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Chapter 18: Problem 140

Calculate \(\mathscr{E}^{\circ}\) and \(\Delta G^{\circ}\) for the reaction $$2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow 2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g)$$ at 298 \(\mathrm{K}\) . Using thermodynamic data in Appendix \(4,\) estimate \(\mathscr{E}^{\circ}\) and \(\Delta G^{\circ}\) at \(0^{\circ} \mathrm{C}\) and $90 .^{\circ} \mathrm{C}\( . Assume \)\Delta H^{\circ}\( and \)\Delta S^{\circ}$ do not depend on temperature.

Short Answer

Expert verified
At 298K, the standard cell potential, \(\mathscr{E}^{\circ}\), is -1.23 V, and the standard Gibbs free energy change, \(\Delta G^{\circ}\), is 2.37 x 10^5 J/mol. With help of the estimated \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) values, we can calculate the values of \(\mathscr{E}^{\circ}\) and \(\Delta G^{\circ}\) at 0°C (273 K) and 90°C (363 K) using the relations \(\Delta G^{\circ}=\Delta H^{\circ}-T\Delta S^{\circ}\) and \(\mathscr{E}^{\circ} = -\dfrac{\Delta G^{\circ}}{nF}\).

Step by step solution

01

Calculate the number of moles of electrons transferred during the reaction

First, we need to determine the number of moles of electrons transferred during the reaction between water and oxygen. In the decomposition of water, two moles of electrons are transferred to form one mole of both hydrogen and oxygen gas. Therefore, n = 2 moles of electrons.
02

Calculate the standard cell potential, \(\mathscr{E}^{\circ}\) at 298 K

Using Redox reaction data from Appendix 4, we can calculate \(\mathscr{E}^{\circ}\) at 298 K: For the half-reactions: \(\mathrm{O}_{2}(g) + 4\mathrm{H}^{+}(aq) + 4e^{-} \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l)\), \(\mathscr{E}^{\circ}_1 = 1.23\,\text{V}\) Since the overall reaction occurs in reverse, the standard cell potential for the reaction at 298 K will be the negative of the value obtained above (\(\mathscr{E}^{\circ}_1\)): \(\mathscr{E}^{\circ} = -\mathscr{E}^{\circ}_1 = -1.23\,\text{V}\)
03

Calculate the standard Gibbs free energy change, \(\Delta G^{\circ}\), at 298 K

Now, we can determine the standard Gibbs free energy change, \(\Delta G^{\circ}\), using the relation: \(\Delta G^{\circ}=-nF\mathscr{E}^{\circ}\) where F is the Faraday constant, equal to \(9.6485 \times 10^4\text{C/mol}\), and n is the number of moles of electrons transferred during the reaction from step 1. \(\Delta G^{\circ}=-2 \times 9.6485 \times 10^4 \text{C/mol} \times (-1.23\,\text{V}) = 2.37 \times 10^5\,\text{J/mol}\)
04

Estimate the \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) values using Appendix 4

According to the thermodynamic data in Appendix 4: \(\Delta H^{\circ} = 2 \Delta H_{\mathrm{H}_{2}(g)} + \Delta H_{\mathrm{O}_{2}(g)}- 2 \Delta H_{\mathrm{H}_{2} \mathrm{O}(l)}\) \(\Delta S^{\circ} = 2 \Delta S_{\mathrm{H}_{2}(g)} + \Delta S_{\mathrm{O}_{2}(g)} - 2 \Delta S_{\mathrm{H}_{2} \mathrm{O}(l)}\)
05

Estimate \(\mathscr{E}^{\circ}\) and \(\Delta G^{\circ}\) at 0°C and 90°C

The temperature in Celsius must be converted to Kelvin: 0°C = 273 K, 90°C = 363 K. Using the approximation formula at the given temperatures: \(\Delta G^{\circ}=\Delta H^{\circ}-T\Delta S^{\circ}\) We can now calculate \(\Delta G^{\circ}\) at the different temperatures, and finally use the relation: \(\mathscr{E}^{\circ} = -\dfrac{\Delta G^{\circ}}{nF}\) to calculate \(\mathscr{E}^{\circ}\) at 0°C (273 K) and 90°C (363 K).
06

Summary

By calculating the standard cell potential and Gibbs free energy change at 298 K and using the given thermodynamic data, we have estimated the values of \(\mathscr{E}^{\circ}\) and \(\Delta G^{\circ}\) at 0°C and 90°C for the decomposition of water into hydrogen and oxygen gas.

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Most popular questions from this chapter

Consider the following half-reactions: $$\begin{array}{ll}{\mathrm{IrCl}_{6}^{3-}+3 \mathrm{e}^{-} \longrightarrow \mathrm{Ir}+6 \mathrm{Cl}^{-}} & {\mathscr{E}^{\circ}=0.77 \mathrm{V}} \\\ {\mathrm{PtCl}_{4}^{2-}+2 \mathrm{e}^{-} \longrightarrow \mathrm{Pt}+4 \mathrm{Cl}^{-}} & {\mathscr{E}^{\circ}=0.73 \mathrm{V}} \\\ {\mathrm{PdCl}_{4}^{2-}+2 \mathrm{e}^{-} \longrightarrow \mathrm{Pd}+4 \mathrm{Cl}^{-}} & {\mathscr{E}^{\circ}=0.62 \mathrm{V}}\end{array}$$ A hydrochloric acid solution contains platinum, palladium, and iridium as chloro-complex ions. The solution is a constant 1.0\(M\) in chloride ion and 0.020\(M\) in each complex ion. Is it feasible to separate the three metals from this solution by electrolysis? (Assume that 99\(\%\) of a metal must be plated out before another metal begins to plate out.)

A galvanic cell consists of a standard hydrogen electrode and a copper electrode immersed in a Cu(NO \(_{3} )_{2}(a q)\) solution. If you wish to construct a calibration curve to show how the cell potential varies with \(\left[\mathrm{Cu}^{2+}\right],\) what should you plot to obtain a straight line? What will be the slope of this line?

Balance the following oxidation–reduction reactions that occur in basic solution using the half-reaction method. a. $\mathrm{PO}_{3}^{3-}(a q)+\mathrm{MnO}_{4}^{-}(a q) \rightarrow \mathrm{PO}_{4}^{3-}(a q)+\mathrm{MnO}_{2}(s)$ b. $\operatorname{Mg}(s)+\mathrm{OCl}^{-}(a q) \rightarrow \mathrm{Mg}(\mathrm{OH})_{2}(s)+\mathrm{Cl}^{-}(a q)$ c. $\mathrm{H}_{2} \mathrm{CO}(a q)+\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}+(a q) \rightarrow$ $$\mathrm{HCO}_{3}(a q)+\mathrm{Ag}(s)+\mathrm{NH}_{3}(a q)$$

Combine the equations $$\Delta G^{\circ}=-n F \mathscr{E}^{\circ} \text { and } \Delta G^{\circ}=\Delta H^{\circ}-T \Delta S^{\circ}$$ to derive an expression for \(\mathscr{E}^{\circ}\) as a function of temperature. Describe how one can graphically determine \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) from measurements of \(\mathscr{E}^{\circ}\) at different temperatures, assuming that \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) do not depend on temperature. What property would you look for in designing a reference half-cell that would produce a potential relatively stable with respect to temperature?

An electrochemical cell consists of a standard hydrogen electrode and a copper metal electrode. a. What is the potential of the cell at \(25^{\circ} \mathrm{C}\) if the copper electrode is placed in a solution in which \(\left[\mathrm{Cu}^{2+}\right]=\) \(2.5 \times 10^{-4} \mathrm{M} ?\) b. The copper electrode is placed in a solution of unknown \(\left[\mathrm{Cu}^{2+}\right] .\) The measured potential at $25^{\circ} \mathrm{C}\( is 0.195 \)\mathrm{V}\( . What is \)\left[\mathrm{Cu}^{2+}\right] ?$ (Assume \(\mathrm{Cu}^{2+}\) is reduced.)
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