Connection 15.2 For a particular reaction in which A → products, a doubling of the concentration of A causes the reaction rate to double. What is the order of the reaction? Example 15.2 Determining the Order and Rate Constant of a Reaction Consider the reaction between nitrogen dioxide and carbon monoxide: The initial rate of the reaction is measured at several different concentrations of the reactants, and the tabulated results are shown here. From the data, determine: a. the rate law for the reaction b. the rate constant (k) for the reaction Solution a. Between the first two experiments, the concentration of NO2 doubles, the concentration of CO stays constant, and the rate quadruples, suggesting that the reaction is second order in NO2. Between the second and third experiments, the concentration of NO2 stays constant, the concentration of CO doubles, and the rate remains constant (the small change in the least significant figure is simply experimental error), suggesting that the reaction is zero order in CO. Write the overall rate expression. Rate = k[NO2]2[CO]0 = k[NO2]2 Determining the Order of a Reaction The rate law must be determined experimentally. We can use the method of initial rates, where data from different experiments with varying starting concentrations of reactants and the corresponding initial rates are given. Determine how rate is impacted by change in a single reactant in two different experiments. Given the following. [A](M) [B] (M) Rate (M/s) 0.1 0.02 0.005 0.2 0.02 0.02 0.1 0.01 0.0025 What is the rate law for the reaction A + 2B → C Possible Answers: Rate = k [A] Rate = k [A] [B] Rate = k [B] Rate = k [A] [B] 2 Rate = k [A]2[B] Correct answer: Rate = k [A]2[B] ▪ When B doubles and A stays the same the rate doubles, making it first order with respect to compound B. [A](M) [B] (M) Rate (M/s) 0.1 0.02 0.005 0.2 0.02 0.02 0.1 0.01 0.0025 Correct answer: Rate = k [A]2[B] ▪ When B doubles and A stays the same the rate doubles, making it first order with respect to compound B. ▪ When compound A doubles and B stays the same, the rate quadruples, making it second order in regards to A. [A](M) [B] (M) Rate (M/s) 0.1 0.02 0.005 0.2 0.02 0.02 0.1 0.01 0.0025 Integrated Rate Laws The rate law shows the relationship between rate and concentration. It is useful to have an equation relating concentration with time. Using calculus, we can obtain the integrated rate law that shows the relationship between the concentration of A and the time of the reaction. Determining the Rate Law When There Are Multiple Reactants (1 of 2) Changing each reactant will affect the overall rate of the reaction. By changing the initial concentration of one reactant at a time, the effect of each reactant’s concentration on the rate can be determined. In examining results, we compare differences in rate for reactions that differ only in the concentration of one reactant. Rate = k A m B n where m is the reaction order with respect to A and n is the reaction order with respect to B Half-Life The half-life, t1/2 , of a reaction is the time required for the concentration of the reactant to fall to half its initial value. The half-life of the reaction depends on the order of the reaction. First-Order Reactions First-order reactions are very common. The hydrolysis of aspirin, the hydrolysis of the anticancer drugs SO2Cl2 → Cl2 + SO2 2N2O5 → O2 + 4NO2 2H2O2 → 2H2O + O2  First-Order Reactions In a first-order reaction, the reaction rate is directly proportional to the concentration of one of the reactants. First-order reactions often have the general form A → products. The differential rate for a first-order reaction is as follows: Δ[A] rate= − = k[A] Δt First-Order Reactions 1 Rate law: rate = 𝑘 A =𝑘 A Integrated rate law: 𝑙𝑛[A] = −k𝑡 + 𝑙𝑛[A]𝑖𝑛𝑖𝑡𝑖𝑎𝑙 [𝐴] ln = −kt [A]𝑖𝑛𝑖𝑡𝑖𝑎𝑙 When rate = M/sec 0.693 t1/2 = k = s −1 k First-Order Reactions Rate law: rate = 𝑘 A 1 = 𝑘 A Integrated rate law: 𝑙𝑛[A] = −k𝑡 + 𝑙𝑛[A]𝑖𝑛𝑖𝑡𝑖𝑎𝑙 This eq. has the form of the algebraic equation for a straight line Graph ln[A] versus time—straight line with slope = −k and y-intercept = ln[A]initial – Used to determine the rate constant First-Order Integrated Rate Law 𝑙𝑛[A] = −k𝑡 + 𝑙𝑛[A]𝑖𝑛𝑖𝑡𝑖𝑎𝑙 Second-Order Reactions Dimerization reactions in which two smaller molecules, each called a monomer, combine to form a larger molecule (a dimer). Decomposition of NO2 to NO and O2 and the decomposition of HI to I2 and H2. Second-Order Reactions The simplest kind of second-order reaction is one whose rate is proportional to the square of the concentration of one reactant. These generally have the form 2A → products. A second kind of second-order reaction has a reaction rate that is proportional to the product of the concentrations of two reactants. Such reactions generally have the form A + B → products. Second-Order Reactions Rate = 𝑘 A 2 1 1 = 𝑘𝑡 + A A initial 1 Graph A versus time—straight line with slope = k and 1 y − intercept = 𝐀 initial – Used to determine the rate constant 1 t½ = k[A]𝑖𝑛𝑖𝑡𝑖𝑎𝑙 When Rate = M/sec, k = M−1 ⋅ s−1 Second-Order Integrated Rate Law 1 1 = 𝑘𝑡 + A A initial Zero-Order Reactions Photochemical reaction between hydrogen and chlorine Decomposition of N2O on hot platinum surface Decomposition of NH3 in presence of molybdenum or tungsten is a zero-order reaction. A zeroth-order reaction that takes place in the human liver is the oxidation of ethanol (from alcoholic beverages) to acetaldehyde, catalyzed by the enzyme alcohol dehydrogenase. Zero-Order Reactions Rate = 𝑘 A 0 =𝑘 – Constant rate reactions [A] = −kt + [A]initial Graph of [A] versus time—straight line with slope = −k and y-intercept = [A]initial Ainitial t½ = 2k When Rate = M/sec, k = M/sec Rate Laws Summarized Graphical Determination of the Rate Law for A → Product 1 Plots of [A] versus time, ln[A] versus time, and A versus time allow determination of whether a reaction is zero, first, or second order. Whichever plot gives a straight line determines the order with respect to [A]. – If linear is [A] versus time, Rate = 𝑘 A 0. – If linear is ln[A] versus time, Rate = 𝑘 A 1. 1 – If linear is A versus time, Rate = 𝑘 A 2. Test Preparation The oxidation of ammonia produces nitrogen and water via the following reaction: 4NH3(g) + 3O2(g) --> 2N2(g) + 6H2O(l) Suppose the rate of formation of H2O( l) is 3.0 mol/(L ·s). Which of the following statements is true? A. The rate of consumption of NH3 is 2.0 mol/(L s). B. The rate of consumption of O2 is 2.0 mol/(L s). C. The rate of formation of N2 is 1.3 mol/(L s). D. The rate of formation of N2 is 2.0 mol/(L s). E. The rate of consumption of NH3 is 0.50 mol/(L s). Test Preparation 4NH3(g) + 3O2(g) --> 2N2(g) + 6H2O(l) Suppose the rate of formation of H 2O( l) is 3.0 mol/(L ·s). Which of the following statements is true? Answer A. The rate of consumption of NH 3 is 2.0 mol/(L s). 3.0 mol/(L ·s)x(4mol NH3/6mol H2O)= 2.0 mol/(L s) Cyclopropane is used as an anesthetic. The isomerization of cyclopropane to propene is a first-order reaction with a rate constant of 9.2/s. If an initial sample of cyclopropane has a concentration of 6.00 M, what will the cyclopropane concentration be after 1.00 s? The rate law is Rate = k[cyclopropane] k = 9.2/s [A]initial = 6.00 M t = 1.00 s [A] = ? The reaction is first order, so the integrated rate law is [𝐴] ln = −kt [A]𝑖𝑛𝑖𝑡𝑖𝑎𝑙  A 9.2 ln =− x1.00 s = −9.2 A 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 s A = e−9.2 A 𝑖𝑛𝑖𝑡𝑖𝑎𝑙  A 9.2 ln =− x1.00 s = −9.2 A 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 s A = e−9.2 = 1.01 × 10−4 A 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 A = 1.01 × 10−4 6.00 M A = 6.1 × 10−4 M Ammonium nitrite is unstable because ammonium ion reacts with nitrite ion to produce nitrogen: NH4+(aq) + NO2−(aq) → N2(g) + 2H2O(l) In a solution that is 10.0 M in NH4+, the reaction is first order in nitrite ion (at low concentrations), and the rate constant at 25°C is k=3.0 × 10−3/s. What is the half-life of the reaction in min? 0.693 t1/2 = k k = 3.0 × 10−3/s For first-order reactions: 0.693 0.693 t1/2 = = k 3.0 × 10−3 s t½ = 2.3 × 102 s = 3.9 min What is the reactant concentration after 78.9 seconds for a second order reaction with a half-life of 3.10 minutes if the initial concentration was 0.555M? 1 1 = kt + A  A initial What is the reactant concentration after 78.9 seconds for a second order reaction with a half-life of 3.10 minutes if the initial concentration was 0.555M? 186 s=1 / k (0.555M) k = 9.687 x 10-3 s-1M-1 What is the reactant concentration after 78.9 seconds for a second order reaction with a half-life of 3.10 minutes if the initial concentration was 0.555M? 1 1 = kt + A  A initial 1 1 = k (78.9s) + [A] 0.555 1 = 2.566 [A] [A] = 0.390M A researcher is running a new chemical reaction that obeys first order kinetics and discovers that after 24 hours only ½ of the reactants are turned into products. How long will it take in hours for 90% of the reactants to be reacted? (Hint: how much is remaining when 90% is reacted?) 0.693 t1/2 = k A researcher is running a new chemical reaction that obeys first order kinetics and discovers that after 24 hours only ½ of the reactants are turned into products. How long will it take in hours for 90% of the reactants to be reacted? (Hint: how much is remaining when 90% is reacted? 10%) 0.693 t1/2 = k 0.693 t1/2 = 24h = k k = 0.029 h-1 A researcher is running a new chemical reaction that obeys first order kinetics and discovers that after 24 hours only ½ of the reactants are turned into products. How long will it take in hours for 90% of the reactants to be reacted? (Hint: how much is remaining when 90% is reacted? 10%) 100−90 ln ( ) = - 0.029 h-1 (t) 100 t = 79 h At a given temperature, a first-order reaction has a rate constant of 3.5 x 10–3 s–1. How long will it take for the reaction to be 24% complete? At a given temperature, a first-order reaction has a rate constant of 3.5 x 10–3 s–1. How long will it take for the reaction to be 24% complete? [A]initial =100% remaining = [A] = 100 -24 = 76% ln(76/100) = - 3.5 x 10–3 x t t = 78.41 sec The Effect of Temperature on Rate The rate constant of the rate law, k, is temperature dependent. The Arrhenius equation shows the relationship: where T is the temperature in kelvin R is the gas constant in energy units, 𝟖. 𝟑𝟏𝟒 𝐉/(𝐦𝐨𝐥 · 𝐊) A is called the frequency factor, the rate the reactant energy approaches the activation energy Ea is the activation energy, the minimum energy needed to start the reaction. Increasing temperature will increase the number of molecules with sufficient energy to overcome the energy barrier. Arrhenius Plots The Arrhenius equation can be algebraically solved to give the following form: 𝐸a 1 ln 𝑘 = − + ln A 𝑅 𝑇 𝑦 = 𝑚𝑥 + 𝑏 1 This equation is in the form y = mx + b, where y = ln(k) and x = T. 1 A graph of ln(k) versus T is a straight line. −8.314 J/mol ⋅ K slope of the line = Ea , in Joules 𝑒 𝑦−intercept = 𝐴 unit is the same as 𝑘  Arrhenius Equation If you only have two (T, k) data points, the following forms of the Arrhenius equation can be used: 𝑘2 𝐸a 1 1 ln = − 𝑘1 𝑅 𝑇1 𝑇2 k1 and k2 are the rate constants at T1 and T2 Ea is the activation energy, the minimum energy needed to start the reaction, kJ/mol. R is the gas constant 8.314 J/molxK T is the temperature in kelvins When a new drug product is being formulated, it is desirable to determine the stability of the drug entity in the drug product so that a shelf life or expiration date may be assigned to the product In a series of experiments on the decomposition of dinitrogen pentoxide, N2O5, rate constants were determined at two different temperatures: At 35°C, the rate constant was 1.4 × 10−4/s. At 45°C, the rate constant was 5.0 × 10−4/s. 1.What is the activation energy? T1 = 35°C = 308 K T2 = 45°C = 318 K k1 = 1.4 × 10−4/s k2 = 5.0 × 10−4/s  k2  Ea  1 1 ln   =  −   k1  R  T1 T2   5.0 × 10−4 /s Ea 1 1 ln = − 1.4 × 10−4 /s J 308 K 318 K 8.314 molxK Ea = 1.0 × 105 J/mol Consider the reaction between nitrogen dioxide and carbon monoxide: The rate constant at 701 K is measured as 2.57 M–1 x s–1 and that at 895 K is measured as 567 M–1 x s–1. Find the activation energy for the reaction in kJ/mol. Consider the reaction between nitrogen dioxide and carbon monoxide: The rate constant at 701 K is measured as 2.57 M–1 x s–1 and that at 895 K is measured as 567 M–1 x s–1. Find the activation energy for the reaction in kJ/mol. Given: T1 = 701 K, k1 = 2.57 M–1 x s–1 T2 = 895 K, k2 = 567 M–1 x s–1 Solve Substitute the two rate constants and the two temperatures into the equation. Solve the equation for Ea, the activation energy, and convert to kJ/mol. If a reaction is zero-order in a reactant, when the concentration of the reactant is decreased by a factor of 2, the reaction rate will A) quadruple. B) decrease by a factor of 1/2. C) remain constant. D) decrease by a factor of 1/4. E) double. If a reaction is zero-order in a reactant, when the concentration of the reactant is decreased by a factor of 2, the reaction rate will A) quadruple. B) decrease by a factor of 1/2. C) remain constant. D) decrease by a factor of 1/4. E) double. A chemical reaction that is first-order in X is observed to have a rate constant of 2.20 x 10–2 s–1. If the initial concentration of X is 1.0 M, what is the concentration of X after 186 s? (Ans. 0.017 M) [𝐴] ln = −kt [A]𝑖𝑛𝑖𝑡𝑖𝑎𝑙 For the hypothetical second-order reaction A -> products, k = 0.319 M–1 s–1. If the initial concentration of A is 0.834 M, how long would it take for A to be 94.8% consumed? (Ans. 68.5 s) The isomerization of cyclopropane follows first order kinetics. The rate constant at 700 K is 6.20x 10 ─ 4 min ─ 1, and the half-life at 760 K is 29.0 min. Calculate the activation energy for this reaction. (R = 8.31 J/(mol · K)) (Ans. 270 kJ/mol)