Problem 109 - Chemistry The Molecular Science

Question

This reaction occurs in a cell with \(\mathrm{H}_{2}(\mathrm{~g})\) pressure of \(1.0 \mathrm{~atm}\) and (conc. \(\left.\mathrm{Cl}^{-}\right)=1.0 \mathrm{M}\) at \(25^{\circ} \mathrm{C} ;\) the measured \(E_{\text {cell }}=0.34 \mathrm{~V}\). Calculate the \(\mathrm{pH}\) of the solution. $$ \mathrm{H}_{2}(\mathrm{~g})+2 \mathrm{AgCl}(\mathrm{s}) \longrightarrow 2 \mathrm{H}^{+}(\mathrm{aq})+2 \mathrm{Cl}^{-}(\mathrm{aq})+2 \mathrm{Ag}(\mathrm{s}) $$

Step-by-Step Solution

Answer

The pH of the solution is approximately 1.74.

Step 1: Write the Nernst Equation

For the given cell reaction, we use the Nernst equation:\[ E_{cell} = E^0_{cell} - \frac{RT}{nF} \ln Q \]Since the cell is operating at standard temperature (25°C or 298 K), \( R = 8.314 \ \text{J} \ \text{mol}^{-1} \ \text{K}^{-1} \), and \( F = 96485 \ \text{C} \ \text{mol}^{-1} \). The reaction involves 2 moles of electrons (\( n = 2 \)).

Step 2: Identify the Standard Cell Potential

The reaction can be split into two half-reactions: the reduction of \( \mathrm{AgCl} \) to \( \mathrm{Ag} \), and the oxidation of \( \mathrm{H}_2 \).- For the reduction of \( \mathrm{AgCl} \), standard potential \( E^0 = +0.222 \ \text{V} \) for one \( \mathrm{AgCl} \) per electron, thus for 2 electrons and 2 \( \mathrm{AgCl} \), it's the same.- For the oxidation of \( \mathrm{H}_2 \), standard potential \( E^0 = 0.000 \ \text{V} \).The overall standard cell potential \( E^0_{cell} = 0.222 \ \text{V} \).

Step 3: Calculate the Reaction Quotient (Q)

The reaction quotient \( Q \) is:\[ Q = \frac{[\mathrm{H}^+]^2 [\mathrm{Cl}^-]^2}{P_{\mathrm{H}_2}} \]Given \([\mathrm{Cl}^-] = 1.0 \ \text{M}\) and \(P_{\mathrm{H}_2} = 1.0 \ \text{atm}\), this simplifies to:\[ Q = [\mathrm{H}^+]^2 \]

Step 4: Rearrange the Nernst Equation to Solve for [H+]

Rearrange the Nernst equation to solve for \([\mathrm{H}^+]\):\[ 0.34 = 0.222 - \frac{0.0591}{2} \ln([\mathrm{H}^+]^2) \]Solving gives:\[ 0.118 = -0.02955 \ln([\mathrm{H}^+]^2) \]\[ \ln([\mathrm{H}^+]^2) = -3.992 \]\[ [\mathrm{H}^+]^2 = e^{-3.992} \]

Step 5: Calculate [H+] Concentration

Solve for \([\mathrm{H}^+]\):\[ [\mathrm{H}^+] = \sqrt{e^{-3.992}} \approx 0.0181 \ \text{M} \]

Step 6: Calculate pH

Finally, calculate the pH using the formula:\[ \text{pH} = -\log([\mathrm{H}^+]) \]\[ \text{pH} = -\log(0.0181) \approx 1.74 \]

Key Concepts

Electrochemistry

Electrochemistry explores the relationship between chemical reactions and electricity. It's the study of how chemical energy is converted into electrical energy and vice versa. In electrochemical cells, a chemical reaction occurs by the transfer of electrons through an external circuit.
These cells can be either:

• Galvanic cells: Produce electric energy through spontaneous reactions.
• Electrolytic cells: Use electric energy to drive non-spontaneous reactions.

In our example, the reaction in a cell involves hydrogen gas and silver chloride, converting chemical energy into measurable voltage (cell potential). Understanding the flow of electrons and the reactions at the electrodes is essential to grasp how these cells work. The Nernst equation is pivotal in predicting the behavior of electrochemical cells based on concentrations and cell potential.

Cell Potential

Cell potential, denoted as \(E_{cell}\), is the electrical potential difference between two electrodes in an electrochemical cell. It measures how much energy per electron can be extracted from the cell reaction. The potential depends on:

• The nature of the substances involved.
• The concentrations of the ionic species.
• The temperature and pressure of the gases involved.

Standard cell potential \(E^0_{cell}\) is measured under standard conditions: 1 M concentration, 1 atm pressure, and 25°C temperature. In the given exercise, it was crucial to distinguish between the standard potential and the measured potential to understand the system's behavior.
The standard reduction potentials for the half-reactions provide insight into the electron flow. The Nernst equation then allows the non-standard situation (actual conditions) to be analyzed by considering concentration changes from standard conditions.

pH Calculation

Calculating pH involves determining the hydrogen ion concentration in a solution. pH is defined as:\[\text{pH} = -\log([\mathrm{H}^+])\]This equation transforms the concentration of hydrogen ions into a more manageable scale. In the exercise, we need to find \([\mathrm{H}^+]\) using the relationship given by the Nernst equation. Rearranging the equation helps uncover the ion concentration based on cell potential.
The steps included solving for hydrogen ion concentration \(([\mathrm{H}^+]))\), extracting it from the logarithmic form given by the Nernst equation, and then calculating the pH. This provides a direct measure of the acidity of the solution under the conditions of the reaction. Understanding this process lets us connect electrochemical properties with chemical equilibrium and acidity.