Question

A voltaic cell utilizes the following reaction: $$ 4 \mathrm{Fe}^{2+}(a q)+\mathrm{O}_{2}(g)+4 \mathrm{H}^{+}(a q) \longrightarrow 4 \mathrm{Fe}^{3+}(a q)+2 \mathrm{H}_{2} \mathrm{O}(l) $$ (a) What is the emf of this cell under standard conditions? (b) What is the emf of this cell when \(\left[\mathrm{Fe}^{2+}\right]=1.3 \mathrm{M},\left[\mathrm{Fe}^{3+}\right]=\) \(0.010 \mathrm{M}, P_{\mathrm{O}_{2}}=0.50 \mathrm{atm}\) , and the \(\mathrm{pH}\) of the solution in the cathode half-cell is 3.50\(?\)

Short answer

The emf of the voltaic cell under standard conditions is 0.46 V, and under the given non-standard conditions, it is approximately 0.632 V.

Step-by-step solution

Write the half-reactions

First, we'll balance the redox reaction by splitting it into half-reactions. 1. Oxidation half-reaction: \(Fe^{2+}(aq) \longrightarrow Fe^{3+}(aq) + e^-\) 2. Reduction half-reaction: \( O_2(g) + 4H^+(aq) + 4e^- \longrightarrow 2H_2O(l) \)

Calculate the standard emf (E°) using standard reduction potentials

Using standard reduction potentials from a reference table, we find the following values: \(E°_{Fe^{3+}/Fe^{2+}} = 0.77 \,V\) and \(E°_{O_2/H_2O} = 1.23\,V \) Now, we can calculate the standard emf of the cell: \(E°_{cell} = E°_{cathode} - E°_{anode} = E°_{O_2/H_2O} - E°_{Fe^{3+}/Fe^{2+}}\) \(E°_{cell} = 1.23 \,V - 0.77\,V = 0.46\,V\) The emf of this cell under standard conditions is 0.46 V.

Calculate the emf under non-standard conditions using the Nernst equation

Now, let's use the Nernst equation to find the emf under non-standard conditions: \(E_{cell} = E°_{cell} - \frac{RT}{nF} \ln Q\) Here, R refers to the gas constant (\(8.314\, J/(K \cdot mol)\)), T is the temperature in Kelvin (assuming room temperature, \(298\,K\)), n is the number of moles of electrons transferred in the redox reaction (in this case, 4), and F is the Faraday constant (\(96485\, C/mol\)). First, let's find the reaction quotient Q: \(Q = \frac{[Fe^{3+}]^4 \cdot [H_2O]^2}{[Fe^{2+}]^4 \cdot [O_2] \cdot [H^+]^4}\) Since the concentrations of liquids, such as \(H_2O\), do not affect the reaction quotient, we can ignore this term: \(Q = \frac{[Fe^{3+}]^4}{[Fe^{2+}]^4 \cdot [O_2] \cdot [H^+]^4}\) Substituting the given values, we find: \(Q = \frac{(0.010\,M)^4}{(1.3\,M)^4 \cdot (0.50\, atm) \cdot (10^{-3.5})^4}\) Now, let's plug this value of Q into the Nernst equation: \(E_{cell} = 0.46\,V - \frac{8.314 \,J/(K \cdot mol) \cdot 298\,K}{4 \cdot 96485\,C/mol} \ln Q\) Solving the above equation, we find: \(E_{cell} \approx 0.632\,V\) So, the emf of this cell under the given non-standard conditions is approximately 0.632 V.

Key concepts

Nernst Equation

The Nernst Equation is a fundamental part of electrochemistry, offering insights into how the electromotive force (emf) of an electrochemical cell changes under non-standard conditions. The equation is expressed as:

\(E_{cell} = E°_{cell} - \frac{RT}{nF} \ln Q\)

where \(E_{cell}\) represents the cell potential under non-standard conditions, \(E°_{cell}\) is the standard emf of the cell, \(R\) is the universal gas constant, \(T\) stands for the temperature in Kelvin, \(n\) denotes the number of moles of electrons exchanged in the redox reaction, \(F\) is the Faraday's constant, and \(Q\) is the reaction quotient reflecting the ratio of product and reactant activities. The Nernst Equation corrects the standard emf for the effects of temperature, pressure, and concentration, making it an essential tool for predicting the behavior of batteries, fuel cells, and natural systems such as neuron signaling.

Standard Reduction Potential

Standard reduction potentials are critical to understanding the driving force behind an electrochemical reaction. These values indicate the tendency of a species to gain electrons and be reduced, under standard conditions (concentration of 1 M, pressure of 1 atm, and a temperature of 25°C or 298 K). These potentials are measured in volts and represent the voltage at which the reduction process occurs relative to the standard hydrogen electrode, which is assigned a potential of 0 volts.

In a voltaic cell, the half-cell with the higher standard reduction potential serves as the cathode (site of reduction), while the one with the lower potential is the anode (site of oxidation). The overall standard emf of the cell is calculated by subtracting the anode's potential from the cathode's. For example, with potentials of 1.23 V for the oxygen reduction and 0.77 V for the iron reduction, the standard cell emf is calculated to be 0.46 V.

Reaction Quotient

The reaction quotient, \(Q\), plays a pivotal role in determining the shift of a reaction away from equilibrium. It is similar in form to the equilibrium constant but applies to systems that have not necessarily reached equilibrium. The reaction quotient is calculated using the current concentrations (or partial pressures) of the products raised to the power of their stoichiometric coefficients divided by those of the reactants similarly raised.

For the reaction in a typical voltaic cell, the reaction quotient is expressed as \(Q = \frac{[Fe^{3+}]^4}{[Fe^{2+}]^4 \cdot [O_2] \cdot [H^+]^4}\), excluding pure liquids and solids, as they have a fixed activity of 1. The value of \(Q\) is then substituted into the Nernst equation to compute the non-standard emf of an electrochemical cell. If \(Q\) is greater than the equilibrium constant \(K\), the cell potential will be lower than the standard emf, as the reaction is shifting to favor the reactants; conversely, if \(Q\) is less than \(K\), the reaction favors products and the cell potential will be higher.

Electrochemical Cell

An electrochemical cell is a device that generates electrical energy from chemical reactions. Voltaic (or galvanic) cells are a type of electrochemical cell that converts the energy released during a spontaneous redox reaction into electrical energy. As demonstrated in the given problem, a voltaic cell can consist of an oxidation half-reaction and a reduction half-reaction, occurring in separate compartments connected by a salt bridge that allows ions to flow and maintain charge neutrality.

The standard emf of a cell describes its potential difference under standard conditions, and its calculation involves standard reduction potentials of the involved half-reactions. However, when the conditions vary from the standard state, values such as concentration and gas partial pressures change, as do the temperature and emf. The Nernst equation enables calculation of the new cell potential under these non-standard conditions, ensuring accurate predictions of cell behavior in everyday applications like batteries and sensors.