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Chapter 20: Problem 97

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.

Short Answer

Expert verified
Both processes, whether employing a fuel cell or using tabulated thermodynamic data, will yield identical values for the standard cell potential \(E_{\text{CO2/C3H8}}^{\circ}\), since it is a fundamental property of the substances involved in the reaction.

Step by step solution

01

Write Half-Cell Reactions and Overall Reaction

First, write the half-cell reactions and the overall reaction to represent the complete combustion process of propane in a fuel cell. Half-cell reactions: Anode: \( C_3H_8(g) + 5O_2(g) \rightarrow 3CO_2(g) + 4H_2O(l) \) Cathode: \( O_2(g) + 4H^+(aq) + 4e^- \rightarrow 2H_2O(l) \) Overall reaction: \( C_3H_8(g) + 5O_2(g) \rightarrow 3CO_2(g) + 4H_2O(l) \)
02

Determine \( \Delta G^{\circ} \) and \( E_{\text{cell }}^{\circ} \)

Use Gibbs free energy equation and Nernst equation for each half-cell reaction and overall reaction. \(\Delta G^{\circ} = -nFE_{\text{cell }}^{\circ} \)Note: n stands for moles of electrons transferred (10 in this case), F is the Faraday constant approximately equal to 96485 C/mol.
03

Calculate \( E_{\text{CO2/C3H8}}^{\circ} \)

Use the calculated \( \Delta G^{\circ} \) and the equation from step 2 to determine \( E_{\text{cell }}^{\circ} \).Then, use the definition for the electromotive force of the cell to find \( E_{\text{CO2/C3H8}}^{\circ} \). \( E_{\text{cell }}^{\circ} \) is equal to \( E_{\text{CO2/C3H8}}^{\circ} \), as it is the reduction of CO2 to C3H8.
04

Obtain \( E_{\text{CO2/C3H8}}^{\circ} \) from Thermodynamic Data

Finally, compare the above result with the \( E_{\text{CO2/C3H8}}^{\circ} \) value obtained directly from standard thermodynamic tables for the half-reaction involving the reduction of CO2 to C3H8. Both must be equal, providing the verification of our chemistry principles working correctly.

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Most popular questions from this chapter

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 \(\mathrm{KI}(\mathrm{aq})\)

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 \(\mathrm{H}_{2} \mathrm{SO}_{4}(\mathrm{aq}) ?\)

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.

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.

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.
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