Chapter 20

From the observations listed, estimate the value of \(E^{\circ}\) for the half- reaction \(\mathrm{M}^{2+}(\mathrm{aq})+2 \mathrm{e}^{-} \longrightarrow \mathrm{M}(\mathrm{s})\) (a) The metal M reacts with HNO \(_{3}(\text { aq })\), but not with \(\mathrm{HCl}(\mathrm{aq}) ; \mathrm{M}\) displaces \(\mathrm{Ag}^{+}(\mathrm{aq}),\) but not \(\mathrm{Cu}^{2+}(\mathrm{aq})\) (b) The metal \(M\) reacts with \(\mathrm{HCl}(\mathrm{aq}),\) producing \(\mathrm{H}_{2}(\mathrm{g}),\) but displaces neither \(\mathrm{Zn}^{2+}(\text { aq })\) nor \(\mathrm{Fe}^{2+}(\mathrm{aq})\).

You must estimate \(E^{\circ}\) for the half-reaction \(\operatorname{In}^{3+}(\mathrm{aq})+\) \(3 \mathrm{e}^{-} \longrightarrow \operatorname{In}(\mathrm{s}) .\) You have no electrical equipment, but you do have all of the metals listed in Table 20.1 and aqueous solutions of their ions, as well as \(\operatorname{In}(\mathrm{s})\) and \(\operatorname{In}^{3+}(\text { aq })\). Describe the experiments you would perform and the accuracy you would expect in your result.

Given that \(E_{\text {cell }}^{\circ}=3.20 \mathrm{V}\) for the reaction $$2 \mathrm{Na}(\mathrm{in} \mathrm{Hg})+\mathrm{Cl}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{Na}^{+}(\mathrm{aq})+2 \mathrm{Cl}^{-}(\mathrm{aq})$$ What is \(E^{\circ}\) for the reduction \(2 \mathrm{Na}^{+}(\mathrm{aq})+2 \mathrm{e}^{-} \longrightarrow\) \(2 \mathrm{Na}(\text { in } \mathrm{Hg}) ?\)

$$E_{\text {cathode }}^{\circ}=(2.71-2.310) V=+0.40 V$$VVV

\(E_{\text {cathode }}^{\circ}=(2.71-2.310) V=+0.40 \mathrm{V}\)

The theoretical \(E_{\text {cell }}^{\circ}\) for the methane-oxygen fuel cell is \(1.06 \mathrm{V} .\) What is \(E^{\circ}\) for the reduction half-reaction \(\mathrm{CO}_{2}(\mathrm{g})+8 \mathrm{H}^{+}(\mathrm{aq})+8 \mathrm{e}^{-} \longrightarrow \mathrm{CH}_{4}(\mathrm{g})+2 \mathrm{H}_{2} \mathrm{O}(1) ?\)

Given these half-reactions and associated standard reduction potentials, answer the questions that follow: $$\begin{aligned} &\left[\mathrm{Zn}\left(\mathrm{NH}_{3}\right)_{4}\right]^{2+}(\mathrm{aq})+2 \mathrm{e}^{-} \longrightarrow \mathrm{Zn}(\mathrm{s})+4 \mathrm{NH}_{3}(\mathrm{aq})\\\ &E^{\circ}=-1.015 \mathrm{V} \end{aligned}$$ $$\begin{array}{c} \mathrm{Ti}^{3+}(\mathrm{aq})+\mathrm{e}^{-} \longrightarrow \mathrm{Ti}^{2+}(\mathrm{aq}) \\ E^{\circ}=-0.37 \mathrm{V} \end{array}$$ $$\begin{aligned} &\mathrm{VO}^{2+}(\mathrm{aq})+2 \mathrm{H}^{+}(\mathrm{aq})+\mathrm{e}^{-} \longrightarrow \mathrm{V}^{3+}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\mathrm{I})\\\ &E^{\circ}=0.340 \mathrm{V} \end{aligned}$$ $$\begin{array}{r} \mathrm{Sn}^{2+}(\mathrm{aq})+2 \mathrm{e}^{-} \longrightarrow \mathrm{Sn}(\mathrm{aq}) \\ E^{\circ}=-0.14 \mathrm{V} \end{array}$$ (a) Determine which pair of half-cell reactions leads to a cell reaction with the largest positive cell potential, and calculate its value. Which couple is at the anode and which is at the cathode? (b) Determine which pair of these half-cell reactions leads to the cell with the smallest positive cell potential, and calculate its value. Which couple is at the anode and which is at the cathode?

\(\mathrm{Ni}^{2+}\) has a more positive reduction potential than \(\mathrm{Cd}^{2+}\) (a) Which ion is more easily reduced to the metal? (b) Which metal, Ni or Cd, is more easily oxidized?

Refer to standard reduction potentials, and predict which metal in each of the following pairs is the stronger reducing agent: (a) sodium or potassium (b) magnesium or barium

For the reduction half-reaction \(\mathrm{Hg}_{2}^{2+}(\mathrm{aq})+2 \mathrm{e}^{-}\) \(\longrightarrow 2 \mathrm{Hg}(1), E^{\circ}=0.797 \mathrm{V} .\) Will \(\mathrm{Hg}(\mathrm{l})\) react with and dissolve in HCl(aq)? in HNO3(aq)? Explain.

Consider the reaction \(\operatorname{Co}(\mathrm{s})+\mathrm{Ni}^{2+}(\mathrm{aq}) \longrightarrow\) \(\mathrm{Co}^{2+}(\mathrm{aq})+\mathrm{Ni}(\mathrm{s}), \quad\) with \(\quad E_{\mathrm{cell}}^{\circ}=0.02 \mathrm{V} . \quad\) If \(\quad \mathrm{Co}(\mathrm{s}) \quad\) is added to a solution with \(\left[\mathrm{Ni}^{2+}\right]=1 \mathrm{M},\) should the reaction go to completion? Explain.

Dichromate ion \(\left(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}\right)\) in acidic solution is a good oxidizing agent. Which of the following oxidations can be accomplished with dichromate ion in acidic solution? Explain. (a) \(\operatorname{sn}^{2+}\left(\text { aq) to } \operatorname{Sn}^{4+}(\text { aq })\right.\) (b) \(\mathrm{I}_{2}(\mathrm{s})\) to \(\mathrm{IO}_{3}^{-}(\mathrm{aq})\) (c) \(\mathrm{Mn}^{2+}(\mathrm{aq})\) to \(\mathrm{MnO}_{4}^{-}(\mathrm{aq})\)

Predict whether the following metals will react with the acid indicated. If a reaction does occur, write the net ionic equation for the reaction. Assume that reactants and products are in their standard states. (a) \(\mathrm{Ag}\) in \(\mathrm{HNO}_{3}(\mathrm{aq}) ;\) (b) \(\mathrm{Zn}\) in \(\mathrm{HI}(\mathrm{aq}) ;\) (c) \(\mathrm{Au}\) in \(\mathrm{HNO}_{3}\) (for the couple \(\left.\mathrm{Au}^{3+} / \mathrm{Au}, E^{\circ}=1.52 \mathrm{V}\right)\).

Predict whether, to any significant extent, (a) \(\mathrm{Fe}(\mathrm{s})\) will displace \(\mathrm{Zn}^{2+}(\mathrm{aq})\) (b) \(\mathrm{MnO}_{4}^{-}(\mathrm{aq})\) will oxidize \(\mathrm{Cl}^{-}(\mathrm{aq})\) to \(\mathrm{Cl}_{2}(\mathrm{g})\) in acidic solution; (c) \(\mathrm{Ag}(\mathrm{s})\) will react with \(1 \mathrm{M} \mathrm{HCl}(\mathrm{aq})\) (d) \(\mathrm{O}_{2}(\mathrm{g})\) will oxidize \(\mathrm{Cl}^{-}(\mathrm{aq})\) to \(\mathrm{Cl}_{2}(\mathrm{g})\) in acidic solution.

Use the data in Appendix D to calculate the standard cell potential for each of the following reactions. Which reactions will occur spontaneously? (a) \(\mathrm{H}_{2}(\mathrm{g})+\mathrm{F}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{H}^{+}(\mathrm{aq})+2 \mathrm{F}^{-}(\mathrm{aq})\) (b) \(\mathrm{Cu}(\mathrm{s})+\mathrm{Ba}^{2+}(\mathrm{aq}) \longrightarrow \mathrm{Cu}^{2+}(\mathrm{aq})+\mathrm{Ba}(\mathrm{s})\) (c) \(3 \mathrm{Fe}^{2+}(\mathrm{aq}) \longrightarrow \mathrm{Fe}(\mathrm{s})+2 \mathrm{Fe}^{3+}(\mathrm{aq})\) (d) \(\mathrm{Hg}(1)+\mathrm{HgCl}_{2}(\mathrm{aq}) \longrightarrow \mathrm{Hg}_{2} \mathrm{Cl}_{2}(\mathrm{s})\)

In each of the following examples, sketch a voltaic cell that uses the given reaction. Label the anode and cathode; indicate the direction of electron flow; write a balanced equation for the cell reaction; and calculate \(E_{\mathrm{cell}}^{\circ}\). (a) \(\mathrm{Cu}(\mathrm{s})+\mathrm{Fe}^{3+}(\mathrm{aq}) \longrightarrow \mathrm{Cu}^{2+}(\mathrm{aq})+\mathrm{Fe}^{2+}(\mathrm{aq})\) (b) \(\mathrm{Pb}^{2+}(\mathrm{aq})\) is displaced from solution by \(\mathrm{Al}(\mathrm{s})\) (c) \(\mathrm{Cl}_{2}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \longrightarrow \mathrm{Cl}^{-}(\mathrm{aq})+\mathrm{O}_{2}(\mathrm{g})+\mathrm{H}^{+}(\mathrm{aq})\) (d) \(\mathrm{Zn}(\mathrm{s})+\mathrm{H}^{+}+\mathrm{NO}_{3}^{-} \longrightarrow \mathrm{Zn}^{2+}+\) \(\mathrm{H}_{2} \mathrm{O}(1)+\mathrm{NO}(\mathrm{g})\)

For the reaction \(2 \mathrm{Cu}^{+}(\mathrm{aq})+\mathrm{Sn}^{4+}(\mathrm{aq}) \longrightarrow\) \(2 \mathrm{Cu}^{2+}(\mathrm{aq})+\mathrm{Sn}^{2+}(\mathrm{aq}), E_{\mathrm{cell}}^{\circ}=-0.0050 \mathrm{V}\) (a) can a solution be prepared at \(298 \mathrm{K}\) that is \(0.500 \mathrm{M}\) in each of the four ions? (b) If not, in which direction will a reaction occur?

Use the Nernst equation and data from Appendix D to calculate \(E_{\text {rell for each of the following cells. }}\) (a) \(\operatorname{Mn}(\mathrm{s}) | \mathrm{Mn}^{2+}(0.40 \mathrm{M}) \| \mathrm{Cr}^{3+}(0.35 \mathrm{M})\) \(\mathrm{Cr}^{2+}(0.25 \mathrm{M}) | \mathrm{Pt}(\mathrm{s})\) (b) \(\operatorname{Mg}\left(\text { s) } | \operatorname{Mg}^{2+}(0.016 \mathrm{M}) \|\left[\mathrm{Al}(\mathrm{OH})_{4}\right]^{-}(0.25 \mathrm{M})\right.\) \(\mathrm{OH}^{-}(0.042 \mathrm{M}) | \mathrm{Al}(\mathrm{s})\)

Write an equation to represent the oxidation of \(\mathrm{Cl}^{-}(\mathrm{aq})\) to \(\mathrm{Cl}_{2}(\mathrm{g})\) by \(\mathrm{PbO}_{2}(\mathrm{s})\) in an acidic solution. Will this reaction occur spontaneously in the forward direction if all other reactants and products are in their standard states and (a) \(\left[\mathrm{H}^{+}\right]=6.0 \mathrm{M} ;\) (b) \(\left[\mathrm{H}^{+}\right]=1.2 \mathrm{M}\) (c) \(\mathrm{pH}=4.25 ?\) Explain.

Can the displacement of \(\mathrm{Pb}(\mathrm{s})\) from \(1.0 \mathrm{M} \mathrm{Pb}\left(\mathrm{NO}_{3}\right)_{2}\) be carried to completion by tin metal? Explain.

A concentration cell is constructed of two hydrogen electrodes: one immersed in a solution with \(\left[\mathrm{H}^{+}\right]=1.0 \mathrm{M}\) and the other in \(0.65 \mathrm{M} \mathrm{KOH}\) (a) Determine \(E_{\text {cell for the reaction that occurs. }}\) (b) Compare this value of \(E_{\text {cell }}\) with \(E^{\circ}\) for the reduction of \(\mathrm{H}_{2} \mathrm{O}\) to \(\mathrm{H}_{2}(\mathrm{g})\) in basic solution, and explain the relationship between them.

A voltaic cell, with \(E_{\text {cell }}=0.180 \mathrm{V},\) is constructed as follows: $$\mathrm{Ag}(\mathrm{s})\left|\mathrm{Ag}^{+}\left(\operatorname{satd} \mathrm{Ag}_{3} \mathrm{PO}_{4}\right) \| \mathrm{Ag}^{+}(0.140 \mathrm{M})\right| \mathrm{Ag}(\mathrm{s})$$ What is the \(K_{\mathrm{sp}}\) of \(\mathrm{Ag}_{3} \mathrm{PO}_{4} ?\)

For the voltaic cell, $$\begin{array}{l} \mathrm{Ag}(\mathrm{s}) | \mathrm{Ag}^{+}(0.015 \mathrm{M}) \| \mathrm{Fe}^{3+}(0.055 \mathrm{M}) \\ \quad \mathrm{Fe}^{2+}(0.045 \mathrm{M}) | \mathrm{Pt}(\mathrm{s}) \end{array}$$ (a) what is \(E_{\text {cell initially? }}\) (b) As the cell operates, will \(E_{\text {cell increase }}\) decrease, or remain constant with time? Explain. (c) What will be \(E_{\text {cell }}\) when \(\left[\mathrm{Ag}^{+}\right]\) has increased to \(0.020 \mathrm{M} ?\) (d) What will be \(\left[\mathrm{Ag}^{+}\right]\) when \(E_{\text {cell }}=0.010 \mathrm{V} ?\) (e) What are the ion concentrations when \(E_{\text {cell }}=0 ?\)

Derive a balanced equation for the reaction occurring in the cell: $$\mathrm{Fe}(\mathrm{s})\left|\mathrm{Fe}^{2+}(\mathrm{aq}) \| \mathrm{Fe}^{3+}(\mathrm{aq}), \mathrm{Fe}^{2+}(\mathrm{aq})\right| \mathrm{Pt}(\mathrm{s})$$ (a) If \(E_{\text {cell }}^{\circ}=1.21 \mathrm{V},\) calculate \(\Delta G^{\circ}\) and the equilibrium constant for the reaction. (b) Use the Nernst equation to determine the potential for the cell: $$\begin{array}{r} \mathrm{Fe}(\mathrm{s}) | \mathrm{Fe}^{2+}\left(\mathrm{aq}, 1.0 \times 10^{-3} \mathrm{M}\right) \| \mathrm{Fe}^{3+}\left(\mathrm{aq}, 1.0 \times 10^{-3} \mathrm{M}\right) \\ \mathrm{Fe}^{2+}(\mathrm{aq}, 0.10 \mathrm{M}) | \mathrm{Pt}(\mathrm{s}) \end{array}$$ (c) In light of (a) and (b), what is the likelihood of being able to observe the disproportionation of \(\mathrm{Fe}^{2+}\) into \(\mathrm{Fe}^{3+}\) and Fe under standard conditions?

Describe how you might construct batteries with each of the following voltages: (a) \(0.10 \mathrm{V} ;\) (b) \(2.5 \mathrm{V} ;\) (c) \(10.0 \mathrm{V}\). Be as specific as you can about the electrodes and solution concentrations you would use, and indicate whether the battery would consist of a single cell or two or more cells connected in series.

A lithium battery, which is different from a lithiumion battery, uses lithium metal as one electrode and carbon in contact with \(\mathrm{MnO}_{2}\) in a paste of \(\mathrm{KOH}\) as the other electrode. The electrolyte is lithium perchlorate in a nonaqueous solvent, and the construction is similar to the silver battery. The half-cell reactions involve the oxidation of lithium and the reaction $$\begin{aligned}\mathrm{MnO}_{2}(\mathrm{s})+2 \mathrm{H}_{2} \mathrm{O}(1)+\mathrm{e}^{-} \longrightarrow \mathrm{Mn}(\mathrm{OH})_{3}(\mathrm{s})+& \\\\\mathrm{OH}^{-}(\mathrm{aq}) &E^{\circ}=-0.20 \mathrm{V}\end{aligned}$$ Draw a cell diagram for the lithium battery, identify the negative and positive electrodes, and estimate the cell potential under standard conditions.

Natural gas transmission pipes are sometimes protected against corrosion by the maintenance of a small potential difference between the pipe and an inert electrode buried in the ground. Describe how the method works.

In the construction of the Statue of Liberty, a framework of iron ribs was covered with thin sheets of copper less than \(2.5 \mathrm{mm}\) thick. A layer of asbestos separated the copper skin and iron framework. Over time, the asbestos wore away and the iron ribs corroded. Some of the ribs lost more than half their mass in the 100 years before the statue was restored. At the same time, the copper skin lost only about \(4 \%\) of its thickness. Use electrochemical principles to explain these observations.

A quantity of electric charge brings about the deposition of \(3.28 \mathrm{g}\) Cu at a cathode during the electrolysis of a solution containing \(\mathrm{Cu}^{2+}(\text { aq })\). What volume of \(\mathrm{H}_{2}(\mathrm{g}),\) measured at \(28.2^{\circ} \mathrm{C}\) and \(763 \mathrm{mm} \mathrm{Hg},\) would be produced by this same quantity of electric charge in the reduction of \(\mathrm{H}^{+}(\) aq) at a cathode?

Which of the following reactions occur spontaneously, and which can be brought about only through electrolysis, assuming that all reactants and products are in their standard states? For those requiring electrolysis, what is the minimum voltage required? (a) \(2 \mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \longrightarrow 2 \mathrm{H}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g})\left[\text { in } 1 \mathrm{M} \mathrm{H}^{+}(\mathrm{aq})\right]\) (b) \(\mathrm{Zn}(\mathrm{s})+\mathrm{Fe}^{2+}(\mathrm{aq}) \longrightarrow \mathrm{Zn}^{2+}(\mathrm{aq})+\mathrm{Fe}(\mathrm{s})\) (c) \(2 \mathrm{Fe}^{2+}(\mathrm{aq})+\mathrm{I}_{2}(\mathrm{s}) \longrightarrow 2 \mathrm{Fe}^{3+}(\mathrm{aq})+2 \mathrm{I}^{-}(\mathrm{aq})\) (d) \(\mathrm{Cu}(\mathrm{s})+\mathrm{Sn}^{4+}(\mathrm{aq}) \longrightarrow \mathrm{Cu}^{2+}(\mathrm{aq})+\mathrm{Sn}^{2+}(\mathrm{aq})\)

An aqueous solution of \(\mathrm{K}_{2} \mathrm{SO}_{4}\) is electrolyzed by means of Pt electrodes. (a) Which of the following gases should form at the anode: \(\mathrm{O}_{2}, \mathrm{H}_{2}, \mathrm{SO}_{2}, \mathrm{SO}_{3} ?\) Explain. (b) What product should form at the cathode? Explain. (c) What is the minimum voltage required? Why is the actual voltage needed likely to be higher than this value?

If a lead storage battery is charged at too high a voltage, gases are produced at each electrode. (It is possible to recharge a lead-storage battery only because of the high overpotential for gas formation on the electrodes.) (a) What are these gases? (b) Write a cell reaction to describe their formation.

Calculate the quantity indicated for each of the following electrolyses. (a) the mass of \(\mathrm{Zn}\) deposited at the cathode in 42.5 min when 1.87 A of current is passed through an aqueous solution of \(\mathrm{Zn}^{2+}\) (b) the time required to produce \(2.79 \mathrm{g} \mathrm{I}_{2}\) at the anode if a current of \(1.75 \mathrm{A}\) is passed through \(\mathrm{KI}(\mathrm{aq})\)

Calculate the quantity indicated for each of the following electrolyses. (a) \(\left[\mathrm{Cu}^{2+}\right]\) remaining in \(425 \mathrm{mL}\) of a solution that was originally \(0.366 \mathrm{M} \mathrm{CuSO}_{4},\) after passage of \(2.68 \mathrm{A}\) for 282 s and the deposition of Cu at the cathode (b) the time required to reduce \(\left[\mathrm{Ag}^{+}\right]\) in \(255 \mathrm{mL}\) of \(\mathrm{AgNO}_{3}(\mathrm{aq})\) from 0.196 to \(0.175 \mathrm{M}\) by electrolyzing the solution between \(\mathrm{Pt}\) electrodes with a current of \(1.84 \mathrm{A}\)

A solution containing both \(\mathrm{Ag}^{+}\) and \(\mathrm{Cu}^{2+}\) ions is subjected to electrolysis. (a) Which metal should plate out first? (b) Plating out is finished after a current of \(0.75 \mathrm{A}\) is passed through the solution for 2.50 hours. If the total mass of metal is \(3.50 \mathrm{g},\) what is the mass percent of silver in the product?

Describe a laboratory experiment that you could perform to evaluate the Faraday constant, \(F,\) and then show how you could use this value to determine the Avogadro constant.

It is sometimes possible to separate two metal ions through electrolysis. One ion is reduced to the free metal at the cathode, and the other remains in solution. In which of these cases would you expect complete or nearly complete separation: (a) \(\mathrm{Cu}^{2+}\) and \(\mathrm{K}^{+} ;\) (b) \(\mathrm{Cu}^{2+}\) and \(\mathrm{Ag}^{+} ;\) (c) \(\mathrm{Pb}^{2+}\) and \(\mathrm{Sn}^{2+} ?\) Explain.

You prepare \(1.00 \mathrm{L}\) of a buffer solution that is \(1.00 \mathrm{M}\) \(\mathrm{NaH}_{2} \mathrm{PO}_{4}\) and \(1.00 \mathrm{M} \mathrm{Na}_{2} \mathrm{HPO}_{4} .\) The solution is divided in half between the two compartments of an electrolytic cell. Both electrodes used are Pt. Assume that the only electrolysis is that of water. If 1.25 A of current is passed for 212 min, what will be the \(\mathrm{pH}\) in each cell compartment at the end of the electrolysis?

A common reference electrode consists of a silver wire coated with \(\mathrm{AgCl}(\mathrm{s})\) and immersed in \(1 \mathrm{M} \mathrm{KCl}\) $$\mathrm{AgCl}(\mathrm{s})+\mathrm{e}^{-} \longrightarrow \mathrm{Ag}(\mathrm{s})+\mathrm{Cl}^{-}(1 \mathrm{M}) E^{\circ}=0.2223 \mathrm{V}$$ (a) What is \(E_{\text {cell }}^{\circ}\) when this electrode is a cathode in combination with a standard zinc electrode as an anode? (b) Cite several reasons why this electrode should be easier to use than a standard hydrogen electrode. (c) By comparing the potential of this silver-silver chloride electrode with that of the silver-silver ion electrode, determine \(K_{\mathrm{sp}}\) for \(\mathrm{AgCl}\).

A test for completeness of electrodeposition of \(\mathrm{Cu}\) from a solution of \(\mathrm{Cu}^{2+}(\mathrm{aq})\) is to add \(\mathrm{NH}_{3}(\mathrm{aq}) .\) A blue color signifies the formation of the complex ion \(\left[\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}\right]^{2+}\left(K_{\mathrm{f}}=1.1 \times 10^{13}\right) .\) Let \(250.0 \mathrm{mL}\) of \(0.1000 \mathrm{M} \mathrm{CuSO}_{4}(\text { aq })\) be electrolyzed with a \(3.512 \mathrm{A}\) current for 1368 s. At this time, add a sufficient quantity of \(\mathrm{NH}_{3}(\text { aq })\) to complex any remaining \(\mathrm{Cu}^{2+}\) and to maintain a free \(\left[\mathrm{NH}_{3}\right]=0.10 \mathrm{M} .\) If \(\left[\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}\right]^{2+}\) is detectable at concentrations as low as \(1 \times 10^{-5} \mathrm{M}\) should the blue color appear?

The electrolysis of \(\mathrm{Na}_{2} \mathrm{SO}_{4}(\mathrm{aq})\) is conducted in two separate half-cells joined by a salt bridge, as suggested by the cell diagram \(\mathrm{Pt}\left|\mathrm{Na}_{2} \mathrm{SO}_{4}(\mathrm{aq})\right|\left|\mathrm{Na}_{2} \mathrm{SO}_{4}(\mathrm{aq})\right| \mathrm{Pt}\) (a) In one experiment, the solution in the anode compartment becomes more acidic and that in the cathode compartment, more basic during the electrolysis. When the electrolysis is discontinued and the two solutions are mixed, the resulting solution has \(\mathrm{pH}=7\). Write half-equations and the overall electrolysis equation. (b) In a second experiment, a 10.00 -mL sample of an unknown concentration of \(\mathrm{H}_{2} \mathrm{SO}_{4}(\mathrm{aq})\) and a few drops of phenolphthalein indicator are added to the \(\mathrm{Na}_{2} \mathrm{SO}_{4}(\mathrm{aq})\) in the cathode compartment. Electrolysis is carried out with a current of \(21.5 \mathrm{mA}\) (milliamperes) for 683 s, at which point, the solution in the cathode compartment acquires a lasting pink color. What is the molarity of the unknown \(\mathrm{H}_{2} \mathrm{SO}_{4}(\mathrm{aq}) ?\)

Silver tarnish is mainly \(\mathrm{Ag}_{2} \mathrm{S}\) : $$\begin{array}{r}\mathrm{Ag}_{2} \mathrm{S}(\mathrm{s})+2 \mathrm{e}^{-} \longrightarrow 2 \mathrm{Ag}(\mathrm{s})+\mathrm{S}^{2-}(\mathrm{aq}) \\\E^{\circ}=-0.691 \mathrm{V}\end{array}$$ A tarnished silver spoon is placed in contact with a commercially available metallic product in a glass baking dish. Boiling water, to which some \(\mathrm{NaHCO}_{3}\) has been added, is poured into the dish, and the product and spoon are completely covered. Within a short time, the removal of tarnish from the spoon begins. (a) What metal or metals are in the product? (b) What is the probable reaction that occurs? (c) What do you suppose is the function of the \(\mathrm{NaHCO}_{3} ?\) (d) An advertisement for the product appears to make two claims: (1) No chemicals are involved, and (2) the product will never need to be replaced. How valid are these claims? Explain.

Your task is to determine \(E^{\circ}\) for the reduction of \(\mathrm{CO}_{2}(\mathrm{g})\) to \(\mathrm{C}_{3} \mathrm{H}_{8}(\mathrm{g})\) in two different ways and to explain why each gives the same result. (a) Consider a fuel cell in which the cell reaction corresponds to the complete combustion of propane gas. Write the half-cell reactions and the overall reaction. Determine \(\Delta G^{\circ}\) and \(E_{\text {cell }}^{\circ}\) for the reaction, then obtain \(E_{\mathrm{CO}_{2} / \mathrm{C}_{3} \mathrm{H}_{8}^{*}}^{\circ}\) (b) Without considering the oxidation that occurs simultaneously, obtain \(E_{\mathrm{CO}_{2} / \mathrm{C}_{3} \mathrm{H}_{8}}^{\circ}\) directly from tabulated thermodynamic data for the reduction half-reaction.

Show that for nonstandard conditions the temperature variation of a cell potential is $$E\left(T_{1}\right)-E\left(T_{2}\right)=\left(T_{1}-T_{2}\right) \frac{\left(\Delta S^{\circ}-R \ln Q\right)}{z F}$$ where \(E\left(T_{1}\right)\) and \(E\left(T_{2}\right)\) are the cell potentials at \(T_{1}\) and \(T_{2},\) respectively. We have assumed that the value of \(Q\) is maintained at a constant value. For the nonstandard cell below, the potential drops from \(0.394 \mathrm{V}\) at \(50.0^{\circ} \mathrm{C}\) to \(0.370 \mathrm{V}\) at \(25.0^{\circ} \mathrm{C} .\) Calculate \(Q\) \(\Delta H^{\circ},\) and \(\Delta S^{\circ}\) for the reaction, and calculate \(K\) for the two temperatures. $$\mathrm{Cu}(\mathrm{s})\left|\mathrm{Cu}^{2+}(\mathrm{aq}) \| \mathrm{Fe}^{3+}(\mathrm{aq}), \mathrm{Fe}^{2+}(\mathrm{aq})\right| \mathrm{Pt}(\mathrm{s})$$ Choose concentrations of the species involved in the cell reaction that give the value of \(Q\) that you have calculated, and then determine the equilibrium concentrations of the species at \(50.0^{\circ} \mathrm{C}\)

Show that for a combination of half-cell reactions that produce a standard reduction potential for a half-cell that is not directly observable, the standard reduction potential is $$E^{\circ}=\frac{\sum n_{i} E_{i}^{\circ}}{\sum n_{i}}$$ where \(n_{i}\) is the number of electrons in each half-reaction of potential \(E_{i}^{\circ} .\) Use the following half-reactions: $$ \begin{array}{c} \mathrm{H}_{5} \mathrm{IO}_{6}(\mathrm{aq})+\mathrm{H}^{+}(\mathrm{aq})+2 \mathrm{e}^{-} \longrightarrow \mathrm{IO}_{3}^{-}(\mathrm{aq})+ \\ 3 \mathrm{H}_{2} \mathrm{O}(1) \quad E^{\circ}=1.60 \mathrm{V} \\ \mathrm{IO}_{3}^{-}(\mathrm{aq})+6 \mathrm{H}^{+}(\mathrm{aq})+5 \mathrm{e}^{-} \longrightarrow \frac{1}{2} \mathrm{I}_{2}(\mathrm{s})+3 \mathrm{H}_{2} \mathrm{O}(1) \\ E^{\circ}=1.19 \mathrm{V} \\ 2 \mathrm{HIO}(\mathrm{aq})+2 \mathrm{H}^{+}(\mathrm{aq})+2 \mathrm{e}^{-} \longrightarrow \mathrm{I}_{2}(\mathrm{s})+2 \mathrm{H}_{2} \mathrm{O}(1) \\ E^{\circ}=1.45 \mathrm{V} \\ \mathrm{I}_{2}(\mathrm{s})+2 \mathrm{e}^{-} \longrightarrow 2 \mathrm{I}^{-}(\mathrm{aq}) \quad \quad E^{\circ}=0.535 \mathrm{V} \end{array} $$ Calculate the standard reduction potential for $$ \mathrm{H}_{6} \mathrm{IO}_{6}+5 \mathrm{H}^{+}+2 \mathrm{I}^{-}+3 \mathrm{e}^{-} \longrightarrow $$ $$ \frac{1}{2} \mathrm{I}_{2}+4 \mathrm{H}_{2} \mathrm{O}=2 \mathrm{HIO} $$

Consider the following electrochemical cell: $$ \operatorname{Pt}(\mathrm{s})\left|\mathrm{H}_{2}(\mathrm{g}, 1 \mathrm{atm})\right| \mathrm{H}^{+}(1 \mathrm{M}) \| \mathrm{Ag}^{+}(x \mathrm{M}) | \mathrm{Ag}(\mathrm{s}) $$ (a) What is \(E_{\text {cell }}^{\circ}-\) that is, the cell potential when \(\left[\mathrm{Ag}^{+}\right]=1 \mathrm{M} ?\) (b) Use the Nernst equation to write an equation for \(E_{\text {cell }}\) when \(\left[\mathrm{Ag}^{+}\right]=x\) (c) Now imagine titrating \(50.0 \mathrm{mL}\) of \(0.0100 \mathrm{M}\) \(\mathrm{AgNO}_{3}\) in the cathode half-cell compartment with 0.0100 M KI. The titration reaction is $$\mathrm{Ag}^{+}(\mathrm{aq})+\mathrm{I}^{-}(\mathrm{aq}) \longrightarrow \mathrm{AgI}(\mathrm{s})$$ Calculate \(\left[\mathrm{Ag}^{+}\right]\) and then \(E_{\text {cell }}\) after addition of the following volumes of \(0.0100 \mathrm{M} \mathrm{KI}:(\mathrm{i}) 0.0 \mathrm{mL} ;(\mathrm{ii}) 20.0 \mathrm{mL}\) (iii) \(49.0 \mathrm{mL} ;(\text { iv }) 50.0 \mathrm{mL} ;(\mathrm{v}) 51.0 \mathrm{mL} ;(\mathrm{vi}) 60.0 \mathrm{mL}\) (d) Use the results of part (c) to sketch the titration curve of \(E_{\text {cell }}\) versus volume of titrant.

Ultimately, \(\Delta G_{\mathrm{f}}^{\mathrm{Q}}\) values must be based on experimental results; in many cases, these experimental results are themselves obtained from \(E^{\circ}\) values. Early in the twentieth century, G. N. Lewis conceived of an experimental approach for obtaining standard potentials of the alkali metals. This approach involved using a solvent with which the alkali metals do not react. Ethylamine was the solvent chosen. In the following cell diagram, \(\mathrm{Na}(\text { amalg, } 0.206 \%)\) represents a solution of \(0.206 \%\) Na in liquid mercury. 1\. \(\mathrm{Na}(\mathrm{s}) | \mathrm{Na}^{+}(\text {in ethylamine }) | \mathrm{Na}(\text { amalg }, 0.206 \%)\) \(E_{\text {cell }}=0.8453 \mathrm{V}\) Although Na(s) reacts violently with water to produce \(\mathrm{H}_{2}(\mathrm{g}),\) at least for a short time, a sodium amalgam electrode does not react with water. This makes it possible to determine \(E_{\text {cell }}\) for the following voltaic cell. 2\. \(\mathrm{Na}(\text { amalg }, 0.206 \%)\left|\mathrm{Na}^{+}(1 \mathrm{M}) \| \mathrm{H}^{+}(1 \mathrm{M})\right|\) $$\mathrm{H}_{2}(\mathrm{g}, 1 \mathrm{atm}) \quad E_{\mathrm{cell}}=1.8673 \mathrm{V}$$ (a) Write equations for the cell reactions that occur in the voltaic cells (1) and (2) (b) Use equation (20.14) to establish \(\Delta G\) for the cell reactions written in part (a). (c) Write the overall equation obtained by combining the equations of part (a), and establish \(\Delta G^{\circ}\) for this overall reaction. (d) Use the \(\Delta G^{\circ}\) value from part (c) to obtain \(E_{\text {cell }}^{\circ}\) for the overall reaction. From this result, obtain \(E_{\mathrm{Na}^{+}}^{\circ} / \mathrm{Na}\) Compare your result with the value listed in Appendix D.

Only a tiny fraction of the diffusible ions move across a cell membrane in establishing a Nernst potential (see Focus On 20: Membrane Potentials), so there is no detectable concentration change. Consider a typical cell with a volume of \(10^{-8} \mathrm{cm}^{3},\) a surface area \((A)\) of \(10^{-6} \mathrm{cm}^{2},\) and a membrane thickness \((l)\) of \(10^{-6} \mathrm{cm}\) Suppose that \(\left[\mathrm{K}^{+}\right]=155 \mathrm{mM}\) inside the cell and \(\left[\mathrm{K}^{+}\right]=4 \mathrm{mM}\) outside the cell and that the observed Nernst potential across the cell wall is \(0.085 \mathrm{V}\). The membrane acts as a charge-storing device called a capacitor, with a capacitance, \(C,\) given by $$C=\frac{\varepsilon_{0} \varepsilon A}{l}$$ where \(\varepsilon_{0}\) is the dielectric constant of a vacuum and the product \(\varepsilon_{0} \varepsilon\) is the dielectric constant of the membrane, having a typical value of \(3 \times 8.854 \times 10^{-12}\) \(\mathrm{C}^{2} \mathrm{N}^{-1} \mathrm{m}^{-2}\) for a biological membrane. The SI unit of capacitance is the firad, \(1 \mathrm{F}=1\) coulomb per volt \(=1 \mathrm{CV}^{-1}=1 \times \mathrm{C}^{2} \mathrm{N}^{-1} \mathrm{m}^{-1}\) (a) Determine the capacitance of the membrane for the typical cell described. (b) What is the net charge required to maintain the observed membrane potential? (c) How many \(\mathrm{K}^{+}\) ions must flow through the cell membrane to produce the membrane potential? (d) How many \(\mathrm{K}^{+}\) ions are in the typical cell? (e) Show that the fraction of the intracellular \(K^{+}\) ions transferred through the cell membrane to produce the membrane potential is so small that it does not change \(\left[\mathrm{K}^{+}\right]\) within the cell.

When deciding whether a particular reaction corresponds to a cell with a positive standard cell potential, which of the following thermodynamic properties would you use to get your answer without performing any calculations? Which would you not use? Explain. (a) \(\Delta G^{\circ} ;\) (b) \(\Delta S^{\circ} ;\) (c) \(\Delta H^{\circ} ;\) (d) \(\Delta U^{\circ} ;\) (e) \(K\).

Consider two cells involving two metals \(X\) and \(Y\) $$\begin{aligned} \mathrm{X}(\mathrm{s})\left|\mathrm{X}^{+}(\mathrm{aq})\right|\left|\mathrm{H}^{+}(\mathrm{aq}), \mathrm{H}_{2}(\mathrm{g}, 1 \mathrm{bar})\right| \mathrm{Pt}(\mathrm{s}) & \\\ \mathrm{X}(\mathrm{s})\left|\mathrm{X}^{+}(\mathrm{aq}) \| \mathrm{Y}^{2+}(\mathrm{aq})\right| \mathrm{Y}(\mathrm{s}) \end{aligned}$$ In the first cell electrons flow from the metal \(X\) to the standard hydrogen electrode. In the second cell electrons flow from metal \(X\) to metal Y. Is \(E_{x^{+} / x}^{\circ_{+}}\) greater orless than zero? Is \(E_{x^{+} / x}^{\circ}>E_{\mathrm{Y}^{2+}},_{\mathrm{Y}} ?\) Explain.