Homework 8: Chemical Kinetics

There are select solutions to these problems here.

Q8.1

A) Atmospheric chemistry in the region below the clouds of Venus appears to be dominated by reactions of sulfur and carbon-containing compounds. Included in representative elementary reactions are the following:

$SO_2 + CO \rightleftharpoons SO + CO_2 \tag{1}$

$SO + CO \rightleftharpoons S + CO_2 \tag{2}$

$SO + SO_2 \rightleftharpoons S + SO_3 \tag{3}$

For each elementary reaction, write an expression for the net rate of reaction in terms of the concentrations of reactants and products (assuming second order kinetics for each direction).

B) What are the equilibrium constants for each reaction in terms of the final equilibrium concetrations?

C) What are the equilibrium constants for each reaction in terms of the reaction rate laws from question A?

Q8.2

In acid, nitriles hydrolyze to produce a carboxylic acid and ammonium ion. For example, acetonitrile, a substance used to extract fatty acids from fish liver oils, is hydrolyzed to acetic acid via the following reaction:

Express the reaction rate in terms of changes in the concentrations of each reactant and each product with time.

Q8.3

Ozone production occurs at lower altitudes according to the elementary reaction $$O + O_2 \rightarrow O_3$$, with an estimated rate of ozone production of 4.86 × 1031 molecules·s−1 worldwide. What is the overall reaction order?

The absorption of high energy ultraviolet light will induce the dissociate of O2 molecules to into two oxygen atoms:

$O_2 (g) \overset {light}{\longrightarrow} 2O (g)$

If the reaction rate of loss of O3 due to absorption of UV light is 0.89 × 1031 molecules·s−1, and 0.06 × 1031 molecules·s−1 of ozone is transported to other atmospheric regions, is ozone being produced faster than it is being destroyed? Measurements show that ozone concentrations are not increasing rapidly. What conclusion can you draw from these data?

Q8.4

The water in a fishery became polluted when toxic waste was dumped into its pond, causing the fish population to substantially decline. The percentage of fish that survived is recorded in the following table.

 Day % survival 1 2 3 4 5 79 55 38 31 19

What is the reaction order of live fish → dead fish? What is the rate constant? If the fish continue to die at this rate, how many fish will be alive after 10 days?

Q8.5

Until 200 yr ago, manufactured iron contained charcoal produced from freshly cut wood that was added during the smelting process. As a result of this practice, older samples of iron can be dated accurately using the carbon-14 method. An archaeologist found a cast iron specimen that she believed dated to the period between 480 and 221 BC in Hunan, China. Radiocarbon dating of the sample indicated a 24% reduction in carbon-14 content. Was the archaeologist correct?

Q8.6

Because of its short half-life, 32P-labeled compounds must be shipped as quickly as possible so that they can be used as radioactive tags in biological studies. A 50 g sample that contained 0.60% 32P by mass was shipped at 11 a.m. on Monday morning. The package was delivered to a chemist via an overnight delivery service such that it arrived the next day.

1. What would be the mass of 32P remaining in the sample if he received the package on Tuesday afternoon but was unable to use it until 9 a.m. on Wednesday?
2. What would be the mass of 32P present in the sample if the shipper had not delivered the sample until Friday afternoon and then it sat on a loading dock until 9 a.m. on Monday morning?
3. The late shipment was used immediately on Monday morning, but the biological samples were not analyzed until Thursday at 5 p.m. What percentage of the sample still consists of 32P?

Q8.7

Tritium (3H) is a radioactive isotope that is commonly used to follow biochemical reactions.

1. Using the data in Table 14.6, calculate the radioactive decay constant (k) for tritium.
2. Use the value of k to determine the mass of tritium that is still present in a 5.00 g sample of NaB3H4 that is 17.57 yr old.

Q8.8

L-Aspartic acid is an amino acid found in fossil bone. It can convert to a geometrically different form (D-aspartic acid) at 20°C, with a half-life corresponding to the conversion of L → D of 14,000–20,000 yr. If the temperature of an archaeological site is constant, then the extent of the conversion can be used to date fossils. In one such case, archaeologists dated the arrival of humans on the North American continent to be 20,000 yr ago, but the conversion of L-aspartic acid to D-aspartic acid in human fossils indicated that Paleo-Indians were living in California at least 48,000 yr ago. What would be the relative concentrations of the L- and D-forms that produced this result? Carbon-14 has a half-life of approximately 5730 yr. What percentage of the carbon-14 originally present would have been found in the bones?

The technique described is frequently used in conjunction with radiocarbon dating. In cases where the results from the two techniques are in gross disagreement, what information can you get by comparing the two results?

Q8.9

Peroxides are able to initiate the radical polymerization of alkenes. Polyethylene, for example, is a high-molecular-weight polymer used as a film in packaging, as kitchenware, and as tubing. It is produced by heating ethylene at high pressure in the presence of oxygen or peroxide. It is formed by the following radical process:

$$\mathrm{RO:OR}\xrightarrow{\Delta}\mathrm{2RO}\cdot$$
$$\mathrm{RO}\cdot + \mathrm{CH_2}\textrm{=CH}_2\rightarrow\mathrm{RO-CH_2-CH_2}\cdot$$

RO―CH2―CH2· + CH2=CH2 → RO―CH2―CH2―CH2―CH2·

1. Label the steps that correspond to initiation and propagation.
2. Show all available chain-terminating steps.
3. The polymerization of styrene (C6H5CH=CH2) occurs by a similar process to produce polystyrene, which is used as a packaging material. Draw the structure of the polymer that results from five propagation cycles.

Q8.10

Lucite and Plexiglas are transparent polymers used as a glass substitute when a plastic material is preferred for safety. The compound used to synthesize Lucite and Plexiglas is methyl methacrylate, which is shown here. During the polymerization reaction, light produces a radical initiator from hydrogen peroxide (H2O2 → 2HO·). Show the mechanism for the polymerization, being sure to include the initiation and propagation steps.

Q8.11

At higher altitudes ozone is converted to O2 by the reaction O + O3 → 2O2, with a rate constant at 220 K of 6.8 × 10−16 cm3·molecule−1·s−1.

1. What is the overall reaction order?
2. What is Ea for this reaction if A = 8 × 10−12 cm3·molecule−1·s−1?

If Cl is present, the rate constant at 220 K becomes 3.7 × 10−11 cm3·molecule−1 · s−1, with A = 4.7 × 10−11 cm3·molecule−1·s−1.

1. Calculate Ea for the depletion of ozone in the presence of Cl.
2. Show an energy-level diagram for these two processes, clearly labeling reactants, products, and activation energies.
3. If you were an environmental scientist using these data to explain the effects of Cl on ozone concentration, what would be your conclusions?

Q8.12

Nitric acid is produced commercially by the catalytic oxidation of ammonia by air over platinum gauze at approximately 900°C. The following reactions occur:

\begin{align}\mathrm{NH_3(g)}+\dfrac{5}{4}\mathrm{O_2(g)}\rightarrow\mathrm{NO(g)}+\dfrac{3}{2}\mathrm{H_2O(g)}\hspace{3mm}\Delta H^\circ&=-226.3\textrm{ kJ/mol} \\ \mathrm{NO(g)}+\dfrac{1}{2}\mathrm{O_2(g)}\rightarrow\mathrm{NO_2(g)}\hspace{3mm}\Delta H^\circ&=-57.1\textrm{ kJ/mol} \\ \mathrm{3NO_2(g)}+\mathrm{H_2O(g)}\rightarrow\mathrm{2HNO_3(l)}+\mathrm{NO(g)}\hspace{3mm}\Delta H^\circ&=-71.7\textrm{ kJ/mol}\end{align}

Why is platinum gauze rather than platinum wire used for the initial reaction? The reaction 4NH3(g) + 3O2(g) → 2N2(g) + 6H2O(g) has ΔH° = −316.6 kJ/mol. What would occur if the catalyst were not present? If the gas leaving the catalyst is not free of NH3, the following reaction takes place: 6NO(g) + 4NH3(g) → 5N2(g) + 6H2O(g). If this occurs, what will be the overall reaction?

Q8.13

Figure 14.27 illustrates the mechanism for the reduction of ethylene on a platinum surface to produce ethane. Industrially important silanes are synthesized using a related mechanism and are used to increase adhesion between layers of glass fiber and between layers of silicone rubber. Predict the products of the following reactions:

Q8.14

In catalysis, if a molecule forms strong bonds to the catalyst, then the catalyst may become poisoned. Experiments on various catalysts showed the following results:

• Fe, Ru, and Os form weak bonds with N2; however, O2, alkynes, alkenes, CO, H2, and CO2 interact more strongly.
• CO2 and H2 form weak bonds with a Co or Ni surface.
• Rh, Pd, Ir, and Pt form weak bonds with H2 but do not bond with CO2.
• Cu, Ag, and Au form weak bonds with CO and ethylene.

1. Explain why Fe was chosen as a catalyst to convert nitrogen and hydrogen to ammonia. Why is Fe more suitable than Ru or Os?
2. Because alkenes generally interact more strongly with metal surfaces than does H2, what catalyst would you choose for hydrogenation of an alkene such as ethylene?
3. Although platinum is used in catalytic converters for automobile exhaust, it was not found to be a particularly effective catalyst for the reaction of H2 with a mixture of carbon monoxide and carbon dioxide to produce methane. Why?
4. If you were interested in developing a catalyst to reversibly bind ethylene, which of the catalysts listed here would you choose?

Q8.15

Nonstoichiometric metal oxides can be effective catalysts for oxidation–reduction reactions. One such catalyst is Ni1−xO, found to be effective for converting CO to CO2 when oxygen is present. Why is it so effective?

Q8.16

The chemical reactions in an organism can be controlled by regulating the activity of certain enzymes. Efficient regulation results in an enzyme being active only when it is needed. For example, if a cell needed histidine, the nine enzymes needed to synthesize histidine would all be active. If the cell had adequate histidine, however, those enzymes would be inactive. The following diagram illustrates a situation in which three amino acids (D, F, H) are all synthesized from a common species, A. The numbers above the arrows refer to the enzymes that catalyze each step. Which enzymes would need to be regulated to produce D? F? H?

Q8.17

Because phosphorus-32 is incorporated into deoxyribonucleic acid (DNA), it can be used to detect DNA fragments. Consequently, it is used extensively in biological research, including the Human Genome Project, whose goal was to determine the complete sequence of human DNA. If you were to start with a 20 g sample of phosphorus that contained 10% 32P by mass, converted it into DNA via several chemical steps that had an overall yield of 75% and took 25 days, and then incorporated it into bacteria and allowed them to grow for 5 more days, what mass of 32P would be available for analysis at the end of this time?

Q8.18

The enzyme urease contains two atoms of nickel and catalyzes the hydrolysis of urea by the following reaction:

H2NC(O)NH2 + H2O → 2NH3 + CO2

Urease is one of the most powerful catalysts known. It lowers the activation energy for the hydrolysis of urea from 137 kJ/mol to only 37 kJ/mol. Calculate the ratio of the reaction rate of the catalyzed reaction to the reaction rate of the uncatalyzed reaction at 37°C. Assume that the frequency factor is the same for both reactions.

Q8.19

The reaction rate for the hydrogenation of ethylene to give ethane can be increased by heterogeneous catalysts such as Pt or Ni:

$$\mathrm{H_2(g)}+\mathrm{H_2}\textrm{C=CH}_2\textrm{(g)}\xrightarrow{\,\textrm{Pt,Ni}\,}\mathrm{H_3C-C}\textrm{H}_3\mathrm{(g)}$$

The activation energy for the uncatalyzed reaction is large (188 kJ/mol), so the reaction is very slow at room temperature. In the presence of finely divided metallic Ni, the activation energy is only 84 kJ/mol. Calculate the ratio of the reaction rate of the catalyzed reaction to the reaction rate of the uncatalyzed reaction at 75°C.

Q8.20

What effect does a catalyst have on the activation energy of a reaction? What effect does it have on the frequency factor (A)? What effect does it have on the change in potential energy for the reaction?

Q8.21

How is it possible to affect the product distribution of a reaction by using a catalyst?

Q8.22

A heterogeneous catalyst works by interacting with a reactant in a process called adsorption. What occurs during this process? Explain how this can lower the activation energy.

Q8.23

What effect does increasing the surface area of a heterogeneous catalyst have on a reaction? Does increasing the surface area affect the activation energy? Explain your answer.

Q8.24

Identify the differences between a heterogeneous catalyst and a homogeneous catalyst in terms of the following.

1. ease of recovery
2. collision frequency
3. temperature sensitivity
4. cost

Q8.25

An area of intensive chemical research involves the development of homogeneous catalysts, even though homogeneous catalysts generally have a number of operational difficulties. Propose one or two reasons why a homogenous catalyst may be preferred.

Q8.26

Consider the following reaction between cerium(IV) and thallium(I) ions:

2Ce4+ + Tl+ → 2Ce3+ + Tl3+

This reaction is slow, but Mn2+ catalyzes it, as shown in the following mechanism:

Ce4+ + Mn2+ → Ce3+ + Mn3+

Ce4+ + Mn3+ → Ce3+ + Mn4+

Mn4+ + Tl+ → Tl3+ + Mn2+

In what way does Mn2+ increase the reaction rate?

Q8.27

The text identifies several factors that limit the industrial applications of enzymes. Still, there is keen interest in understanding how enzymes work for designing catalysts for industrial applications. Why?

Q8.28

Most enzymes have an optimal pH range; however, care must be taken when determining pH effects on enzyme activity. A decrease in activity could be due to the effects of changes in pH on groups at the catalytic center or to the effects on groups located elsewhere in the enzyme. Both examples are observed in chymotrypsin, a digestive enzyme that is a protease that hydrolyzes polypeptide chains. Explain how a change in pH could affect the catalytic activity due to (a) effects at the catalytic center and (b) effects elsewhere in the enzyme. (Hint: remember that enzymes are composed of functional amino acids.)

Q8.29

At some point during an enzymatic reaction, the concentration of the activated complex, called an enzyme–substrate complex (ES), and other intermediates involved in the reaction is nearly constant. When a single substrate is involved, the reaction can be represented by the following sequence of equations:

$$\textrm{enzyme (E)} + \textrm{substrate (S)}\rightleftharpoons \\ \textrm{enzyme-substrate complex (ES)}\rightleftharpoons \\ \textrm{enzyme (E)} + \textrm{product (P)}$$

This can also be shown as follows:

$$\mathrm{E+S}\overset{\large{k_1}}{\underset{\large{k_{-1}}}{\rightleftharpoons}}\mathrm{ES}\overset{\large{k_2}}{\underset{\large{k_{-2}}}{\rightleftharpoons}}\mathrm{E+P}$$

Using molar concentrations and rate constants, write an expression for the rate of disappearance of the enzyme–substrate complex. Typically, enzyme concentrations are small, and substrate concentrations are high. If you were determining the rate law by varying the substrate concentrations under these conditions, what would be your apparent reaction order?

Q8.30

A particular reaction was found to proceed via the following mechanism:

A + B → C + D

2C → E

E + A → B + F

What is the overall reaction? Is this reaction catalytic, and if so, what species is the catalyst? Identify the intermediates.

Q8.31

A particular reaction has two accessible pathways (A and B), each of which favors conversion of X to a different product (Y and Z, respectively). Under uncatalyzed conditions pathway A is favored, but in the presence of a catalyst pathway B is favored. Pathway B is reversible, whereas pathway A is not. Which product is favored in the presence of a catalyst? without a catalyst? Draw a diagram illustrating what is occurring with and without the catalyst.

Q8.32

The kinetics of an enzyme-catalyzed reaction can be analyzed by plotting the reaction rate versus the substrate concentration. This type of analysis is referred to as a Michaelis–Menten treatment. At low substrate concentrations, the plot shows behavior characteristic of first-order kinetics, but at very high substrate concentrations, the behavior shows zeroth-order kinetics. Explain this phenomenon.

Q8.33

Consider the reaction

$X_2 + Z_2 \rightarrow 2XZ$

with the experimentally determined rate law

$Rate = k[X_2][Z_2]$

Which of the following two mechanisms is feasible for this reaction?

Mechanism 1 (single step):

$X_2 + Z_2 \rightarrow 2XZ \tag{single step}$

Mechanism 2 (three steps):

$Z_2 \rightleftharpoons 2Z \tag{Step 1: fast}$

$Z + X_2 \rightleftharpoons ZX_2 \tag{Step 2: fast}$

$ZX_2 + Z \rightarrow 2XZ \tag{Step 3: slow}$

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