The hydrolysis of the sugar sucrose to the sugars glucose and fructose, $\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}+\mathrm{H}_2 \mathrm{O} \longrightarrow \mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6+\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6$ follows a first-order rate law for the disappearance of sucrose: rate $=k\left[\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right]$ (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, $k=2.1 \times 10^{-11} \mathrm{~s}^{-1}$ at $27^{\circ} \mathrm{C}$ and $8.5 \times 10^{-11} \mathrm{~s}^{-1}$ at $37^{\circ} \mathrm{C}$. Determine the activation energy, the frequency factor, and the rate constant for this equation at $47^{\circ} \mathrm{C}$ (assuming the kinetics remain consistent with the Arrhenius equation at this temperature). (b) When a solution of sucrose with an initial concentration of $0.150 \mathrm{M}$ reaches equilibrium, the concentration of sucrose is $1.65 \times 10^{-7} \mathrm{M}$. How long will it take the solution to reach equilibrium at $27^{\circ} \mathrm{C}$ 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)?

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 $30^{\circ} \mathrm{C}$ to $37^{\circ} \mathrm{C}$. What is the activation energy for the ALP-catalyzed conversion of PNPP to PNP and phosphate?

Consider the following reaction in aqueous solution:

The rate constant for the decomposition of acetaldehyde, $\mathrm{CH}_3 \mathrm{CHO}$, to methane, $\mathrm{CH}_4$, and carbon monoxide, $\mathrm{CO}$, in the gas phase is $1.1 \times 10^{-2} \mathrm{~L} \mathrm{~mol}^{-1} \mathrm{~s}^{-1}$ at $703 \mathrm{~K}^{\text {and }} 4.95 \mathrm{~L} \mathrm{~mol}^{-1} \mathrm{~s}^{-1}$ at $865 \mathrm{~K}$. Determine the activation energy for this decomposition.

The rate constant at $325^{\circ} \mathrm{C}$ for the decomposition reaction $\mathrm{C}_4 \mathrm{H}_8 \longrightarrow 2 \mathrm{C}_2 \mathrm{H}_4$ is $6.1 \times 10^{-8} \mathrm{~s}$, and the activation energy is $261 \mathrm{~kJ}$ per mole of $\mathrm{C}_4 \mathrm{H}_8$. Determine the frequency factor for the reaction.

In an experiment, a sample of $\mathrm{NaClO}_3$ was $90 \%$ decomposed in $48 \mathrm{~min}$. Approximately how long would this decomposition have taken if the sample had been heated $20^{\circ} \mathrm{C}$ higher? (Hint: Assume the rate doubles for each 10 ${ }^{\circ} \mathrm{C}$ rise in temperature.)

The rate of a certain reaction doubles for every $10^{\circ} \mathrm{C}$ rise in temperature. (a) How much faster does the reaction proceed at $45^{\circ} \mathrm{C}$ than at $25^{\circ} \mathrm{C}$ ? (b) How much faster does the reaction proceed at $95^{\circ} \mathrm{C}$ than at $25^{\circ} \mathrm{C}$ ?

How does an increase in temperature affect rate of reaction? Explain this effect in terms of the collision theory of the reaction rate.

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.

Account for the relationship between the rate of a reaction and its activation energy.

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

Fluorine-18 is a radioactive isotope that decays by positron emission to form oxygen-18 with a half-life of 109.7 min. (A positron is a particle with the mass of an electron and a single unit of positive charge; the equation is $_{518} \mathrm{F} \longrightarrow_{18}^{8} \mathrm{O}+\mathrm{e}^{-}$.) Physicians use $^{18} \mathrm{F}$ to study the brain by injecting a quantity of fluoro-substituted glucose into the blood of a patient. The glucose accumulates in the regions where the brain is active and needs nourishment. (a) What is the rate constant for the decomposition of fluorine-18? (b) If a sample of glucose containing radioactive fluorine-18 is injected into the blood, what percent of the radioactivity will remain after 5.59 h? (c) How long does it take for 99.99$\%$ of the $^{18} \mathrm{F}$ to decay?

When every collision between reactants leads to a reaction, what determines the rate at which the reaction occurs?

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