THE RATES OF REACTIONS Chapter 13

Reaction Rate The reaction rate is defined as the change in concentration of a species with time. Consider the reaction 2HI(g)  H 2 (g) + I 2 (g) As the HI decomposes its concentration in the flask decreases. Rate=change in conc. of reactant time interval - ∆[HI]/∆t We put a negative sign because the concentration of HI is decreasing.

Instantaneous reaction rate The instantaneous reaction rate is the (positive) slope of a tangent drawn to the graph of concentration as a function of time, it varies as the reaction proceeds.

Rate Laws The rate of a chemical reaction is the amount of substance reacted or produced per unit time. The rate law is an expression indicating how the rate depends on the concentrations of the reactants and catalysts. The power of the concentration in the rate law expression is called the order with respect to the reactant or catalyst.

Order of a reaction The order of a chemical reaction with respect to a reactant is an indication of how the reaction takes place, not its stoichiometry.

Identifying the order of the reactions. First Order Rate Law : The simplest reaction mechanism is that of uni molecular decomposition. In such a process, a single reactant undergoes a transformation at a constant probability per unit time. Such a mechanism leads to a first-order reaction rate law. Let's assume the reaction has a simple stoichiometry: A  B A First-Order Rate Law is called such because the rate of product formation is proportional to the first power of the number of available reactants (or reactant concentration): rate=k[A] where [A] represents the concentration of species A in the sample.

Second Order Rate Law If two molecules undergo a bimolecular reaction such as a reaction that involves a collisional encounter to produce products, and has a stoichiometry like this: A+A  B+C we expect a reaction rate law that is Second Order in the concentration of [A]) rate=k[A] 2

Zero Order Rate Law If a reaction is catalyzed by a surface and has enough (excess) reactant, the rate of the reaction depends on the area of the catalyst, not on how much reactant is present. This is an unusual circumstance outside of the realm of catalyzed reactions and is described by a Zero Order rate law: rate = k

Many reactions can be classified according to their order in a particular species. The order is the power to which a species is raised in the rate law. The overall order is the sum of all the individual orders.

Integrated rate law The differential rate law describes how the rate of reaction varies with the concentrations of various species, usually reactants, in the system. The rate of reaction is proportional to the rates of change in concentrations of the reactants and products; that is, the rate is proportional to a derivative of a concentration. To illustrate this point, consider the reaction A  B The rate of reaction, r, is given by r = - d [A]/dt Suppose this reaction obeys a first-order rate law: r = k [A] This rate law can also be written as r = - d [A]/d t = k [A]

This previous equation is a differential equation that relates the rate of change in a concentration to the concentration itself. Integration of this equation produces the corresponding integrated rate law, which relates the concentration to time. d [A]/[A] = - k d t At t = 0, the concentration of A is [A] 0. The integrated rate law is thus [A] = [A] 0 e - k t [A] molar concentration of A.

Initial concentration is [A ₀];k is the rate constant, t is the elapsed time; exponential e, is the base of natural logarithm. The variation of concentration with time predicted by this equation is shown in the fig. The behavior shown in the illustration is called an exponential decay. The change in concentration is initially rapid but the concentration changes more slowly as time goes on and the reactant is used up.

Calculate the concentration of N 2 O₅ remaining 600.s(10.0 min) after the decomposition at 65⁰C when its concentration was mol/L. The reaction and its rate law are 2N 2 O₅(g)  4NO 2 (g)+O 2 (g) rate of disappearance of N 2 O₅=k[N 2 O₅] with k=5.2x10 -3 /s

In a first order reaction the concentration of reactant decays exponentially with time. To verify that a reaction is first order plot the natural logarithm of its concentration as a function of time and expect a straight line, the slope of the straight line is –k.

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Half lives for First –Order Reactions The half life of a reaction is defined as the time it takes for one half of a reactant to disappear. The half life is given the symbol t 1/2 to denote that it is the time at which the concentration of reactant is one half its initial value. For the first order reaction, you can plug the definition of the half life into the concentration-time reaction to obtain a neat relationship: ln ([A] 0 /[A] t) = kt [A] t1/2 = 1/2[A] 0 ln ([A] 0 / 1/2[A] 0 ) = kt 1/2 ln 2 = = kt 1/2

The half life of a first order reaction is the characteristic of the reaction and independent of the initial concentration. A reaction with a large rate constant has a short half life.

Second order integrated rate law. For second order reaction with rate law Rate=k[A] 2 The integrated rate law is [A] t =[A] ₀/t+[A] t kt or 1/[A] t =1/[A]₀+kt For a second order reaction the plot of 1/[A] against t is linear with a slope k.

For first order reaction For second order reaction

First order reaction Second order reaction Rate equation Rate=k[A] Integrated form [A] t =[A] ₀e -kt or ln[A] t =ln[A]₀-kt Half life of a first-order reaction is independent of the starting concentration and is given t 1/2 =ln2/k =0.693/k Rate equation Rate=k[A] 2 Integrated form [A] t =[A] ₀/1+[A]₀kt or 1/[A] t =1/[A]₀ +kt Half life equation for a second-order reaction dependent on one second- order reactant is t 1/2 =1/k[A] ₀ For a second order reaction half lives progressively doubles.

Class practice A pollutant escapes into a local picnic site. Students had shown that the pollutant decays by a first order reaction with rate constant 3.8x10 -3 /h. calculate the time needed for the concentration to fall to one half and one fourth its initial value.

Controlling reaction rates Effect of temperature: Most reactions go faster as the temperature is raised. An increase in 10 ⁰C from room temperature doubles the rate of reaction. An Arrhenius plot displays the logarithm of a rate ln(k) plotted against inverse temperature 1 / T. Arrhenius plots are often used to analyze the effect of temperature on the rates of chemical reactions. For a single rate-limited thermally activated process, an Arrhenius plot gives a straight line, from which the activation energy and the pre-exponential factor can both be determined.

The Arrhenius equation given in the form:k=Ae -Ea/RT can be written equivalently as: ln(k)=ln(A)-Ea/R(1/T) When plotted in the manner described above, the value of the "y-intercept" will correspond to ln(A), and the gradient of the line will be equal to − E a / R. The pre-exponential factor, A, is a constant of proportionality that takes into account a number of factors such as the frequency of collision between and the orientation of the reacting particles. The expression − E a / RT represents the fraction of the molecules present in a gas which have energies equal to or in excess of activation energy at a particular temperature.

Slope=-Ea/R An Arrhenius plot of ln k vs 1/T is used to determine the Arrhenius parameters of a reaction, a large activation energy signifies a high sensitivity of the rate to change in temperature.

Activation energy A factor that influences whether reaction will occur is the energy the molecules carry when they collide. Not all of the molecules have the same kinetic energy, as shown in the figure below. This is important because the kinetic energy molecules carry when they collide is the principal source of the energy that must be invested in a reaction to get it started.

ClNO 2 (g) + NO(g)  NO 2 (g) + ClNO(g) ∆ G o = kJ/mol Before the reactants can be converted into products, the free energy of the system must overcome the activation energy for the reaction, as shown in the figure below. The vertical axis in this diagram represents the free energy of a pair of molecules as a chlorine atom is transferred from one to the other. The horizontal axis represents the sequence of infinitesimally small changes that must occur to convert the reactants into the products of this reaction.

According to the collision theory of gas phase reactions, molecules react when they collide. In collision theory the activation energy is the energy needed for the reaction to occur. It is represented as a energy barrier between the reactants and the products, with a height equal to the activation energy. In activated complex theory, a reaction occurs only if two molecules acquire enough energy, perhaps from the surrounding solvent, to form an activated complex and cross an energy barrier.

Catalysis and rate of reaction Catalysts increase the rates of reactions by providing a new mechanism that has a smaller activation energy, as shown in the figure below. A larger proportion of the collisions that occur between reactants now have enough energy to overcome the activation energy for the reaction. As a result, the rate of reaction increases.

To illustrate how a catalyst can decrease the activation energy for a reaction by providing another pathway for the reaction, let's take the mechanism for the decomposition of hydrogen peroxide catalyzed by the I - ion. In the presence of this ion, the decomposition of H 2 O 2 doesn't have to occur in a single step. It can occur in two steps, both of which are easier and therefore faster. In the first step, the I - ion is oxidized by H 2 O 2 to form the hypoiodite ion, OI -. H 2 O 2 (aq) + I - (aq)  H 2 O(aq) + OI - (aq) In the second step, the OI - ion is reduced to I - by H 2 O 2. OI - (aq) + H 2 O 2 (aq)  H 2 O(aq) + O 2 (g) + I - (aq)

Because there is no net change in the concentration of the I - ion as a result of these reactions, the I - ion satisfies the criteria for a catalyst. Because H 2 O 2 and I - are both involved in the first step in this reaction, and the first step in this reaction is the rate-limiting step, the overall rate of reaction is first-order in both reagents.

Rate laws and elementary reactions An elementary reaction is a chemical reaction in which one or more chemical species react directly to form products in a single reaction step and with a single transition state. In a unimolecular elementary reaction a molecule, A, dissociates to form the product(s). A  products The rate of such a reaction, at constant temperature, is proportional to the concentration of the species A d[A]/dt=-k[A]

In a bimolecular elementary reaction, two atoms, molecules, ions or radicals, A and B, react together to form the product (s). A+B  products The rate of such a reaction, at constant temperature, is proportional to the product of the concentrations of the species A and B. d[A]/ dt=d [B]/ dt =-k [A] [B]

Home work Page ,13.39, 13.49,13.57,13.61,13.69