Exercises

What is the difference between average rate, initial rate, and instantaneous rate?

In the nuclear industry, chlorine trifluoride is used to prepare uranium hexafluoride, a volatile compound of uranium used in the separation of uranium isotopes. Chlorine trifluoride is prepared by the reaction ${\text{Cl}}_2(g)+{\text{3F}}_2(g)⟶{\text{2ClF}}_3(g).$ Write the equation that relates the rate expressions for this reaction in terms of the disappearance of Cl₂ and F₂ and the formation of ClF₃.

A study of the rate of the reaction represented as $2A⟶B$ gave the following data:

(a) Determine the average rate of disappearance of A between 0.0 s and 10.0 s, and between 10.0 s and 20.0 s.

(b) Estimate the instantaneous rate of disappearance of A at 15.0 s from a graph of time versus [A]. What are the units of this rate?

(c) Use the rates found in parts (a) and (b) to determine the average rate of formation of B between 0.00 s and 10.0 s, and the instantaneous rate of formation of B at 15.0 s.

Describe the effect of each of the following on the rate of the reaction of magnesium metal with a solution of hydrochloric acid: the molarity of the hydrochloric acid, the temperature of the solution, and the size of the pieces of magnesium.

Go to the PhET Reactions & Rates interactive. Use the Single Collision tab to represent how the collision between monatomic oxygen (O) and carbon monoxide (CO) results in the breaking of one bond and the formation of another. Pull back on the red plunger to release the atom and observe the results. Then, click on “Reload Launcher” and change to “Angled shot” to see the difference.

(a) What happens when the angle of the collision is changed?

(b) Explain how this is relevant to rate of reaction.

In the PhET Reactions & Rates interactive, on the Many Collisions tab, set up a simulation with 15 molecules of A and 10 molecules of BC. Select “Show Bonds” under Options.

(a) Leave the Initial Temperature at the default setting. Observe the reaction. Is the rate of reaction fast or slow?

(b) Click “Pause” and then “Reset All,” and then enter 15 molecules of A and 10 molecules of BC once again. Select “Show Bonds” under Options. This time, increase the initial temperature until, on the graph, the total average energy line is completely above the potential energy curve. Describe what happens to the reaction.

Doubling the concentration of a reactant increases the rate of a reaction four times. With this knowledge, answer the following questions:

(a) What is the order of the reaction with respect to that reactant?

(b) Tripling the concentration of a different reactant increases the rate of a reaction three times. What is the order of the reaction with respect to that reactant?

How much and in what direction will each of the following affect the rate of the reaction: $\text{CO(}g)+{\text{NO}}_2(g)⟶{\text{CO}}_2(g)+\text{NO(}g)$ if the rate law for the reaction is $\text{rate}=k{\left[{\text{NO}}_2\right]}^2?$

(a) Decreasing the pressure of NO₂ from 0.50 atm to 0.250 atm.

(b) Increasing the concentration of CO from 0.01 M to 0.03 M.

Regular flights of supersonic aircraft in the stratosphere are of concern because such aircraft produce nitric oxide, NO, as a byproduct in the exhaust of their engines. Nitric oxide reacts with ozone, and it has been suggested that this could contribute to depletion of the ozone layer. The reaction $\text{NO}+{\text{O}}_3⟶{\text{NO}}_2+{\text{O}}_2$ is first order with respect to both NO and O₃ with a rate constant of 2.20 $\times$ 10⁷ L/mol/s. What is the instantaneous rate of disappearance of NO when [NO] = 3.3 $\times$ 10⁻⁶ M and [O₃] = 5.9 $\times$ 10⁻⁷ M?

The rate constant for the radioactive decay of ¹⁴C is 1.21 $\times$ 10⁻⁴ year⁻¹. The products of the decay are nitrogen atoms and electrons (beta particles):
${}_6^{14}\text{C}⟶{}_7^{14}\text{N}+{\text{e}}^{-}$
$\text{rate}=k\left[{}_6^{14}\text{C}\right]$

What is the instantaneous rate of production of N atoms in a sample with a carbon-14 content of 6.5 $\times$ 10⁻⁹ M?

Alcohol is removed from the bloodstream by a series of metabolic reactions. The first reaction produces acetaldehyde; then other products are formed. The following data have been determined for the rate at which alcohol is removed from the blood of an average male, although individual rates can vary by 25–30%. Women metabolize alcohol a little more slowly than men:

Determine the rate equation, the rate constant, and the overall order for this reaction.

Nitrosyl chloride, NOCl, decomposes to NO and Cl₂.
$\text{2NOCl(}g)⟶\text{2NO(}g)+{\text{Cl}}_2(g)$

Determine the rate equation, the rate constant, and the overall order for this reaction from the following data:

Nitrogen monoxide reacts with chlorine according to the equation:
$\text{2NO(}g)+{\text{Cl}}_2(g)⟶\text{2NOCl(}g)$

The following initial rates of reaction have been observed for certain reactant concentrations:

What is the rate equation that describes the rate’s dependence on the concentrations of NO and Cl₂? What is the rate constant? What are the orders with respect to each reactant?

For the reaction $A⟶B+C,$ the following data were obtained at 30 °C:

(a) What is the order of the reaction with respect to [A], and what is the rate equation?

(b) What is the rate constant?

The rate constant for the first-order decomposition at 45 °C of dinitrogen pentoxide, N₂O₅, dissolved in chloroform, CHCl₃, is 6.2 $\times$ 10⁻⁴ min⁻¹.
${\text{2N}}_2{\text{O}}_5⟶{\text{4NO}}_2+{\text{O}}_2$

What is the rate of the reaction when [N₂O₅] = 0.40 M?

The following data have been determined for the reaction:
${\text{I}}^{\text{−}}+{\text{OCl}}^{\text{−}}⟶{\text{IO}}^{\text{−}}+{\text{Cl}}^{\text{−}}$

Determine the rate equation and the rate constant for this reaction.

Use the data provided to graphically determine the order and rate constant of the following reaction: ${\text{SO}}_2{\text{Cl}}_2⟶{\text{SO}}_2+{\text{Cl}}_2$

Pure ozone decomposes slowly to oxygen, ${\text{2O}}_3(g)⟶{\text{3O}}_2(g).$ Use the data provided in a graphical method and determine the order and rate constant of the reaction.

What is the half-life for the first-order decay of phosphorus-32? $({}_{15}^{32}\text{P}⟶{}_{16}^{32}\text{S}+{\text{e}}^{-})$ The rate constant for the decay is 4.85 $\times$ 10⁻² day⁻¹.

What is the half-life for the decomposition of NOCl when the concentration of NOCl is 0.15 M? The rate constant for this second-order reaction is 8.0 $\times$ 10⁻⁸ L/mol/s.

The reaction of compound  A to give compounds  C and  D was found to be second-order in  A. The rate constant for the reaction was determined to be 2.42 L/mol/s. If the initial concentration is 0.500 mol/L, what is the value of t_1/2?

Some bacteria are resistant to the antibiotic penicillin because they produce penicillinase, an enzyme with a molecular weight of 3 $\times$ 10⁴ g/mol that converts penicillin into inactive molecules. Although the kinetics of enzyme-catalyzed reactions can be complex, at low concentrations this reaction can be described by a rate equation that is first order in the catalyst (penicillinase) and that also involves the concentration of penicillin. From the following data: 1.0 L of a solution containing 0.15 µg (0.15 $\times$ 10⁻⁶ g) of penicillinase, determine the order of the reaction with respect to penicillin and the value of the rate constant.

There are two molecules with the formula C₃H₆. Propene, ${\text{CH}}_3\text{CH}={\text{CH}}_2,$ is the monomer of the polymer polypropylene, which is used for indoor-outdoor carpets. Cyclopropane is used as an anesthetic:

When heated to 499 °C, cyclopropane rearranges (isomerizes) and forms propene with a rate constant of
5.95 $\times$ 10⁻⁴ s⁻¹. What is the half-life of this reaction? What fraction of the cyclopropane remains after 0.75 h at 499 °C?

Suppose that the half-life of steroids taken by an athlete is 42 days. Assuming that the steroids biodegrade by a first-order process, how long would it take for $\frac{1}{64}$ of the initial dose to remain in the athlete’s body?

Nitroglycerine is an extremely sensitive explosive. In a series of carefully controlled experiments, samples of the explosive were heated to 160 °C and their first-order decomposition studied. Determine the average rate constants for each experiment using the following data:

Chemical reactions occur when reactants collide. What are two factors that may prevent a collision from producing a chemical reaction?

What is the activation energy of a reaction, and how is this energy related to the activated complex of the reaction?

Describe how graphical methods can be used to determine the activation energy of a reaction from a series of data that includes the rate of reaction at varying temperatures.

The rate of a certain reaction doubles for every 10 °C rise in temperature.

(a) How much faster does the reaction proceed at 45 °C than at 25 °C?

(b) How much faster does the reaction proceed at 95 °C than at 25 °C?

The rate constant at 325 °C for the decomposition reaction ${\text{C}}_4{\text{H}}_8⟶{2\text{C}}_2{\text{H}}_4$ is 6.1 $\times$ 10⁻⁸ s⁻¹, and the activation energy is 261 kJ per mole of C₄H₈. Determine the frequency factor for the reaction.

An elevated level of the enzyme alkaline phosphatase (ALP) in human serum is an indication of possible liver or bone disorder. The level of serum ALP is so low that it is very difficult to measure directly. However, ALP catalyzes a number of reactions, and its relative concentration can be determined by measuring the rate of one of these reactions under controlled conditions. One such reaction is the conversion of p-nitrophenyl phosphate (PNPP) to p-nitrophenoxide ion (PNP) and phosphate ion. Control of temperature during the test is very important; the rate of the reaction increases 1.47 times if the temperature changes from 30 °C to 37 °C. What is the activation energy for the ALP–catalyzed conversion of PNPP to PNP and phosphate?

Hydrogen iodide, HI, decomposes in the gas phase to produce hydrogen, H₂, and iodine, I₂. The value of the rate constant, k, for the reaction was measured at several different temperatures and the data are shown here:

What is the value of the activation energy (in kJ/mol) for this reaction?

The hydrolysis of the sugar sucrose to the sugars glucose and fructose,
${\text{C}}_{12}{\text{H}}_{22}{\text{O}}_{11}+{\text{H}}_2\text{O}⟶{\text{C}}_6{\text{H}}_{12}{\text{O}}_6+{\text{C}}_6{\text{H}}_{12}{\text{O}}_6$

follows a first-order rate equation for the disappearance of sucrose: Rate = k[C₁₂H₂₂O₁₁] (The products of the reaction, glucose and fructose, have the same molecular formulas but differ in the arrangement of the atoms in their molecules.)

(a) In neutral solution, k = 2.1 $\times$ 10⁻¹¹ s⁻¹ at 27 °C and 8.5 $\times$ 10⁻¹¹ s⁻¹ at 37 °C. Determine the activation energy, the frequency factor, and the rate constant for this equation at 47 °C (assuming the kinetics remain consistent with the Arrhenius equation at this temperature).

(b) When a solution of sucrose with an initial concentration of 0.150 M reaches equilibrium, the concentration of sucrose is 1.65 $\times$ 10⁻⁷ M. How long will it take the solution to reach equilibrium at 27 °C in the absence of a catalyst? Because the concentration of sucrose at equilibrium is so low, assume that the reaction is irreversible.

(c) Why does assuming that the reaction is irreversible simplify the calculation in part (b)?

Use the PhET Reactions & Rates interactive simulation to simulate a system. On the “Single collision” tab of the simulation applet, enable the “Energy view” by clicking the “+” icon. Select the first $A+BC⟶AB+C$ reaction (A is yellow, B is purple, and C is navy blue). Using the “angled shot” option, try launching the A atom with varying angles, but with more Total energy than the transition state. What happens when the A atom hits the BC molecule from different directions? Why?

In general, can we predict the effect of doubling the concentration of A on the rate of the overall reaction $A+B⟶C$? Can we predict the effect if the reaction is known to be an elementary reaction?

What is the rate equation for the elementary termolecular reaction $A+2B⟶\text{products?}$ For $3A⟶\text{products?}$

Write the rate equation for each of the following elementary reactions:

(a) ${\text{O}}_3\stackrel{\text{sunlight}}{\to }{\text{O}}_2+\text{O}$

(b) ${\text{O}}_3+\text{Cl}⟶{\text{O}}_2+\text{ClO}$

(c) $\text{ClO}+\text{O}⟶\text{Cl}+{\text{O}}_2$

(d) ${\text{O}}_3+\text{NO}⟶{\text{NO}}_2+{\text{O}}_2$

(e) ${\text{NO}}_2+\text{O}⟶\text{NO}+{\text{O}}_2$

Experiments were conducted to study the rate of the reaction represented by this equation.³
$\text{2NO}(g)+2{\text{H}}_2(g)⟶{\text{N}}_2(g)+2{\text{H}}_2\text{O}(g)$

Initial concentrations and rates of reaction are given here.

Consider the following questions:

(a) Determine the order for each of the reactants, NO and H₂, from the data given and show your reasoning.

(b) Write the overall rate law for the reaction.

(c) Calculate the value of the rate constant, k, for the reaction. Include units.

(d) For experiment 2, calculate the concentration of NO remaining when exactly one-half of the original amount of H₂ had been consumed.

(e) The following sequence of elementary steps is a proposed mechanism for the reaction.

Step 1: $\text{NO}+\text{NO}\rightleftharpoons {\text{N}}_2{\text{O}}_2$

Step 2: ${\text{N}}_2{\text{O}}_2+{\text{H}}_2\rightleftharpoons {\text{H}}_2\text{O}+{\text{N}}_2\text{O}$

Step 3: ${\text{N}}_2\text{O}+{\text{H}}_2\rightleftharpoons {\text{N}}_2+{\text{H}}_2\text{O}$

Based on the data presented, which of these is the rate determining step? Show that the mechanism is consistent with the observed rate law for the reaction and the overall stoichiometry of the reaction.

Account for the increase in reaction rate brought about by a catalyst.

Consider this scenario and answer the following questions: Chlorine atoms resulting from decomposition of chlorofluoromethanes, such as CCl₂F₂, catalyze the decomposition of ozone in the atmosphere. One simplified mechanism for the decomposition is:
$\begin{array}{}\\ \\ {\text{O}}_3\stackrel{\text{sunlight}}{\to }{\text{O}}_2+\text{O}\\ {\text{O}}_3+\text{Cl}⟶{\text{O}}_2+\text{ClO}\\ \text{ClO}+\text{O}⟶\text{Cl}+{\text{O}}_2\end{array}$

(a) Explain why chlorine atoms are catalysts in the gas-phase transformation:
$2{\text{O}}_3⟶3{\text{O}}_2$

(b) Nitric oxide is also involved in the decomposition of ozone by the mechanism:
$\begin{array}{}\\ {\text{O}}_3\stackrel{\text{sunlight}}{\to }{\text{O}}_2+\text{O}\\ {\text{O}}_3+\text{NO}⟶{\text{NO}}_2+{\text{O}}_2\\ {\text{NO}}_2+\text{O}⟶\text{NO}+{\text{O}}_2\end{array}$

Is NO a catalyst for the decomposition? Explain your answer.

Water gas is a 1:1 mixture of carbon monoxide and hydrogen gas and is called water gas because it is formed from steam and hot carbon in the following reaction: ${\text{H}}_2\text{O}(g)+\text{C}(s)\rightleftharpoons {\text{H}}_2(g)+\text{CO}(g).$ Methanol, a liquid fuel that could possibly replace gasoline, can be prepared from water gas and hydrogen at high temperature and pressure in the presence of a suitable catalyst. What will happen to the concentrations of H₂, CO, and CH₃OH at equilibrium if more catalyst is added?

For each of the following pairs of reaction diagrams, identify which of the pairs is catalyzed:

(a)

(b)

For each of the following reaction diagrams, estimate the activation energy (E_a) of the reaction:

(a)

(b)

Based on the diagrams in Exercise 17.85, which of the reactions has the fastest rate? Which has the slowest rate?