Problem 93 - Chemistry Principles and Practice

Question

What is the reduction potential of the hydrogen electrode at \(298 \mathrm{~K}\) if the pressure of gaseous hydrogen is \(2.5 \mathrm{~atm}\) in a solution of \(\mathrm{pH} 6.00 ?\)

Step-by-Step Solution

Answer

The reduction potential is approximately \(-0.3665\) V.

Step 1: Understanding the Nernst Equation for Hydrogen Electrode

The reduction potential of the hydrogen electrode can be calculated using the Nernst Equation: \(E = E^0 - \frac{0.0591}{n} \log Q\), where \(E^0\) for the Standard Hydrogen Electrode (SHE) is \(0.00\) V, \(n\) is the number of moles of electrons transferred (\(n=2\) for hydrogen), and \(Q\) is the reaction quotient. In this case, \(Q = \frac{1}{[H^+]^2} \cdot P_{H_2}\).

Step 2: Calculating Concentration of Hydrogen Ions from pH

The pH of the solution is 6.00, so the concentration of hydrogen ions \([H^+]\) can be calculated using the formula: \([H^+] = 10^{-\text{pH}}\). Substituting \(\text{pH} = 6.00\) gives \([H^+] = 10^{-6}\) M.

Step 3: Substituting Values into Reaction Quotient

Substitute \([H^+] = 10^{-6}\) M and \(P_{H_2} = 2.5\) atm into \(Q\):\[Q = \frac{1}{(10^{-6})^2} \cdot 2.5 = \frac{1}{10^{-12}} \cdot 2.5 = 2.5 \times 10^{12}\].

Step 4: Calculating the Reduction Potential

Substitute \(E^0 = 0.00\) V, \(n = 2\), and \(Q = 2.5 \times 10^{12}\) into the Nernst equation:\[E = 0.00 - \frac{0.0591}{2} \log (2.5 \times 10^{12})\]Calculate the logarithm and then solve for \(E\):\[E \approx 0.00 - 0.02955 \times 12.39794 \approx -0.3665 \text{ V}\].

Key Concepts

Standard Hydrogen Electrode

The Standard Hydrogen Electrode (SHE) is a fundamental reference point in electrochemistry. It serves as the benchmark for measuring the electrode potentials of other half-cells. The SHE comprises a platinum electrode in contact with 1 M H⁺ ions and bathed in hydrogen gas at 1 atm and 298 K. This setup represents an ideal condition where the reduction potential is defined as 0.00 volts.

The SHE is so essential because it provides a consistent baseline to compare and calculate other electrode potentials across different conditions. In electrochemical reactions, the SHE is noted for its simplicity and reproducibility:

• It allows for a direct comparison of how easily different substances gain or lose electrons.
• Acts as a base in the Nernst Equation, simplifying the calculations of real-life electrochemical potentials.

In practical applications, making precise observations for electrode potential deviations from the standard state conditions established by the SHE creates a more dynamic and applicable approach to electrochemical reactions.

Reduction Potential

Reduction potential, also known as redox potential, is a measure of the tendency of a chemical species to acquire electrons and thereby be reduced. The reduction potential indicates how likely a substance will gain electrons compared to the hydrogen electrode.

It is measured in volts and can be either positive or negative. A more positive reduction potential means a higher affinity for electrons, suggesting it is more likely to gain electrons. Conversely, a negative potential indicates a lesser tendency. The reduction potential is critical for understanding chemical stability and reactivity in electrochemical cells.

• Helps in calculating the electrochemical potential differences in various systems.
• Used in the context of the Nernst Equation to predict how potential changes with concentration, pressure, and other variables.

Analyzing the reduction potential helps in determining the energetics of a reaction, especially crucial when moving between standard state conditions and the conditions stated in real-world problems, as demonstrated when solving the given exercise.

Reaction Quotient

The reaction quotient \(Q\) is a mathematical representation of the concentrations of the reactants and products during a reaction. It helps predict the direction of the reaction at any point in time. In the context of a redox reaction using the Nernst Equation, \(Q\) adjusts the electrode potential from the standard state to actual conditions.

For the hydrogen electrode, the reaction quotient formula \( Q = \frac{1}{[H^+]^2} \cdot P_{H_2} \) incorporates the concentration of hydrogen ions and the partial pressure of hydrogen gas. These variables provide a clear understanding of how non-standard concentrations can modify electrode potential.

• Serves as a critical component in the calculation of actual cell potentials using the Nernst Equation.
• Helps in predicting the shift in reactions under various pressures and concentrations, aiding in a comprehensive assessment of their outcomes.

Understanding the calculation of \(Q\) and its integration into potential calculations is crucial, especially in determining the reduction potential of the hydrogen electrode in non-standard conditions, as demonstrated in the exercise.