Chapter 15

Why are reaction rates important (both practically and theoretically)?

What units are typically used to express the rate of a reaction?

Why is the reaction rate for reactants defined as the negative of the change in reactant concentration with respect to time, whereas for products it is defined as the change in reactant concentration with respect to time (with a positive sign)?

Explain the difference between the average rate of reaction and the instantaneous rate of reaction.

Consider a simple reaction in which reactant A forms products: \(\mathrm{A} \longrightarrow\) products What is the rate law if the reaction is zero order with respect to A? First order? Second order? For each case, explain how a doubling of the concentration of A would affect the rate of reaction.

For a reaction with multiple reactants, how is the overall order of the reaction defined?

Explain the difference between the rate law for a reaction and the integrated rate law for a reaction. What relationship does each kind of rate law express?

Write integrated rate laws for zero-order, first-order, and secondorder reactions of the form \(\mathrm{A} \longrightarrow\) products.

What does the term half-life mean? Write the expressions for the half-lives of zero-order, first-order, and second-order reactions.

How do reaction rates typically depend on temperature? What part of the rate law is temperature dependent?

Explain the meaning of each term within the Arrhenius equation: activation energy, frequency factor, and exponential factor. Use these terms and the Arrhenius equation to explain why small changes in temperature can result in large changes in reaction rates.

What is an Arrhenius plot? Explain the significance of the slope and intercept of an Arrhenius plot.

Explain how a chemical reaction occurs according to the collision model. Explain the meaning of the orientation factor in this model.

Explain the difference between a normal chemical equation for a chemical reaction and the mechanism of that reaction.

In a reaction mechanism, what is an elementary step? Write down the three most common elementary steps and the corresponding rate law for each one.

What are the two requirements for a proposed mechanism to be valid for a given reaction?

What is an intermediate within a reaction mechanism?

What is a catalyst? How does a catalyst increase the rate of a chemical reaction?

Explain the difference between homogeneous catalysis and heterogeneous catalysis.

What are the four basic steps involved in heterogeneous catalysis?

What are enzymes? What is the active site of an enzyme? What is a substrate?

What is the general two-step mechanism by which most enzymes work?

Consider the reaction: $$ 2 \mathrm{~N}_{2} \mathrm{O}(g) \longrightarrow 2 \mathrm{~N}_{2}(g)+\mathrm{O}_{2}(g) $$ a. Express the rate of the reaction in terms of the change in concentration of each of the reactants and products. b. In the first \(15.0 \mathrm{~s}\) of the reaction, \(0.015 \mathrm{~mol}\) of \(\mathrm{O}_{2}\) is produced in a reaction vessel with a volume of \(0.500 \mathrm{~L}\). What is the average rate of the reaction during this time interval? c. Predict the rate of change in the concentration of \(\mathrm{N}_{2} \mathrm{O}\) during this time interval. In other words, what is \(\Delta\left[\mathrm{N}_{2} \mathrm{O}\right] / \Delta t ?\)

For the reaction \(2 \mathrm{~A}(g)+\mathrm{B}(g) \longrightarrow 3 \mathrm{C}(g),\) a. determine the expression for the rate of the reaction in terms of the change in concentration of each of the reactants and products. b. when \(A\) is decreasing at a rate of \(0.100 \mathrm{M} / \mathrm{s},\) how fast is \(\mathrm{B}\) decreasing? How fast is C increasing?

For the reaction \(\mathrm{A}(g)+\frac{1}{2} \mathrm{~B}(g) \longrightarrow 2 \mathrm{C}(g)\) a. determine the expression for the rate of the reaction in terms of the change in concentration of each of the reactants and products. b. when \(C\) is increasing at a rate of \(0.025 \mathrm{M} / \mathrm{s},\) how fast is \(\mathrm{B}\) decreasing? How fast is A decreasing?

Consider the reaction: $$ 8 \mathrm{H}_{2} \mathrm{~S}(g)+4 \mathrm{O}_{2}(g) \longrightarrow 8 \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{S}_{8}(g) $$ Complete the table. $$ \begin{array}{lllll} \Delta\left[\mathrm{H}_{2} S\right] / \Delta t & \Delta\left[\mathrm{O}_{2}\right] / \Delta t & \Delta\left[\mathrm{H}_{2} \mathrm{O}\right] / \Delta t & \Delta\left[\mathrm{S}_{8}\right] / \Delta t & \text { Rate } \\ -0.080 \mathrm{M} / \mathrm{s} & & & & \\ \hline \end{array} $$

Consider the reaction: \(\mathrm{C}_{4} \mathrm{H}_{8}(g) \longrightarrow 2 \mathrm{C}_{2} \mathrm{H}_{4}(g)\) The tabulated data were collected for the concentration of \(\mathrm{C}_{4} \mathrm{H}_{8}\) as a function of time: $$ \begin{array}{cc} \text { Time (s) } & {\left[\mathrm{C}_{4} \mathrm{H}_{8}\right] \text { (M) }} \\ \hline 0 & 1.000 \\ \hline 10 & 0.913 \\ \hline 20 & 0.835 \\ \hline 30 & 0.763 \\ \hline 40 & 0.697 \\ \hline 50 & 0.637 \\ \hline \end{array} $$ a. What is the average rate of the reaction between 0 and 10 s? Between 40 and 50 s? b. What is the rate of formation of \(\mathrm{C}_{2} \mathrm{H}_{4}\) between 20 and \(30 \mathrm{~s}\) ?

Consider the reaction: $$ \mathrm{NO}_{2}(g) \longrightarrow \mathrm{NO}(g)+\frac{1}{2} \mathrm{O}_{2}(g) $$ The tabulated data were collected for the concentration of \(\mathrm{NO}_{2}\) as a function of time: $$ \begin{array}{cc} \text { Time (s) } & {\left[\mathrm{NO}_{2}\right] \text { (M) }} \\ \hline 0 & 1.000 \\ \hline 10 & 0.951 \\ \hline 20 & 0.904 \\ \hline 30 & 0.860 \\ \hline 40 & 0.818 \\ \hline 50 & 0.778 \\ \hline 60 & 0.740 \\ \hline 70 & 0.704 \\ \hline 80 & 0.670 \\ \hline 90 & 0.637 \\ \hline 100 & 0.606 \\ \hline \end{array} $$ a. What is the average rate of the reaction between 10 and 20 s? Between 50 and 60 s? b. What is the rate of formation of \(\mathrm{O}_{2}\) between 50 and \(60 \mathrm{~s}\) ?

What are the units of \(k\) for each type of reaction? a. first-order reaction b. second-order reaction c. zero-order reaction

This reaction is first order in \(\mathrm{N}_{2} \mathrm{O}_{5}\) $$ \mathrm{N}_{2} \mathrm{O}_{5}(g) \longrightarrow \mathrm{NO}_{3}(g)+\mathrm{NO}_{2}(g) $$ The rate constant for the reaction at a certain temperature is \(0.053 / \mathrm{s}\) a. Calculate the rate of the reaction when \(\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]=0.055 \mathrm{M}\). b. What would the rate of the reaction be at the concentration indicated in part a if the reaction were second order? Zero order? (Assume the same numerical value for the rate con- stant with the appropriate units.)

A reaction in which \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C}\) react to form products is first order in A, second order in B, and zero order in C. a. Write a rate law for the reaction. b. What is the overall order of the reaction? c. By what factor does the reaction rate change if [A] is doubled (and the other reactant concentrations are held constant)? d. By what factor does the reaction rate change if [B] is doubled (and the other reactant concentrations are held constant)? e. By what factor does the reaction rate change if [C] is doubled (and the other reactant concentrations are held constant)? f. By what factor does the reaction rate change if the concentrations of all three reactants are doubled?

A reaction in which \(A, B,\) and \(C\) react to form products is zero order in A, one-half order in B, and second order in C. a. Write a rate law for the reaction. b. What is the overall order of the reaction? c. By what factor does the reaction rate change if [A] is doubled (and the other reactant concentrations are held constant)? d. By what factor does the reaction rate change if [B] is doubled (and the other reactant concentrations are held constant)? e. By what factor does the reaction rate change if [C] is doubled (and the other reactant concentrations are held constant)? f. By what factor does the reaction rate change if the concentrations of all three reactants are doubled?

Consider the data showing the initial rate of a reaction (A \longrightarrow products) at several different concentrations of A. What is the order of the reaction? Write a rate law for the reac- tion, including the value of the rate constant, \(k\). $$ \begin{array}{cc} {[\mathrm{A}](\mathrm{M})} & \text { Initial Rate }(\mathrm{M} / \mathrm{s}) \\\ \hline 0.100 & 0.053 \\ \hline 0.200 & 0.210 \\ \hline 0.300 & 0.473 \\ \hline \end{array} $$

Consider the data showing the initial rate of a reaction (A \(\longrightarrow\) products) at several different concentrations of A. What is the order of the reaction? Write a rate law for the reaction, includ- ing the value of the rate constant, \(k\). $$ \begin{array}{cc} {[\mathrm{A}](\mathrm{M})} & \text { Initial Rate }(\mathrm{M} / \mathrm{s}) \\\ 0.15 & 0.008 \\ \hline 0.30 & 0.016 \\ \hline 0.60 & 0.032 \\ \hline \end{array} $$

Consider the tabulated data showing the initial rate of a reaction (A \(\longrightarrow\) products) at several different concentrations of A. What is the order of the reaction? Write a rate law for the reac- tion, including the value of the rate constant, \(k\). $$ \begin{array}{cc} {[\mathrm{A}](\mathrm{M})} & \text { Initial Rate }(\mathrm{M} / \mathrm{s}) \\\ 0.12 & 0.0078 \\ \hline 0.16 & 0.0104 \\ \hline 0.20 & 0.0130 \\ \hline \end{array} $$

Consider the tabulated data showing the initial rate of a reaction (A \(\longrightarrow\) products) at several different concentrations of A. What is the order of the reaction? Write a rate law for the reac- tion, including the value of the rate constant, \(k\). $$ \begin{array}{cc} {[\mathrm{A}](\mathrm{M})} & \text { Initial Rate }(\mathrm{M} / \mathrm{s}) \\\ 0.12 & 3.89 \times 10^{-4} \\ \hline 0.18 & 8.75 \times 10^{-4} \\ \hline 0.28 & 2.12 \times 10^{-3} \\ \hline \end{array} $$

Indicate the order of reaction consistent with each observation. a. A plot of the concentration of the reactant versus time yields a straight line. b. The reaction has a half-life that is independent of initial concentration. c. A plot of the inverse of the concentration versus time yields a straight line.

Indicate the order of reaction consistent with each observation. a. The half-life of the reaction gets shorter as the initial concentration is increased. b. A plot of the natural log of the concentration of the reactant versus time yields a straight line. c. The half-life of the reaction gets longer as the initial concentration is increased.

This reaction was monitored as a function of time: \(A \longrightarrow B+C\) A plot of \(\ln [\mathrm{A}]\) versus time yields a straight line with slope \(-0.0045 / \mathrm{s}\) a. What is the value of the rate constant \((k)\) for this reaction at this temperature? b. Write the rate law for the reaction. c. What is the half-life? d. If the initial concentration of \(\mathrm{A}\) is \(0.250 \mathrm{M},\) what is the concentration after 225 s?

This reaction was monitored as a function of time: $$ \mathrm{AB} \longrightarrow \mathrm{A}+\mathrm{B} $$ A plot of \(1 /[\mathrm{AB}]\) versus time yields a straight line with a slope of \(+0.55 / \mathrm{M} \cdot \mathrm{s}\) a. What is the value of the rate constant ( \(k\) ) for this reaction at this temperature? b. Write the rate law for the reaction. c. What is the half-life when the initial concentration is \(0.55 \mathrm{M} ?\) d. If the initial concentration of AB is \(0.250 \mathrm{M}\) and the reaction mixture initially contains no products, what are the concentrations of A and B after 75 s?

The decomposition of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is first order in \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) and has a rate constant of \(1.42 \times 10^{-4} \mathrm{~s}^{-1}\) at a certain temperature. a. What is the half-life for this reaction? b. How long will it take for the concentration of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) to decrease to \(25 \%\) of its initial concentration? c. If the initial concentration of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is \(1.00 \mathrm{M}\), how long will it take for the concentration to decrease to \(0.78 \mathrm{M} ?\) d. If the initial concentration of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is \(0.150 \mathrm{M},\) what is the concentration of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) after \(2.00 \times 10^{2} \mathrm{~s}\) ? After \(5.00 \times 10^{2} \mathrm{~s} ?\)

The decomposition of \(\mathrm{XY}\) is second order in \(\mathrm{XY}\) and has a rate constant of \(7.02 \times 10^{-3} \mathrm{M}^{-1} \cdot \mathrm{s}^{-1}\) at a certain temperature. a. What is the half-life for this reaction at an initial concentra- tion of \(0.100 \mathrm{M} ?\) b. How long will it take for the concentration of XY to decrease to \(12.5 \%\) of its initial concentration when the ini- tial concentration is \(0.100 \mathrm{M}\) ? When the initial concentra- tion is \(0.200 \mathrm{M} ?\) c. If the initial concentration of \(\mathrm{XY}\) is \(0.150 \mathrm{M}\), how long will it take for the concentration to decrease to \(0.062 \mathrm{M} ?\) d. If the initial concentration of \(\mathrm{XY}\) is \(0.050 \mathrm{M},\) what is the concentration of XY after \(5.0 \times 10^{1}\) s? After \(5.50 \times 10^{2}\) s?

The half-life for the radioactive decay of U-238 is 4.5 billion years and is independent of initial concentration. How long will it take for \(10 \%\) of the U- 238 atoms in a sample of U- 238 to decay? If a sample of U-238 initially contained \(1.5 \times 10^{18}\) atoms when the universe was formed 13.8 billion years ago, how many U- 238 atoms does it contain today?

The half-life for the radioactive decay of \(\mathrm{C}-14\) is 5730 years and is independent of the initial concentration. How long does it take for \(25 \%\) of the \(\mathrm{C}-14\) atoms in a sample of \(\mathrm{C}-14\) to decay? If a sample of C-14 initially contains 1.5 mmol of C-14, how many millimoles are left after 2255 years?

A chemical reaction is endothermic and has an activation energy that is twice the value of the enthalpy change of the reaction. Draw a diagram depicting the energy of the reaction as it progresses. Label the position of the reactants and products and indicate the activation energy and enthalpy of reaction.

The activation energy of a reaction is \(56.8 \mathrm{~kJ} / \mathrm{mol}\), and the frequency factor is \(1.5 \times 10^{11} / \mathrm{s}\). Calculate the rate constant of the reaction at \(25^{\circ} \mathrm{C}\).

The rate constant \((k)\) for a reaction was measured as a function of temperature. A plot of In \(k\) versus \(1 / T\) (in \(\mathrm{K}\) ) is linear and has a slope of -7445 K. Calculate the activation energy for the reaction.

The rate constant \((k)\) for a reaction was measured as a function of temperature. A plot of \(\ln k\) versus \(1 / T(\) in \(\mathrm{K})\) is linear and has a slope of \(-1.01 \times 10^{4} \mathrm{~K}\). Calculate the activation energy for the reaction.

The data shown here were collected for the first-order reaction: \(\mathrm{N}_{2} \mathrm{O}(g) \longrightarrow \mathrm{N}_{2}(g)+\mathrm{O}(g)\) Use an Arrhenius plot to determine the activation barrier and frequency factor for the reaction. $$ \begin{array}{cc} \text { Temperature (K) } & \text { Rate Constant (1/s) } \\ 800 & 3.24 \times 10^{-5} \\ \hline 900 & 0.00214 \\ \hline 1000 & 0.0614 \\ \hline 1100 & 0.955 \\ \hline \end{array} $$

The tabulated data show the rate constant of a reaction measured at several different temperatures. Use an Arrhenius plot to determine the activation barrier and frequency factor for the reaction. $$ \begin{array}{cc} \text { Temperature (K) } & \text { Rate Constant (1/s) } \\ 300 & 0.0134 \\ \hline 310 & 0.0407 \\ \hline 320 & 0.114 \\ \hline 330 & 0.303 \\ \hline 340 & 0.757 \\ \hline \end{array} $$