Question

Given \(\mathrm{E}^{\circ} \mathrm{Cr}^{3+} / \mathrm{Cr}=-0.72 \mathrm{~V}, \mathrm{E}^{\circ} \mathrm{Fe}^{2+} / \mathrm{Fe}=-0.42 \mathrm{~V}\). The potential for the cell \(\mathrm{Cr}\left|\mathrm{Cr}^{3+}(0.1 \mathrm{M}) \| \mathrm{Fe}^{2+}(0.01 \mathrm{M})\right| \mathrm{Fe}\) is (a) \(0.26 \mathrm{~V}\) (b) \(0.399 \mathrm{~V}\) (c) \(-0.339 \mathrm{~V}\) (d) \(-0.26 \mathrm{~V}\)

Short answer

Option (a) 0.26 V is correct.

Step-by-step solution

Write the Nernst Equation

The Nernst equation is used to calculate the cell potential under non-standard conditions. It is given by: \[E_{cell} = E^{ ext{°}}_{cell} - \frac{RT}{nF}\ln Q\]For simplicity at room temperature (25°C), it becomes:\[E_{cell} = E^{ ext{°}}_{cell} - \frac{0.0592}{n}\log Q\] where\(E^{ ext{°}}_{cell}\) is the standard cell potential, \(n\) is the number of moles of electrons exchanged, and \(Q\) is the reaction quotient.

Determine Standard Cell Potential

The standard cell potential \(E^{ ext{°}}_{cell}\) is calculated from the standard potentials of the two half-cells:\[E^{ ext{°}}_{cell} = E^{ ext{°}}_{ ext{cathode}} - E^{ ext{°}}_{ ext{anode}}\]With iron as the cathode and chromium as the anode, this becomes:\[E^{ ext{°}}_{cell} = (-0.42 \, \text{V}) - (-0.72 \, \text{V}) = 0.30 \, \text{V}\]

Write the Cell Reaction

To calculate \(Q\), we need the balanced cell reaction:Anode (oxidation): \(\text{Cr} \rightarrow \text{Cr}^{3+} + 3e^-\)Cathode (reduction): \(\text{Fe}^{2+} + 2e^- \rightarrow \text{Fe}\)The balanced equation combining the oxidation and reduction reactions:\(2\text{Cr} + 3\text{Fe}^{2+} \rightarrow 2\text{Cr}^{3+} + 3\text{Fe} \)This shows \(n = 6\), meaning 6 electrons are exchanged.

Calculate the Reaction Quotient \(Q\)

The reaction quotient \(Q\) is determined from the concentrations:\[Q = \frac{[\text{Cr}^{3+}]^2}{[\text{Fe}^{2+}]^3} = \frac{(0.1)^2}{(0.01)^3}\]This simplifies to:\[Q = \frac{0.01}{0.000001} = 10,000\]

Substitute into Nernst Equation

Using the calculated values:\[E_{cell} = 0.30 \, \text{V} - \frac{0.0592}{6} \log(10,000)\]Evaluate the logarithmic term:\[\log(10,000) = 4\]

Complete the Calculation

Substitute and solve:\[E_{cell} = 0.30 - \frac{0.0592 \cdot 4}{6}\]Calculate the adjustments:\[E_{cell} = 0.30 - \frac{0.2368}{6} = 0.30 - 0.03947;\]\[E_{cell} \approx 0.26 \, \text{V}\]

Select Correct Answer

The correct calculated cell potential for the given conditions is \(0.26 \, \text{V}\), which corresponds to option (a).

Key concepts

Nernst Equation

The Nernst equation is essential in electrochemistry to determine the electric potential of an electrochemical cell under non-standard conditions. It is represented by the formula:\[E_{cell} = E^{\text{°}}_{cell} - \frac{RT}{nF}\ln Q\]For most practical purposes at room temperature (25°C), the equation simplifies to:\[E_{cell} = E^{\text{°}}_{cell} - \frac{0.0592}{n}\log Q\]Here is what each part means:- **\(E_{cell}\)**: The cell potential under non-standard conditions.- **\(E^{\text{°}}_{cell}\)**: The standard cell potential, or the potential difference between the cathode and anode when all reactants and products are in their standard states.- **\(n\)**: The number of moles of electrons being transferred in the cell reaction.- **\(Q\)**: The reaction quotient, expressing the ratio of product concentrations to reactant concentrations raised to their respective stoichiometric coefficients.Utilizing the Nernst equation lets us predict how changes in concentration impact the potential of an electrochemical cell. It is particularly useful when concentrations deviate from those defined as standard, such as 1 M for solutions.

Standard Cell Potential

The standard cell potential, denoted \(E^{\text{°}}_{cell}\), is a measure of the driving force behind an electrochemical reaction when all components are at their standard states. Standard conditions typically mean 1 M concentration for dissolved species, 1 atm pressure for gases, and pure solids or liquids for other phases.It is calculated using the equation:\[E^{\text{°}}_{cell} = E^{\text{°}}_{\text{cathode}} - E^{\text{°}}_{\text{anode}}\]Where:- **\(E^{\text{°}}_{\text{cathode}}\)**: The standard reduction potential for the cathode reaction.- **\(E^{\text{°}}_{\text{anode}}\)**: The standard reduction potential for the anode reaction.These potentials are found in tables of standard electrode potentials, which provide the potential of each half-cell compared to the standard hydrogen electrode (SHE). A positive \(E^{\text{°}}_{cell}\) indicates a spontaneous reaction under standard conditions. This value is fundamental to understanding the electrochemical behavior of cells and is used as a reference point in the Nernst equation.

Reaction Quotient

The reaction quotient, denoted as \(Q\), is a number that provides a momentary glimpse into the direction that a reaction may move towards equilibrium. It is calculated similarly to the equilibrium constant \(K\), but for non-equilibrium conditions.For any general reaction:\(aA + bB \rightleftharpoons cC + dD\) The reaction quotient is formulated as:\[Q = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]This formula considers the activities or concentrations of the products and reactants, with each raised to the power corresponding to its stoichiometric coefficient.- **If \(Q < K\)**: The reaction proceeds in the forward direction towards equilibrium.- **If \(Q = K\)**: The system is at equilibrium, and no net reaction occurs.- **If \(Q > K\)**: The reaction proceeds in the reverse direction to form more reactants.In the context of the Nernst equation, \(Q\) affects the cell potential, showing how changes in concentration alter the driving force of the electrochemical cell.

Electrode Potentials

Electrode potentials reflect the voltage associated with an electrochemical reaction at each electrode within the cell. Each half-cell in an electrochemical cell has its own potential, influenced by its specific reduction or oxidation process.- **Anode Potential**: Involves oxidation, where electrons are released, and the potential is often less than that of the cathode.- **Cathode Potential**: Involves reduction, where electrons are gained. The potential for reduction reactions is typically more positive, attracting electrons naturally.The standard electrode potential, given by \(E^{\text{°}}\), measures how easily a species is reduced compared to the hydrogen electrode, which is set as the baseline with a potential of 0 volts.Half-cell potentials are vital for determining the overall potential, \(E^{\text{°}}_{cell}\), of a galvanic cell. It is the difference between the cathode's and the anode's potentials:\[E^{\text{°}}_{cell} = E^{\text{°}}_{\text{cathode}} - E^{\text{°}}_{\text{anode}}\]Understanding electrode potentials enables us to predict and manipulate the direction and extent of electrochemical reactions, crucial in designing batteries and other devices relying on redox reactions.