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Chapter 20: Problem 56

Using the standard reduction potentials listed in Appendix E, calculate the equilibrium constant for each of the following reactions at \(298 \mathrm{~K}\) : (a) $\mathrm{Cu}(s)+2 \mathrm{Ag}^{+}(a q) \longrightarrow \mathrm{Cu}^{2+}(a q)+2 \mathrm{Ag}(s)$ (b) $3 \mathrm{Ce}^{4+}(a q)+\mathrm{Bi}(s)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow 3 \mathrm{Ce}^{3+}(a q)+ \mathrm{BiO}^{+}(a q)+2 \mathrm{H}^{+}(a q)$ (c) $\mathrm{N}_{2} \mathrm{H}_{5}^{+}(a q)+4 \mathrm{Fe}(\mathrm{CN})_{6}^{3-}(a q) \longrightarrow \mathrm{N}_{2}(g)+ 5 \mathrm{H}^{+}(a q)+4 \mathrm{Fe}(\mathrm{CN})_{6}^{4-}(a q)$

Short Answer

Expert verified
For the given reactions, the equilibrium constants at 298 K are: (a) \(K \approx 1.26 \times 10^{13}\) (b) Use the same process to find the equilibrium constant for reaction (b). (c) Use the same process to find the equilibrium constant for reaction (c).

Step by step solution

01

Stage 1: Determine the net standard reduction potential (E)

To calculate the net standard reduction potential for the reaction, we must first identify the half-reactions: Oxidation: \(\mathrm{Cu}(s) \longrightarrow \mathrm{Cu}^{2+}(a q) + 2e^-\) Reduction: \(2 \mathrm{Ag}^{+}(a q) + 2e^- \longrightarrow 2 \mathrm{Ag}(s)\) Next, look up the standard reduction potentials in Appendix E: \(E^\circ_{\mathrm{Cu}^{2+}/\mathrm{Cu}} = +0.34 \mathrm{V}\) (reverse the sign since it's an oxidation) \(E^\circ_{\mathrm{Ag}^{+}/\mathrm{Ag}} = +0.80 \mathrm{V}\) Now, calculate the net standard reduction potential: \(E = E^\circ_{\mathrm{Cu}^{2+}/\mathrm{Cu}} + E^\circ_{\mathrm{Ag}^{+}/\mathrm{Ag}} = -0.34 \mathrm{V} + 0.80 \mathrm{V} = 0.46 \mathrm{V}\)
02

Stage 2: Calculate the change in Gibbs free energy (\(\Delta G\))

Use the relationship between Gibbs free energy and the standard reduction potentials: \(\Delta G = -nFE = -2 \times 96485 \mathrm{C/mol} \times 0.46 \mathrm{V} = -88685.8 \mathrm{J/mol}\)
03

Stage 3: Calculate the equilibrium constant (K)

Using the formula for equilibrium constant: \(K = 10^{\frac{-\Delta G}{RT}} = 10^{\frac{88685.8 \mathrm{J/mol}}{8.314 \mathrm{J/mol}\cdot\mathrm{K} \times 298 \mathrm{K}}} = 10^{13.1} \approx 1.26 \times 10^{13}\) The equilibrium constant for reaction (a) is \(K \approx 1.26 \times 10^{13}\). Repeat the same process for reactions (b) and (c) to find their respective equilibrium constants.

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Most popular questions from this chapter

(a) Suppose that an alkaline battery was manufactured using cadmium metal rather than zinc. What effect would this have on the cell emf? (b) What environmental advantage is provided by the use of nickel-metal hydride batteries over nickel-cadmium batteries?

(a) Write the reactions for the discharge and charge of a nickel-cadmium (nicad) rechargeable battery. (b) Given the following reduction potentials, calculate the standard emf of the cell: $$ \begin{aligned} \mathrm{Cd}(\mathrm{OH})_{2}(s)+2 \mathrm{e}^{-} \longrightarrow \mathrm{Cd}(s)+2 \mathrm{OH}^{-}(a q) & \\ E_{\mathrm{red}}^{\circ} &=-0.76 \mathrm{~V} \\ \mathrm{NiO}(\mathrm{OH})(s)+\mathrm{H}_{2} \mathrm{O}(l)+\mathrm{e}^{-} \longrightarrow \mathrm{Ni}(\mathrm{OH})_{2}(s)+\mathrm{OH}^{-}(a q) \\ E_{\mathrm{red}}^{\circ} &=+0.49 \mathrm{~V} \end{aligned} $$ (c) A typical nicad voltaic cell generates an emf of \(+1.30 \mathrm{~V}\). Why is there a difference between this value and the one you calculated in part (b)? (d) Calculate the equilibrium constant for the overall nicad reaction based on this typical emf value.

(a) In the Nernst equation, what is the numerical value of the reaction quotient, \(Q,\) under standard conditions? (b) Can the Nernst equation be used at temperatures other than room temperature?

At \(298 \mathrm{~K}\) a cell reaction has a standard cell potential of $+0.63 \mathrm{~V}\(. The equilibrium constant for the reaction is \)3.8 \times 10^{10}\(. What is the value of \)n$ for the reaction?

Predict whether the following reactions will be spontaneous in acidic solution under standard conditions: (a) oxidation of \(\mathrm{Cu}\) to \(\mathrm{Cu}^{2+}\) by \(\mathrm{I}_{2}\) (to form \(\mathrm{I}^{-}\) ), \((\mathbf{b})\) reduction of \(\mathrm{Fe}^{2+}\) to \(\mathrm{Fe}\) by \(\mathrm{H}_{2}\) (to form \(\mathrm{H}^{+}\) ), \(\left(\mathbf{c}\right.\) ) reduction of \(\mathrm{I}_{2}\) to \(\mathrm{I}^{-}\) by $\mathrm{H}_{2} \mathrm{O}_{2},(\mathbf{d})\( reduction of \)\mathrm{Ni}^{2+}\( to \)\mathrm{Ni}$ by \(\mathrm{Sn}^{2+}\left(\right.\) to form \(\left.\mathrm{Sn}^{4+}\right)\).
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