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A C TIVITY 13-1

Ra te s o f C he m ic a l Re a c tio ns W HY? Chemical kinetics is the part of chemistry that looks at the speed at which reactants are converted into products. An experimental rate law can be determined from such measurements, and this rate law can be used to decide how the reaction occurs. Knowledge of the reaction mechanism and the factors that affect the rate of a reaction makes it possible for the chemist to plan the efficient and cost-effective production of industrial, pharmaceutical, and consumer chemicals. It is essential to understand reaction rates in order to understand how reactions occur.

LEARNING O BJEC TIVE •

Understand reaction rates and rate laws

SUC C ESS C RITERIA •

Use kinetic data to identify a reaction as zero, first order, or second order

•

Determine the rate constant for a reaction from kinetic data

•

Graph the concentrations of reactants and products as the reaction progresses

PREREQ UISITES •

Activity 04-1: Balanced Chemical Reaction Equations

•

Activity 05-2: Solution Concentration and Dilution

•

Activity 12-1: Solutions

INFO RMATIO N The average rate of a reaction is given by the change in the concentration of a reactant, ∆cr, or a product, ∆cp over some time interval (∆t), provided that the stoichiometric coefficient for the product and reactant is 1. If the stoichiometric coefficient for some species is not 1, then the rate is determined by dividing the concentration of that species by its stoichiometric coefficient. (1) reaction rate = −

∆cr ∆c p = ∆t ∆t

Since the reaction rate is always given as a positive number and the reactant concentration is ∆cr has been multiplied by –1 in Equation 1. decreasing, ∆t The instantaneous rate of a reaction at a particular time is the limit of the average rate at that time as ∆t approaches 0. If the time interval is small, the average rate is a good approximation to the instantaneous rate. The initial rate of a reaction is the instantaneous rate at the beginning of the reaction, when the reactants are mixed together.

234

M O DEL 1: KINETIC S O F O ZO NE DEC O MPO SITIO N Under certain conditions, ozone in the atmosphere decomposes by dissociation: (O3 O2 + O) The concentration of ozone as a function of time is given by the graph and table of data below. Note that the curved line in the graph refers to the left scale, which gives the ozone concentration. The straight line refers to the right scale, which gives the natural logarithm of the ozone concentration divided by the initial ozone concentration. Figure 1

Ozone Ozon Deco Ozo n e Decomposition Deco m p o s i t i o n 0

90

-0.4

80

-0.8 -1.2

70

-1.6

60

-2 50 -2.4 40

-2.8

30

-3.2

20

-3.6

10

-4

7000

6000

5000

4000

3000

2000

1000

-4.4 0

0

L Ln([O n ([O ] /in [O ]) ([ O3]/initial /i n i t i al [O [ O3])

[[O O 3] co concentration c o n c ent en t r ati ati o n in in m micromoles/liter i c r o m o l es/l es/li /l i t er

100

ttime i m e in eco i n sec sseconds eco n d s

Data on Ozone Decomposition Time (s)

0

100

900

1000

6900

7000

concentration (10–6 M)

89.63

84.87

54.84

51.92

2.07

1.96

ln([O3]/[O3]o)

0.000

– 0.0546

– 0.4914

– 0.5460

– 3.767

– 3.822

KEY Q UESTIO NS 1. What quantities are plotted in the graph shown in Model 1?

Activity 13-1 Rates of Chemical Reactions

235

Foundations of Chemistry

2. What is the relationship between the data in the table and the graph?

3. In the model, how long does it take for half of the ozone to decompose? What is the concentration of ozone after that time? This time is designated as the half-lifetime or the half-life of ozone.

4. What is the initial reaction rate of the ozone decomposition reaction? Provide both the magnitude and the units.

5. What is the rate of the reaction after 6900 seconds have passed?

6. What happens to the reaction rate as the concentration of ozone decreases?

INFO RMATIO N A rate law for a reaction indicates how the rate of the reaction depends on the concentrations of the chemical species involved in the reaction. Generally, rate laws have the following form: the rate of the reaction equals some constant multiplied by a product of concentrations [A], [B], and [C], with exponents x, y, and z, respectively. In equation form: rate = k [A ] x [B] y [C ] z An exponent gives the order of the reaction with respect to that chemical species. For example, if x and z equal 1, then the reaction is first order with respect to A and C; and if y = 2, then the reaction is second order with respect to B. The overall order of the reaction is the sum of the exponents in the rate law, which, in this example, would be: 1 + 2 + 1 = 4. The order of a reaction often differs from the stoichiometric coefficients, which relate the numbers of reactant and product molecules involved in the reaction. The order of a reaction depends on the reaction mechanism, not on the number of molecules involved in the overall reaction. 236

Chapter 13: Chemical Kinetics

M O DEL 2: RA TE LA W S FO R C HEMIC A L REA C TIO NS In the equations below, the reactant is represented by A; [A]t represents the concentration of A at time t; [A]0 represents the initial concentration of A; k is the rate constant; and ln is the natural logarithm. The rate constant is the constant of proportionality in the rate law. The integrated rate law is obtained from the rate law by integration of the differential equation.

Zero-order reaction A

products rate law

rate = –

∆[ A] ∆t

= k[A]0

integrated rate law [A]t = – kt + [A]0

First-order reaction A

products rate law

rate = –

∆[ A] ∆t

= k [A]

integrated rate law, exponential form [A]t = [A]0 e –kt integrated rate law, logarithmic form ln([A]t) = – kt + ln[A]0  [ A]t  Or, equivalently, ln    kt  [ A]0 

Second-order reaction A

products rate law integrated rate law

rate = –

∆[ A] ∆t

= k [A]2

1 1 = kt + [ A] t [ A]0

KEY Q UESTIO NS 7. What are four similarities between zero, first, and second order reactions as represented by Model 2?

8. What are three differences between zero, first, and second order reactions as represented by Model 2?

Activity 13-1 Rates of Chemical Reactions

237

Foundations of Chemistry

9. If the rate constant k increases when the temperature increases, what happens to the rate of the reaction according to the rate laws in Model 2? Is your answer the same for zero, first, and second order reactions?

10. If the concentration of reactant A increases, what happens to the rate of the reaction, according to the rate laws in Model 2? Is your answer the same for zero, first, and second-order reactions?

11. What do you need to plot on the y-axis if you plot time on the x-axis in order to obtain a straight line for each of the reaction orders in Model 2? To answer this question, compare the integrated rate laws to the general equation for a straight line, y = mx + b, where m is the slope, and b is the intercept on the y-axis when x = 0. a) a zero-order reaction.

b) a first-order reaction.

c) a second-order reaction.

12. What are the slope and y-intercept of the straight line that you identifi ed in your answer to Key Question 11 for: a) a zero-order reaction?

b) a first-order reaction?

c) a second-order reaction?

EXERC ISES 1. On the graph in Model 1, draw a vertical line at the half-life time, and label this line t½. Draw a horizontal line on the graph in Model 1 at half the initial concentration, and label this line C½.

238

Chapter 13: Chemical Kinetics

2. Compare your answers to Key Question 11 with the graph in Model 1, and identify the order of the ozone decomposition reaction.

3. Write the rate law for the ozone decomposition reaction.

4. Write the integrated rate law in both exponential and logarithmic forms for the ozone decomposition reaction, and describe how they are related to the graph in Model 1.

5. Identify the order of the decomposition reaction with respect to ozone.

6. Identify the overall order of the ozone decomposition reaction.

7. Determine what would happen to the rate of decomposition if the concentration of ozone were doubled.

8. Determine the rate constant (magnitude and units) for the ozone decomposition reaction.

Activity 13-1 Rates of Chemical Reactions

239

Foundations of Chemistry

G O T IT! 1. Given the following rate law: Rate = k [CHCl3] [Cl2]½ a) Write the reaction order with respect to chloroform.

b) Write the reaction order with respect to chlorine.

c) Write the overall reaction order.

d) Identify what happens to the reaction rate if the concentration of chloroform is cut in half.

e) Identify what happens to the reaction rate if the concentration of chlorine is increased by four times.

conc

ln(conc/initial conc)

2. Label the following graphs appropriately as zero-order, first-order, or second-order reactions.

0

0

t

t order:

conc

1/conc

order:

t order:

t order:

3. The statement, the rate decreases as the reaction proceeds, does not apply to which one of the following? a) a zero-order reaction

240

b) a first-order reaction

c) a second-order reaction

Chapter 13: Chemical Kinetics

PRO BLEMS The data in the following table (molar concentration of N2O5 versus time) were obtained for the decomposition of dinitrogen pentoxide: 2 N2O5(g)

4 NO2(g) + O2(g)

time in s

0

1000

2000

8000

9000

15000

16000

20000

21000

conc in M

0.75

0.65

0.57

0.24

0.21

0.092

0.080

0.046

0.040

1. Determine, for example by making the appropriate graphs, whether this decomposition reaction is zero, first, or second order with respect to N2O5.

Activity 13-1 Rates of Chemical Reactions

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Foundations of Chemistry

2. The stoichiometric coefficient for N2O5 in the reaction equation is 2. Depending on your answer to Problem 1, explain why the reaction order you obtained in Problem 1 is also 2, or explain why it can be different from 2.

3. Determine the rate constant for this reaction (magnitude and units).

4. Sketch a graph showing the relative concentrations of N2O5, NO2, and O2 as a function of time as the reaction proceeds. Your graph should show these concentrations on the same scale, and should show the final concentrations relative to each other after equilibrium has been reached.

5. Write a paragraph explaining why the concentrations vary as shown by your graph.

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Chapter 13: Chemical Kinetics...