A voltaic cell utilizes the following reaction and operates at 298 K: 3 Ce⁴⁺(aq) + Cr(s) → 3 Ce³⁺(aq) + Cr³⁺(aq) (b) What is the emf of this cell when [Ce⁴⁺] = 3.0 M, [Ce³⁺] = 0.10 M, and [Cr³⁺] = 0.010 M? (c) What is the emf of the cell when [Ce⁴⁺] = 0.010 M, [Ce³⁺] = 2.0 M, and [Cr³⁺] = 1.5 M?

Verified step by step guidance

1. Step 1: Identify the half-reactions and their standard reduction potentials. The given reaction is 3 Ce⁴⁺(aq) + Cr(s) → 3 Ce³⁺(aq) + Cr³⁺(aq). The half-reactions are: Ce⁴⁺ + e⁻ → Ce³⁺ and Cr → Cr³⁺ + 3e⁻. Look up the standard reduction potentials (E°) for these half-reactions in a standard reduction potential table.
2. Step 2: Calculate the standard cell potential (E°cell). Use the formula E°cell = E°cathode - E°anode. The cathode is where reduction occurs (Ce⁴⁺ to Ce³⁺), and the anode is where oxidation occurs (Cr to Cr³⁺). Substitute the standard reduction potentials into the formula to find E°cell.
3. Step 3: Use the Nernst equation to calculate the cell potential (Ecell) under non-standard conditions. The Nernst equation is Ecell = E°cell - (RT/nF) * ln(Q), where R is the gas constant (8.314 J/mol·K), T is the temperature in Kelvin (298 K), n is the number of moles of electrons transferred (3 in this case), F is Faraday's constant (96485 C/mol), and Q is the reaction quotient.
4. Step 4: Calculate the reaction quotient (Q) for the given concentrations. For the reaction 3 Ce⁴⁺(aq) + Cr(s) → 3 Ce³⁺(aq) + Cr³⁺(aq), Q = ([Ce³⁺]^3 * [Cr³⁺]) / [Ce⁴⁺]^3. Substitute the given concentrations into this expression to find Q for each scenario.
5. Step 5: Substitute the values of E°cell, R, T, n, F, and Q into the Nernst equation to calculate the emf of the cell for each set of conditions. This will give you the emf for both scenarios: (b) [Ce⁴⁺] = 3.0 M, [Ce³⁺] = 0.10 M, [Cr³⁺] = 0.010 M and (c) [Ce⁴⁺] = 0.010 M, [Ce³⁺] = 2.0 M, [Cr³⁺] = 1.5 M.

Key Concepts

Electrochemical Cells

Electrochemical cells, such as voltaic cells, convert chemical energy into electrical energy through redox reactions. In a voltaic cell, oxidation occurs at the anode and reduction at the cathode, generating a flow of electrons that can be harnessed for electrical work. Understanding the components and functioning of these cells is crucial for calculating their electromotive force (emf).

Nernst Equation

The Nernst equation relates the emf of an electrochemical cell to the concentrations of the reactants and products involved in the redox reaction. It allows for the calculation of the cell potential under non-standard conditions by incorporating the reaction quotient, Q, which reflects the current concentrations of the species. This equation is essential for determining how changes in concentration affect the cell's emf.

Standard Electrode Potentials

Standard electrode potentials are measured values that indicate the tendency of a species to be reduced, measured under standard conditions (1 M concentration, 1 atm pressure, and 25°C). These values are used to calculate the standard emf of a cell and are critical for understanding the driving force behind the redox reactions in the cell. The difference in standard potentials between the cathode and anode determines the overall cell potential.