91Ó°ÊÓ

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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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Anode and Cathode

Electrolytic cells have two electrodes: an anode and a cathode. In simple terms, the anode is where oxidation occurs, and the cathode is where reduction takes place. In an electrolytic cell:

• Anode is the positive electrode. Here, the electrode material loses electrons (oxidation). For the given Co and Mg setup, Mg acts as the anode with the reaction: \( Mg \rightarrow Mg^{2+} + 2e^- \).


• Cathode is the negative electrode. At this site, the ions gain electrons (reduction). Co acts as the cathode with the reaction: \( Co^{2+} + 2e^- \rightarrow Co \).

A useful mnemonic is "An Ox" (anode oxidation) and "Red Cat" (reduction cathode). The flow of electrons goes from anode to cathode, driven by an external voltage source.

Half-Cell Reactions

Half-cell reactions depict what happens at each electrode - essentially breaking down the total reaction into oxidation and reduction components. They are crucial in understanding the electron transfer process:

• Oxidation at the Anode: This represents the loss of electrons. As shown in the example, magnesium undergoes oxidation: \( Mg \rightarrow Mg^{2+} + 2e^- \).


• Reduction at the Cathode: Here, reduction involves the gain of electrons. Cobalt experiences this reaction: \( Co^{2+} + 2e^- \rightarrow Co \).

These reactions help determine the direction of electron flow and are fundamental in calculating cell potential.

Nernst Equation

The Nernst Equation is a vital tool for calculating the cell potential under non-standard conditions. It adjusts the standard electrode potentials to account for ion concentrations:

• The equation is expressed as: \( E = E^0 - \frac{RT}{zF} \ln Q \), where \( E^0 \) is the standard electrode potential, \( R \) is the gas constant, \( T \) is the temperature in Kelvin, \( z \) is the number of electrons transferred, \( F \) is Faraday’s constant, and \( Q \) is the reaction quotient.


• Standard Conditions: When all ion concentrations are 1 M, the Nernst Equation simplifies to the standard potentials.


• This equation allows for the calculation of cell potential when conditions differ from standard state, like varying concentrations which influence the reaction quotient \( Q \).

Standard Electrode Potentials

Standard electrode potentials \( E^0 \) are measures of the tendency of a chemical species to be reduced, compared to the standard hydrogen electrode. These potentials are crucial as they:

• Help predict the direction of electron flow in electrochemical cells.


• Determine the feasibility of an electrochemical reaction.


• Calculated as \( E_{cell}^0 = E_{cathode}^0 - E_{anode}^0 \). In the example, Co and Mg have potentials \(-0.28 \text{ V}\) and \(-2.37 \text{ V}\) respectively, leading to \( E_{cell}^0 = 2.09 \text{ V} \).

These aid in the understanding of how readily a reaction will occur in an electrochemical cell.