Question

Calculate the potential for half-cell containing 0.01 M K\(_2\)Cr\(_2\)O\(_7\)(aq), 0.01 M Cr\(^{3+}\)(aq), and 1.0 x 10\(^{-4}\) M H\(^+\)(aq).

Hint:

Use the Nernst equation to adjust the standard potential based on the concentration of the species involved in the reaction.

Answer 1

The objective is to determine the potential of a half-cell with the following species and concentrations: 0.01 M K₂Cr₂O₇ (aq), 0.01 M Cr³⁺ (aq), and 1.0 × 10⁻⁴ M H⁺ (aq).

1. Reaction and Nernst Equation Identification:
The half-cell involves the redox couple Cr₂O₇²⁻ and Cr³⁺. The relevant half-reaction is:\[ \text{Cr}_2\text{O}_7^{2-}(aq) + 14H^+(aq) + 6e^- \rightarrow 2\text{Cr}^{3+}(aq) + 7H_2O(l) \]The potential is calculated using the Nernst equation:\[ E = E^\circ - \frac{0.0591}{n} \log Q \]where:- \( E^\circ \) signifies the standard electrode potential,- \( n \) represents the number of electrons transferred,- \( Q \) is the reaction quotient.

2. Standard Electrode Potential:
The standard electrode potential (\( E^\circ \)) for this half-reaction is provided as:\[ E^\circ = +1.33 \, \text{V} \]

3. Reaction Quotient (Q) Formulation:
The reaction quotient \( Q \) is defined as:\[ Q = \frac{[\text{Cr}^{3+}]^2 [\text{H}_2\text{O}]^7}{[\text{Cr}_2\text{O}_7^{2-}] [\text{H}^+]^{14}} \]As water (\( \text{H}_2\text{O} \)) is a pure liquid, its concentration is constant and excluded from the \( Q \) expression:\[ Q = \frac{[\text{Cr}^{3+}]^2}{[\text{Cr}_2\text{O}_7^{2-}] [\text{H}^+]^{14}} \]

4. Concentration Substitution into Q:
The given concentrations are:- \( [\text{Cr}_2\text{O}_7^{2-}] = 0.01 \, \text{M} \)- \( [\text{Cr}^{3+}] = 0.01 \, \text{M} \)- \( [\text{H}^+] = 1.0 \times 10^{-4} \, \text{M} \)Substituting these values into the \( Q \) expression yields:\[ Q = \frac{(0.01)^2}{(0.01)(1.0 \times 10^{-4})^{14}} \]

5. Nernst Equation Application:
The Nernst equation with the relevant values is:\[ E = 1.33 - \frac{0.0591}{6} \log Q \]Calculation of \( Q \) and subsequent application of the Nernst equation will yield the half-cell potential under the specified conditions.

Final Calculation:
The potential of the half-cell can be determined by calculating \( E \) using these derived equations and provided constants.