Question

Use the Nernst equation and data from Appendix D to calculate \(E_{\text {rell for each of the following cells. }}\) (a) \(\operatorname{Mn}(\mathrm{s}) | \mathrm{Mn}^{2+}(0.40 \mathrm{M}) \| \mathrm{Cr}^{3+}(0.35 \mathrm{M})\) \(\mathrm{Cr}^{2+}(0.25 \mathrm{M}) | \mathrm{Pt}(\mathrm{s})\) (b) \(\operatorname{Mg}\left(\text { s) } | \operatorname{Mg}^{2+}(0.016 \mathrm{M}) \|\left[\mathrm{Al}(\mathrm{OH})_{4}\right]^{-}(0.25 \mathrm{M})\right.\) \(\mathrm{OH}^{-}(0.042 \mathrm{M}) | \mathrm{Al}(\mathrm{s})\)

Short answer

The final answer will be given by substituting the values calculated in steps 2, 3 and 4 into the Nernst Equation.

Step-by-step solution

Determine the half reactions for each cell

The half reactions for the given two cells are: The standard electrode potentials (E0) for the reactions can be obtained from Appendix D.

Calculate the number of moles of electrons(n) transferred in each reaction

The half reactions allow for the determination of the number of moles of electrons that are transferred during each half reaction. This piece of information is required as it is one of the variables present in the Nernst equation.

Calculate the reaction quotient (Q) for each cell

The reaction quotient (Q) is a measure of the ratio of the concentrations of the products to the reactants. The values can be calculated using the given concentrations for the species in each cell.

Calculate the cell potential (E) for each cell

Now, we have to substitute the values of the standard electrode potential, number of moles, reaction quotient, Faraday's constant and temperature into the Nernst equation and calculate the cell potential for each cell.

Key concepts

Electrochemical Cells

Electrochemical cells are fascinating systems where chemical energy is converted into electrical energy or vice versa. They consist of two electrodes, each in contact with an electrolyte, which may be a liquid or a solid. These cells are categorized mainly into two types: galvanic (or voltaic) cells, which are spontaneous, and electrolytic cells, which are non-spontaneous. The most common example of a galvanic cell is a simple battery that powers many of the devices we use today.

The fundamental rule governing electrochemical cells is that oxidation reactions occur at the anode, whereas reduction reactions take place at the cathode. Electrons flow from the anode to the cathode through an external circuit. This flow of electrons is what constitutes an electric current. The two electrodes in an electrochemical cell are immersed in solutions of ions that participate in the redox reactions, enabling the transfer of electrons.

• The anode is where oxidation occurs, and it releases electrons into the external circuit.
• The cathode is where reduction happens, and it accepts electrons from the external circuit.

In the context of this exercise, the Nernst equation is crucial for calculating the cell potential under non-standard conditions by factoring in the concentration of ions at each electrode.

Standard Electrode Potential

The standard electrode potential, denoted as E°, is a measure of the individual potential of a reversible electrode at standard state, which is 1 M concentration, 1 atm pressure, and 25°C temperature. This value is pivotal in determining the cell potential of electrochemical reactions under standard conditions. Every half reaction has a specific E° value that is determined experimentally and tabulated in databases such as Appendix D in a textbook.

Standard electrode potentials provide insight into which species is more easily oxidized or reduced. The more positive the E° value, the stronger the oxidizing agent, that is, the tendency to gain electrons. Conversely, a more negative E° suggests a good reducing agent with a tendency to lose electrons.

• Standard conditions: 1 M for solutions, 1 atm for gases, and 25°C.
• Measured in volts, it provides a basis for calculating the cell voltage.
• Used as a reference point for determining the direction of electron flow in a cell.

The participation of standard electrode potentials in the Nernst equation allows for calculating the cell potential under actual, non-standard conditions, as it adjusts for the deviations in concentration and pressures from standard values.

Reaction Quotient

The reaction quotient, symbolized as Q, is an expression that represents the ratio of the products to the reactants at any point during a reaction. The reaction quotient is crucial because it allows us to check how far a reaction has proceeded toward equilibrium. It shares a similar mathematical form with the equilibrium constant (K), but Q can be calculated at any point during the reaction whereas K is calculated at equilibrium.

In electrochemical cells, Q is used in the Nernst equation to calculate the cell potential when the concentrations of the participating ions are not at standard conditions. The formula for the reaction quotient varies depending on the reaction being considered. Typically, it includes concentrations of the products raised to the power of their coefficients divided by the concentrations of reactants, also raised to their respective coefficients.

• Q = [Products]^coefficients / [Reactants]^coefficients.
• Used to determine the direction in which a reaction will proceed.

By comparing Q to K, we can predict the direction of reaction shift:
- If Q < K, the reaction will proceed forward, producing more products.
- If Q > K, the reaction will proceed in reverse, generating more reactants.
Through understanding Q, we gain insight into how changes in concentration and other conditions affect the cell potential and the spontaneity of reactions in electrochemical cells.