Question

The following cell has a potential of \(0.27 \mathrm{~V}\) at \(25^{\circ} \mathrm{C}\) : $$ \operatorname{Pt}(s)\left|\mathrm{H}_{2}(1 \mathrm{~atm})\right| \mathrm{H}^{+}(? \mathrm{M}) \| \mathrm{Ni}^{2+}(1 \mathrm{M}) \mid \mathrm{Ni}(s) $$ What is the \(\mathrm{pH}\) of the solution in the anode compartment?

Short answer

The pH of the solution in the anode compartment is approximately 4.25.

Step-by-step solution

Identify the Cell Components

The given cell consists of a hydrogen half-cell and a nickel half-cell. The hydrogen half-cell is an anode where hydrogen gas at 1 atm is oxidized, and the nickel half-cell is the cathode with a nickel ion solution at 1 molarity.

Write Half-Reactions

The half-reactions are as follows: - Anode (hydrogen): \[ ext{H}_2(g) ightarrow 2 ext{H}^+(aq) + 2e^- \] - Cathode (nickel): \[ ext{Ni}^{2+}(aq) + 2e^- ightarrow ext{Ni}(s) \]

Determine Standard Potentials

The standard reduction potentials for the electrodes are: - Hydrogen ( ext{E}^{{ ext{H}^+/ ext{H}_2}}) = 0.00 V- Nickel ( ext{E}^{ ext{Ni}^{2+}/ ext{Ni}}) = -0.25 VThe cell potential is the difference: \[ ext{E}_{ ext{cell}}^{ ext{standard}} = ext{E}^{ ext{cathode}} - ext{E}^{ ext{anode}} = (-0.25) - (0) = -0.25 ext{ V}\]

Apply the Nernst Equation

The Nernst equation is used to find the cell potential at non-standard conditions: \[ ext{E}_{ ext{cell}} = ext{E}_{ ext{cell}}^{ ext{standard}} - \frac{RT}{nF} \ln Q\]Where \(R\) is the universal gas constant \(8.314 \times 10^{-3} \text{ kJ/mol K}\), \(T\) is the temperature \(298 \text{ K}\), \(n\) is the number of moles of electrons transferred \(2\), and \(F\) is Faraday's constant \(96,485 \text{ C/mol}\).Given \( ext{E}_{ ext{cell}} = 0.27 ext{ V}\), equate and solve for \(Q\): \[0.27 = -0.25 - \frac{(8.314 \times 10^{-3} \times 298)}{2 imes 96,485} \ln Q\]

Solve for Q

Rearrange the above equation to solve for \(Q\):\[\ln Q = \frac{2 imes 96,485}{8.314 \times 298} (0.27 + 0.25) = 19.57\]Therefore, \[Q = e^{19.57}\]Since \(Q\) in this case is the concentration of \( ext{H}^+\) ions, \[Q = \frac{[\text{Ni}^{2+}]}{{[\text{H}^+]^2}} = [\text{H}^+]^{-2}\]

Calculate [H⁺] and pH

From the value of \(Q\) obtained, solve:\[ [\text{H}^+] = \frac{1}{\sqrt{Q}} = \frac{1}{e^{9.785}} \approx 5.6 \times 10^{-5} \text{ M}\]pH is calculated using:\[\text{pH} = -\log([\text{H}^+]) = -\log(5.6 \times 10^{-5}) \approx 4.25\]

Key concepts

Nernst Equation

The Nernst Equation is crucial in understanding how cell potentials work under non-standard conditions. This equation takes into account the effect of concentration, temperature, and pressure on the electrochemical cell potential. The standard equation is given as: \[ E_{\text{cell}} = E_{\text{cell}}^{\text{standard}} - \frac{RT}{nF} \ln Q \]Here,

• \(E_{\text{cell}}\) is the cell potential at non-standard conditions.
• \(E_{\text{cell}}^{\text{standard}}\) represents the standard cell potential.
• \(R\) is the gas constant, \(8.314 \times 10^{-3} \text{ kJ/mol K}\).
• \(T\) is the temperature in Kelvin.
• \(n\) denotes the number of moles of electrons transferred.
• \(F\) is Faraday's constant, equal to \(96,485 \text{ C/mol}\).
• \(Q\) is the reaction quotient, which is calculated from the concentrations of the products and reactants.

The Nernst Equation is particularly useful because it allows us to predict the cell potential at a given set of conditions, which differ from the standard state. This lets scientists and engineers decide how the reactions will proceed based on existing conditions.

pH Calculation

Calculating the pH of a solution, especially in electrochemistry, involves understanding the concentration of hydrogen ions present in that solution. The pH is found using the formula:\[\text{pH} = -\log_{10}([\text{H}^+])\]
Where \([\text{H}^+]\) is the concentration of hydrogen ions in the solution. Let's break it down further:

• Suppose \([\text{H}^+]\) is known or calculated using chemical reaction data. In this context, it was calculated from the equilibrium expression for the electrochemical cell.
• The logarithmic function \(-\log_{10}\) works to convert the hydrogen ion concentration to a pH value, making it easier to comprehend the acidity or basicity of the solution in simpler terms.
• A lower pH indicates a more acidic solution, while a higher pH suggests a more basic solution.

It’s essential to calculate the pH correctly, especially in fields like electrochemistry, where ionic concentrations greatly influence cell potentials and overall reaction feasibility.

Standard Electrode Potential

Standard Electrode Potential, often denoted as \(E^0\), measures the tendency of a chemical species to acquire electrons and be reduced. It is determined under standard conditions: all solutes at 1 M concentration, gases at 1 atm, and the temperature at 25°C.

• For the hydrogen electrode, \(E^0\) is defined as 0 V because it serves as a reference point for all other electrode potentials.
• Nickel's standard electrode potential \(E^{\text{Ni}^{2+}/\text{Ni}}\) is given as \(-0.25 \text{ V}\), indicating that metallic nickel is less likely to undergo reduction compared to hydrogen ions.
• The difference between the cathode and anode standard potentials gives the standard cell potential \(E_{\text{cell}}^{\text{standard}}\) for the electrochemical cell.

Standard electrode potentials are vital as they help predict the direction of electron flow in electrochemical reactions, and thus, whether a reaction can be spontaneously carried out under given conditions.