The diagram here shows an electrolytic cell consisting of a Co electrode in a \(2.0 \mathrm{M} \mathrm{Co}\left(\mathrm{NO}_{3}\right)_{2}\) solution and a Mg electrode in a \(2.0 \mathrm{M} \mathrm{Mg}\left(\mathrm{NO}_{3}\right)_{2}\) solution. (a) Label the anode and cathode and show the halfcell reactions. Also label the signs \((+\) or \(-)\) on the battery terminals. (b) What is the minimum voltage to drive the reaction? (c) After the passage of \(10.0 \mathrm{~A}\) for \(2.00 \mathrm{~h}\) the battery is replaced with a voltmeter and the electrolytic cell now becomes a galvanic cell. Calculate \(E_{\text {cell. }}\) Assume volumes to remain constant at \(1.00 \mathrm{~L}\) in each compartment.

Step by step solution

01

Anode/Cathode and Half-Cell Reactions

Firstly, identify the cations, which are \( Mg^{2+} \) and \( Co^{2+} \). The electrode in contact with \( Mg^{2+} \) will get oxidized (lose electrons) and thus will act as anode. The anode half-cell reaction would be \( Mg \rightarrow Mg^{2+} + 2e^- \). Conversely, \( Co^{2+} \) ions will be reduced (gain electrons) at cathode. The cathode half-cell reaction would be \( Co^{2+} + 2e^- \rightarrow Co \).

02

Minimum Voltage for Reaction

To determine the minimum voltage, find the standard electrode potentials. For \( Co^{2+}/Co \), the standard potential \( E^0 \) is -0.28V and for \( Mg^{2+}/Mg \), \( E^0 \) is -2.37V. The cell potential or minimum voltage, \( E_{cell} \), can be calculated using the Nernst Equation, \( E_{cell} = E_{cathode}^0 - E_{anode}^0 = -0.28 - (-2.37) = 2.09V.

03

Galvanic Cell Potential

To calculate the new cell potential for a galvanic cell, first calculate the number of moles of \( Mg^{2+} \) and \( Co^{2+} \). Initialize them to 2 moles each (2.0M * 1 L). The current of 10A for 2 hours would oxidize 2.41 moles of \( Mg^{2+} \) and reduce equivalent \( Co^{2+} \) (using Faraday's law). Accordingly, adjust the amounts and apply the Nernst Equation, \( E_{cell} = E_{cell}^0 - \frac{RT}{zF} \ln \frac{Q}, where the reduced form of reaction quotient \( Q = [Co^{2+}]/[Mg^{2+}] = (2 + 2.41)/(2 - 2.41) \).

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