Question

Consider the reduction of nitrate ion in acidic solution to nitrogen oxide \(\left(E_{\text {red }}^{\circ}=0.964 \mathrm{~V}\right)\) by sulfur dioxide that is oxidized to sulfate ion \(\left(E_{\text {red }}^{\circ}=0.155 \mathrm{~V}\right) .\) Calculate the voltage of a cell involving this reaction in which all the gases have pressures of \(1.00 \mathrm{~atm},\) all the ionic species (except \(\left.\mathrm{H}^{+}\right)\) are at \(0.100 \mathrm{M},\) and the \(\mathrm{pH}\) is \(4.30 .\)

Short answer

Given the reduction of nitrate ion (NO3-) to nitrogen oxide (NO) and the oxidation of sulfur dioxide (SO2) to sulfate ion (SO42-) under specific non-standard conditions, the cell potential (E) is found to be 1.046 V using the Nernst equation.

Step-by-step solution

Write the half-reactions

First, we have to write the half-reactions for the reduction of nitrate ion (NO3-) and the oxidation of sulfur dioxide (SO2). Nitrate Reduction: NO3- + 3e- -> NO + 2H2O Sulfur Dioxide Oxidation: 2SO2 + 2H2O -> SO42- + 4H+ + 2e-

Balance the equations and find the overall redox reaction

Combine the two half-reactions to create the balanced redox reaction. Since there are 3 electrons transferred in the reduction and 2 electrons transferred in the oxidation, we will need a lowest common multiple of both, which is 6. Multiple each half-reaction by the necessary factor to balance the electrons, and then combine the half-reactions to get the overall redox reaction. 3 (Nitrate Reduction): 3NO3- + 9e- -> 3NO + 6H2O 4 (Sulfur Dioxide Oxidation): 8SO2 + 8H2O -> 4SO42- + 16H+ + 8e- Balanced Redox Reaction: 8SO2 + 3NO3- + 16H+ -> 4SO42- + 3NO + 6H2O

Calculate the standard cell potential (E°)

Now that we have the balanced equation, we can calculate the standard cell potential (E°) by using the standard reduction potentials (E°red) given in the problem: E°cell = E°red(cathode) - E°red(anode) E°cell = 0.964 V - 0.155 V = 0.809 V

Calculate the non-standard cell potential (E) using the Nernst equation

We can now use the Nernst equation to find the cell potential (E) under non-standard conditions. Nernst Equation: E = E° - (RT/nF) * ln(Q) Here, R is the gas constant (8.314 J/mol·K), T is the temperature in Kelvin (use 298 K for room temperature), n is the number of electrons transferred (6 in this case), F is the Faraday constant (96485 C/mol), and Q is the reaction quotient. Given the conditions in the problem, we have: Q = ([SO42-]^4 * [NO]^3 * [H2O]^6) / ([NO3-]^3 * [SO2]^8 * [H+]^16) Q = (0.1^4 * (1)^3 * (1)^6) / (0.1^3 * (1)^8 * (10^-4.3)^16) Q = 10^(-4) Now, plug in the values into the Nernst equation: E = 0.809 V - ( (8.314 J/mol·K * 298 K) / (6 * 96485 C/mol) ) * ln(10^-4) E = 0.809 V - ( (2.303 * 8.314 J/mol·K * 298 K) / (6 * 96485 C/mol) ) * -4 E = 0.809 V - ( (59.16 mV) * -4 ) E = 0.809 V + 0.23664 V = 1.046 V

State the answer

The voltage of a cell involving the reduction of nitrate ion in acidic solution to nitrogen oxide and the oxidation of sulfur dioxide to sulfate ion, under the given conditions, is 1.046 V.

Key concepts

Understanding Redox Reactions

A redox reaction, short for reduction-oxidation reaction, is a chemical process involving the transfer of electrons between two species. It's composed of two half-reactions: one for reduction, where a species gains electrons, and one for oxidation, where a species loses electrons.

The reduction half-reaction in our exercise involves the conversion of nitrate ion (NO3-) to nitrogen oxide (NO), while the oxidation half-reaction involves sulfur dioxide (SO2) being converted to the sulfate ion (SO42-). The electrons lost by the sulfur dioxide during oxidation are the same electrons gained by the nitrate ion during reduction. It's critical to balance these electrons to ensure that the number of electrons lost is equal to the number gained, maintaining electrical neutrality.

This balance is achieved by multiplying the half-reactions by appropriate coefficients, yielding a reaction where the total charge and mass are conserved. Understanding the fundamentals of redox reactions is fundamental for tackling exercises involving electrochemistry, as these reactions are the foundation upon which electrochemical cells operate.

Applying the Nernst Equation

The Nernst equation is a fundamental equation in electrochemistry, allowing us to calculate the electric potential of an electrochemical cell under non-standard conditions. It links the measurable cell potential to the temperature, reaction quotient (Q), number of electrons transferred (n), the gas constant (R), and the Faraday constant (F).

The Nernst equation can be written as:

\[\begin{equation}E = E^° - \left(\frac{RT}{nF}\right) \ln(Q)\end{equation}\]

Here, the reaction quotient (Q) reflects the concentration of reactants and products at a given moment. For the given exercise, we calculate Q based on the concentrations of ionic and gaseous species involved in the reaction. By understanding and applying the Nernst equation correctly, we can predict how changes in conditions such as concentration and pH will affect the cell potential.

Dynamics of Electrochemical Cells

Electrochemical cells are the devices that convert chemical energy into electrical energy through redox reactions. They consist of two electrodes, an anode and a cathode, connected by an external circuit and a salt bridge that completes the internal circuit and maintains charge balance.

In the context of the exercise, the voltage of a cell is determined by the difference in standard reduction potentials of the cathode and anode, as shown in step 3 of the solution. The standard cell potential gives us a prediction of the cell's ability to do electrical work under standard conditions.

To use an electrochemical cell for practical applications, we must consider factors like temperature, concentration, and gas pressures. Adjusting any of these factors can alter the voltage that the cell generates. A thorough understanding of these variables and their impact on the cell's output is essential for harnessing the full potential of electrochemical cells in real-world situations.