Kinetics: Rates and Mechanism of Chemical Reactions

Ina kinetics experiment, a chemist places crystals of iodine in a closed reaction vessel, introduces a given quantity of hydrogen gas, and obtains data to calculate the rate of hydrogen iodide formation. In a second experiment, she uses the same amounts of iodine and hydrogen, but first warms the flask to 130 degrees Celsius, a temperature above the sublimation point of iodine. In which of these experiments does the reaction proceed at a higher rate? Explain.

How does an increase in temperature affect the rate of a reaction? Explain the two factors involved.

A gas reacts with a solid presence in large chunks. Then the reaction is run again with the solid pulverized. How does the increase in the surface area of the solid affect the rate of its reaction with the gas? Explain.

Define reaction rate, assuming constant temperature and a closed reaction vessel, why does the rate change with time?

(a) What is the difference between an average rate and an instantaneous rate? (b) What is the difference between an initial rate and an instantaneous rate?

Give two reasons to measure initial rates in a kinetic study.

For the reaction , sketch two curves on the same set of axes that show  (a) The formation of product as a function of time(b) The consumption of reactant as a function of time

For the reaction , [C] vs. time is plotted: 

How do you determine each of the following?

(a) The average rate over the entire experiment

(b) The reaction rate at time x 

(c) The initial reaction rate 

(d) Would the values in parts (a), (b) and (c) be different if you plotted [D] vs. time? Explain.

Reaction rate is expressed in terms of changes in the concentration of reactants and products. Write a balanced equation for $Rate=-\frac{1}{2}\frac{∆\left[N_2{O}_5\right]}{∆t}=\frac{1}{4}\frac{∆\left[N{O}_2\right]}{∆t}=\frac{∆\left[O_2\right]}{∆t}$

Reaction rate is expressed in terms of changes in the concentration of reactants and products. Write a balanced equation for

Rate$\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}$$\frac{\mathbf{∆}\mathbf{\left[}{\mathbf{N}}_{\mathbf{2}}{\mathbf{O}}_{\mathbf{5}}\mathbf{\right]}}{\mathbf{∆}\mathbf{t}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}}\frac{\mathbf{∆}\mathbf{\left[}{\mathbf{NO}}_{\mathbf{2}}\mathbf{\right]}}{\mathbf{∆}\mathbf{t}}\mathbf{=}\frac{\mathbf{∆}\mathbf{\left[}{\mathbf{O}}_{\mathbf{2}}\mathbf{\right]}}{\mathbf{∆}\mathbf{t}}$

Reaction rate is expressed in terms of changes in concentration of reactants and products. Write a balanced equation for

$\mathbf{Rate}\mathbf{=}\mathbf{-}\frac{\mathbf{∆}\left[\mathrm{CH}4\right]}{\mathbf{∆}\mathbf{t}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{∆}\left[O2\right]}{\mathbf{∆}\mathbf{t}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{∆}\left[H2O\right]}{\mathbf{∆}\mathbf{t}}\mathbf{=}\frac{\mathbf{∆}\left[\mathrm{CO}2\right]}{\mathbf{∆}\mathbf{t}}$

Question: Reaction rate is expressed in terms of changes in concentration of reactants and products. Write a balanced equation for

The formation of ammonia is one of the most important processes in the chemical industry: 

${\mathbf{N}}_{2\left(g\right)}\mathbf{+}\mathbf{3}{\mathbf{H}}_{\mathbf{2}\left(g\right)}\mathbf{\to }\mathbf{2}{\mathbf{NH}}_{3\left(g\right)}$

Express the rate in terms of changes in $\left[N_2\right]$, $\left[H_2\right]$, and $\left[{\mathrm{NH}}_3\right]$.

By what factor does the rate change in each of the following cases (assuming constant temperature)? 

(a) A reaction is first order in reactant A, and [A] is doubled. 

(b) A reaction is second order in reactant B, and [B] is halved.

(c) A reaction is second order in reactant C, and [C] is tripled.

To determine its rate law. Assuming that you have a valid experimental procedure for obtaining [A2] and [B2] at various times, explain how you determine 

${\mathbf{A}}_{\mathbf{2}\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\mathbf{+}{\mathbf{B}}_{\mathbf{2}\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\mathbf{\to }\mathbf{2}\mathbf{AB}\mathbf{\left(}\mathbf{g}\mathbf{\right)}$

(a) the initial rate, 

(b) the reaction orders, and 

(c) the rate constant. 

The rate law for the general reaction

$\mathit{a}\mathit{A}\mathbf{+}\mathit{b}\mathit{B}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\to }\mathit{c}\mathit{C}\mathbf{+}\mathit{d}\mathit{D}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}$

 is rate = $\mathit{k}{\left[A\right]}^{\mathbf{m}}{\left[B\right]}^{\mathbf{n}}\mathbf{.}\mathbf{.}\mathbf{.}$

 (a) Explain the meaning of k. 

(b) Explain the meanings of m and n. Does m a and n b? Explain. 

(c) If the reaction is first order in A and second order in B, and time is measured in minutes (min), what are the units for k?

Although the depletion of stratospheric ozone threatens life on Earth today, its accumulation was one of the crucial processes that allowed life to develop in prehistoric times: 

$\mathbf{3}{\mathit{O}}_{\mathbf{2}\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\mathbf{\to }\mathbf{2}{\mathit{O}}_{\mathbf{3}\mathbf{\left(}\mathbf{g}\mathbf{\right)}}$

(a) Express the reaction rate in terms $\left[O_2\right]$and $\left[O_3\right]$

(b) At a given instant, the reaction rate in terms of  is 2.17105 mol/L s. What is it in terms of $\left[O_3\right]$?

The decomposition of NOBr is studied manometrically because the number of moles of gas changes; it cannot be studied colorimetrically because both NOBr and Br2 are reddish-brown: 

$\mathbf{2}\mathit{N}\mathit{O}\mathit{B}{\mathit{r}}_{\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\mathbf{\to }\mathbf{2}\mathit{N}{\mathit{O}}_{\left(\mathit{g}\right)}\mathbf{+}\mathit{B}{\mathit{r}}_{\mathbf{2}\mathbf{\left(}\mathbf{g}\mathbf{\right)}}$

Use the data below to answer the following: 

(a) Determine the average rate over the entire experiment.

 (b) Determine the average rate between 2.00 and 4.00 s. 

(c) Use graphical methods to estimate the initial reaction rate.

 (d) Use graphical methods to estimate the rate at 7.00 s. 

(e) At what time does the instantaneous rate equal the average rate over the entire experiment?

Reaction rate is expressed in terms of changes in the concentration of reactants and products. Write a balanced equation for

$\mathit{R}\mathit{a}\mathit{t}\mathit{e}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{\Delta }\mathbf{\left[}{\mathbf{N}}_{\mathbf{2}}{\mathbf{O}}_{\mathbf{5}}\mathbf{\right]}}{\mathbf{\Delta }\mathbf{t}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}}\frac{\mathbf{\Delta }\mathbf{\left[}\mathbf{N}{\mathbf{O}}_{\mathbf{2}}\mathbf{\right]}}{\mathbf{\Delta }\mathbf{t}}\mathbf{=}\frac{\mathbf{\Delta }\mathbf{\left[}{\mathbf{O}}_{\mathbf{2}}\mathbf{\right]}}{\mathbf{\Delta }\mathbf{t}}$

Express the rate of reaction in terms of the change in concentration of each of the reactants and products: 

$\mathbf{2}\mathbf{D}\mathbf{\left(}\mathbf{g}\mathbf{\right)}\mathbf{+}\mathbf{3}\mathbf{E}\mathbf{\left(}\mathbf{g}\mathbf{\right)}\mathbf{+}\mathbf{F}\mathbf{\left(}\mathbf{g}\mathbf{\right)}\mathbf{\to }\mathbf{2}\mathbf{G}\mathbf{\left(}\mathbf{g}\mathbf{\right)}\mathbf{+}\mathbf{H}\mathbf{\left(}\mathbf{g}\mathbf{\right)}$

When [D] is decreasing at 0.1 mol/L.s, how fast is [H] increasing?

Give the individual reaction orders for all substances and the overall reaction order from the following rate law:

$\mathbf{Rate}\mathbf{=}\mathbf{k}\left[{{\mathrm{BrO}}_3}^{-}\right]\left[{\mathrm{Br}}^{-}\right]{\left[H^{+}\right]}^{\mathbf{2}}$

( a) For a reaction with a given Ea, how does an increase in T affect the rate? 

(b) For a reaction at a given T, how does a decrease in Ea affect the rate?

For the reaction , how many unique collisions between A and B are possible if 1.01 mol of A(g) and 2.12 mol of B(g) are present in the vessel?

At ${\mathbf{25}}^{\mathbf{\circ }}\mathbf{C}$ , what is the fraction of collisions with energy equal to or greater than an activation energy of 100. kJ/mol?

If the temperature in Problem 16.60 is increased to ${\mathbf{50}}^{\mathbf{\circ }}\mathbf{C}$, by what factor does the fraction of collisions with energy equal to or greater than the activation energy change?

Is the rate of an overall reaction lower, higher, or equal to the average rate of the individual steps? Explain.

Explain why the coefficients of an elementary step equal the reaction orders of its rate law but those of an overall reaction do not.

For the reaction $\mathbf{A}\mathbf{\left(}\mathbf{g}\mathbf{\right)}\mathbf{+}\mathbf{B}\mathbf{\left(}\mathbf{g}\mathbf{\right)}\mathbf{\to }\mathbf{A}\mathbf{B}\mathbf{\left(}\mathbf{g}\mathbf{\right)}$, how many unique collisions between A and B are possible if 1.01 mol of A(g) and 2.12 mol of B(g) are present in the vessel?

Question: At ${\mathbf{25}}^{\mathbf{0}}\mathit{C}$ , what is the fraction of collisions with energy equal to or greater than an activation energy of 100. kJ/mol?

If the temperature in Problem 16.60 is increased to ${\mathbf{50}}^{\mathbf{0}}\mathit{C}$, by what factor does the fraction of collisions with energy equal to or greater than the activation energy change?

a) For a reaction with a given Ea, how does an increase in T affect the rate? 

(b) For a reaction at a given T, how does a decrease in Ea affect the rate?

For the reaction , $\mathbf{A}\mathbf{B}\mathbf{C}\mathbf{+}\mathbf{D}\mathbf{\rightleftharpoons }\mathbf{A}\mathbf{B}\mathbf{+}\mathbf{C}\mathbf{D}$, $\mathbf{∆}{{\mathbf{H}}^{\mathbf{0}}}_{\mathbf{r}\mathbf{\times }\mathbf{n}}\mathbf{=}\mathbf{ }\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{55}\mathit{k}\mathit{J}\mathbf{/}\mathit{m}\mathit{o}\mathit{l}$ ${\mathit{E}}_{\mathbf{a}\left(fwd\right)}\mathbf{=}\mathbf{215}\mathit{k}\mathit{J}\mathbf{/}\mathit{m}\mathit{o}\mathit{l}$ Assuming a one-step reaction,

(a) Draw a reaction energy diagram;

 (b) Calculate Ea(rev); and 

(c) Sketch a possible transition state if ABC is V-shaped

For the reaction ${\mathbf{A}}_{\mathbf{2}}\mathbf{+}{\mathbf{B}}_{\mathbf{2}}\mathbf{\to }\mathbf{2}\mathbf{A}\mathbf{B}$, $E_{a\left(fwd\right)}=125kJ/mol$ and $E_{a\left(rev\right)}=85kJ/mol$ . Assuming the reaction occurs in one step, 

(a) Draw a reaction energy diagram; 

(b) Calculate $\mathbf{∆}{{\mathbf{H}}^{\mathbf{0}}}_{\mathbf{r}\mathbf{\times }\mathbf{n}}$ ; and 

(c) Sketch a possible transition state.

Is the rate of an overall reaction lower, higher, or equal to the average rate of the individual steps? Explain.

Explain why the coefficients of an elementary step equal the reaction orders of its rate law but those of an overall reaction do not.

The rate constant of a reaction is $\mathbf{4}\mathbf{.}\mathbf{7}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{ }{\mathbf{s}}^{\mathbf{-}\mathbf{1}}$ at $\mathbf{25}{\mathbf{ }}^{\mathbf{0}}\mathbf{C}$ , and the activation energy is 33.6 kJ/mol. What is k at $\mathbf{75}{\mathbf{ }}^{\mathbf{0}}\mathbf{C}$?

The rate constant of a reaction is $\mathbf{4}\mathbf{.}\mathbf{50}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{ }\mathbf{L}\mathbf{/}\mathbf{mol}\mathbf{.}\mathbf{s}$ at  $\mathbf{195}{\mathbf{ }}^{\mathbf{0}}\mathbf{c}$and $\mathbf{3}\mathbf{.}\mathbf{20}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{ }\mathbf{L}\mathbf{/}\mathbf{mol}\mathbf{.}\mathbf{s}$  at $\mathbf{258}{\mathbf{ }}^{\mathbf{0}}\mathbf{C}$ . What is the activation energy of the reaction?

 What is the central idea of collision theory? How does this idea explain the effect of concentration on reaction rate?

While developing a catalytic process to make ethylene glycol from synthesis gas $(\mathrm{CO}+H_2)$, a chemical engineer finds the rate is fourth-order in gas pressure. The uncertainty in the pressure reading is 5%. When the catalyst is modified, the rate increases by 10%. If you were the company patent attorney, would you file for a patent on this catalyst modification? Explain.

Question:For the decomposition of gaseous dinitrogen pentaoxide,

$2{\mathrm{N}}_{5\left(\mathrm{g}\right)} \to 4{\mathrm{NO}}_{2\left(\mathrm{g}\right)}+{\mathrm{O}}_{2\left(\mathrm{g}\right),}$the rate constant is k $=2.8\times {10}^{-3}{\mathrm{s}}^{-1 }\mathrm{at} 60°\mathrm{C}. \mathrm{The}$initial concentration  of ${\mathrm{N}}_2{\mathrm{O}}_5$is 1.58 mol/L. 

(a) What $\left[{\mathrm{N}}_2{\mathrm{O}}_5\right]$after 5.00 min?

(b) What fraction of the ${\mathrm{N}}_2{\mathrm{O}}_5$has decomposed after 5.00 min?

For the decomposition of gaseous dinitrogen pentaoxide, $\mathbf{2}{\mathbf{N}}_{\mathbf{2}}{\mathbf{O}}_{\mathbf{5}\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\mathbf{\to }\mathbf{4}\mathbf{N}{\mathbf{O}}_{\mathbf{2}\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\mathbf{+}{\mathbf{O}}_{\mathbf{2}\mathbf{\left(}\mathbf{g}\mathbf{\right)}}$ , the rate constant is $\mathit{k}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{8}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}{\mathit{s}}^{\mathbf{-}\mathbf{1}}$ at ${\mathbf{60}}^{\mathbf{0}}\mathit{C}$ . The initial concentration of ${\mathit{N}}_{\mathbf{2}}{\mathit{O}}_{\mathbf{5}}$ is 1.58 mol/L. 

(a) What is $\left[N_2{O}_5\right]$ after 5.00 min? 

(b) What fraction of the ${\mathit{N}}_{\mathbf{2}}{\mathit{O}}_{\mathbf{5}}$ has decomposed after 5.00 min?

Even when a mechanism is consistent with the rate law, later work may show it to be incorrect. For example, the reaction between hydrogen and iodine has this rate law: $\mathit{r}\mathit{a}\mathit{t}\mathit{e}\mathbf{=}\mathit{k}\left[H_2\right]\left[I_2\right]$ . The long-accepted mechanism had a single bimolecular step; that is, the overall reaction was thought to be elementary: 

${\mathbf{H}}_{\mathbf{2}\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\mathbf{+}{\mathbf{I}}_{\mathbf{2}\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\mathbf{\to }\mathbf{2}\mathbf{H}{\mathbf{I}}_{\mathbf{\left(}\mathbf{g}\mathbf{\right)}}$

In the 1960s, however, spectroscopic evidence showed the presence of free I atoms during the reaction. Kineticists have since proposed a three-step mechanism:

$$\begin{gathered} \left(1\right){\mathbf{I}}_{\mathbf{2}\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\mathbf{\rightleftharpoons }\mathbf{2}{\mathbf{I}}_{\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\left[fast\right] \\ \left(2\right){\mathbf{H}}_{\mathbf{2}\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\mathbf{+}{\mathbf{I}}_{\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\mathbf{\rightleftharpoons }{\mathbf{H}}_{\mathbf{2}}{\mathbf{I}}_{\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\left[fast\right] \\ \left(3\right){\mathbf{H}}_{\mathbf{2}}{\mathbf{I}}_{\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\mathbf{+}{\mathbf{I}}_{\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\mathbf{\rightleftharpoons }\mathbf{2}\mathbf{H}{\mathbf{I}}_{\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\left[slow\right] \end{gathered}$$

Show that this mechanism is consistent with the rate law.

Suggest an experimental method for measuring the change in concentration with time for each of the following reactions:
$$\begin{gathered} \left(a\right)\mathbf{C}{\mathbf{H}}_{\mathbf{3}}\mathbf{C}{\mathbf{H}}_{\mathbf{2}}\mathbf{B}{\mathbf{r}}_{\mathbf{\left(}\mathbf{l}\mathbf{\right)}}\mathbf{+}{\mathbf{H}}_{\mathbf{2}}{\mathbf{O}}_{\mathbf{\left(}\mathbf{l}\mathbf{\right)}}\mathbf{\to }\mathbf{C}{\mathbf{H}}_{\mathbf{3}}\mathbf{C}{\mathbf{H}}_{\mathbf{2}}\mathbf{O}{\mathbf{H}}_{\mathbf{\left(}\mathbf{l}\mathbf{\right)}}\mathbf{+}\mathbf{H}\mathbf{B}{\mathbf{r}}_{\mathbf{\left(}\mathbf{a}\mathbf{q}\mathbf{\right)}} \\ \left(b\right)\mathbf{2}\mathbf{N}{\mathbf{O}}_{\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\mathbf{+}\mathbf{C}{\mathbf{l}}_{\mathbf{2}\mathbf{\left(}\mathbf{g}\mathbf{\right)}}\mathbf{\to }\mathbf{2}\mathbf{N}\mathbf{O}\mathbf{C}{\mathbf{l}}_{\mathbf{\left(}\mathbf{g}\mathbf{\right)}} \end{gathered}$$

An atmospheric chemist fills a container with gaseous ${\mathit{N}}_{\mathbf{2}}{\mathit{O}}_{\mathbf{5}}$ to a pressure of 125 kPa, and the gas decomposes to $\mathit{N}{\mathit{O}}_{\mathbf{2}}$ and ${\mathit{O}}_{\mathbf{2}}$. What is the partial pressure of ${\mathit{P}}_{\mathbf{N}{\mathbf{O}}_{\mathbf{2}}}$, (in kPa), when the total pressure is 178 kPa?

While developing a catalytic process to make ethylene glycol from synthesis gas $\left(CO+H_2\right)$ , a chemical engineer finds the rate is fourth-order in gas pressure. The uncertainty in the pressure reading is 5%. When the catalyst is modified, the rate increases by 10%. If you were the company patent attorney, would you file for a patent on this catalyst modification? Explain.

Assume water boils at ${\mathbf{100}}^{\mathbf{0}}\mathit{C}$  in Houston (near sea level), and at ${\mathbf{90}}^{\mathbf{0}}\mathit{C}$  in Cripple Creek, Colorado (near 9500 ft). If it takes 4.8 min to cook an egg in Cripple Creek and 4.5 min in Houston, what is ${\mathbf{E}}_{\mathbf{a}}$ for this process?

Chlorine is commonly used to disinfect drinking water, and the inactivation of pathogens by chlorine follows first-order kinetics. The following data show E. coli inactivation:

(a) Determine the first-order inactivation constant, k. [Hint: % inactivation  

$\mathbf{=}\mathbf{100}\mathbf{\times }\left(1-\frac{{\left[A\right]}_t}{{\left[A\right]}_0}\right)$

(b) How much contact time is required for 95% inactivation?

Question:Many drugs decompose in blood by a first-order process. 

(a) Two tablets of aspirin supply 0.60 g of the active compound. After 30 min, this compound reaches a maximum concentration of 2 mg/100 mL of blood. If the half-life for its breakdown is 90 min, what is its concentration (in mg/100 mL) 2.5 h after it reaches its maximum concentration? 

(b) For the decomposition of an antibiotic in a person with a normal temperature $\mathbf{(}\mathbf{98}\mathbf{.}\mathbf{6}\mathbf{°}\mathbf{ }\mathbf{F}\mathbf{)}\mathbf{,}\mathbf{ }\mathbf{k}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{1}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}{\mathbf{s}}^{\mathbf{-}\mathbf{1}}$ ; for a person with a fever at $\mathbf{101}\mathbf{.}\mathbf{9}\mathbf{°}\mathbf{ }\mathbf{F}$, $\mathbf{k}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{9}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}{\mathbf{s}}^{\mathbf{-}\mathbf{1}}$ . If the person with the fever must take another pill when of the first pill has decomposed, how many hours should she wait to take a second pill? A third pill? (Assume the pill is effective immediately.)

(c) Calculate Ea for decomposition of the antibiotic in part (b).

Iodide ion reacts with chloromethane to displace chloride ion in a common organic substitution reaction: 

${\mathbf{I}}^{\mathbf{-}}\mathbf{+}\mathbf{CH}\mathbf{3}\mathbf{CI}\mathbf{\to }\mathbf{CH}\mathbf{3}\mathbf{I}\mathbf{+}{\mathbf{CI}}^{\mathbf{-}}$

(a) Draw a wedge-bond structure of chloroform and indicate the most effective direction of ${\mathbf{I}}^{\mathbf{-}}$ attack.

(b) The analogous reaction with 2-chlorobutane [Figure P16.107(b)] results in a major change in specific rotation as measured by polarimetry. Explain, showing a wedge-bond structure of the product. 

(c) Under different conditions, 2-chlorobutane loses ${\mathbf{CI}}^{\mathbf{-}}$ in a rate-determining step to form a planar intermediate [Figure P16.107(c)]. This cationic species reacts with HI and then loses H to form a product that exhibits no optical activity. Explain, showing a wedge-bond structure.

Question:Sulfonation of benzene has the following mechanism: 

$$\begin{gathered} \mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{2}{\mathbf{H}}_{\mathbf{2}}{\mathbf{SO}}_{\mathbf{4}}\mathbf{\to }{\mathbf{H}}_{\mathbf{3}}{\mathbf{O}}^{\mathbf{+}}\mathbf{+}{{\mathbf{HSO}}_{\mathbf{4}}}^{\mathbf{-}}\mathbf{+}{\mathbf{SO}}_{\mathbf{3}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\left[}\mathbf{fast}\mathbf{\right]} \\ \mathbf{\left(}\mathbf{2}\mathbf{\right)}{\mathbf{SO}}_{\mathbf{3}}\mathbf{+}{\mathbf{C}}_{\mathbf{6}}{\mathbf{H}}_{\mathbf{6}}\mathbf{\to }\mathbf{H}\mathbf{\left(}{\mathbf{C}}_{\mathbf{6}}{{\mathbf{H}}_{\mathbf{5}}}^{\mathbf{+}}\mathbf{\right)}{{\mathbf{SO}}_{\mathbf{3}}}^{\mathbf{-}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\left[}\mathbf{slow}\mathbf{\right]} \\ \mathbf{\left(}\mathbf{3}\mathbf{\right)}\mathbf{H}\mathbf{(}{\mathbf{C}}_{\mathbf{6}}{\mathbf{H}}_{\mathbf{5}}\mathbf{+}\mathbf{)}{{\mathbf{SO}}_{\mathbf{3}}}^{\mathbf{-}}\mathbf{+}{{\mathbf{HSO}}_{\mathbf{4}}}^{\mathbf{-}}\mathbf{\to }{\mathbf{C}}_{\mathbf{6}}{\mathbf{H}}_{\mathbf{5}}{{\mathbf{SO}}_{\mathbf{3}}}^{\mathbf{-}}\mathbf{+}{\mathbf{H}}_{\mathbf{2}}{\mathbf{SO}}_{\mathbf{4}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\left[}\mathbf{fast}\mathbf{\right]} \\ \mathbf{\left(}\mathbf{4}\mathbf{\right)}{\mathbf{C}}_{\mathbf{6}}{\mathbf{H}}_{\mathbf{5}}{{\mathbf{SO}}_{\mathbf{3}}}^{\mathbf{-}}\mathbf{+}{\mathbf{H}}_{\mathbf{3}}{\mathbf{O}}^{\mathbf{+}}\mathbf{\to }{\mathbf{C}}_{\mathbf{6}}{\mathbf{H}}_{\mathbf{5}}{\mathbf{SO}}_{\mathbf{3}}\mathbf{H}\mathbf{+}{\mathbf{H}}_{\mathbf{2}}\mathbf{O}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\left[}\mathbf{fast}\mathbf{\right]} \end{gathered}$$

(a) Write an overall equation for the reaction. 

(b) Write the overall rate law for the initial rate of the reaction