Question

Consider the cell described below: $$\mathrm{Zn}\left|\mathrm{Zn}^{2+}(1.00 M)\right|\left|\mathrm{Ag}^{+}(1.00 M)\right| \mathrm{Ag}$$ Calculate the cell potential after the reaction has operated long enough for the \(\left[\mathrm{Zn}^{2+}\right]\) to have changed by 0.20 \(\mathrm{mol} / \mathrm{L}\) . (Assume \(T=25^{\circ} \mathrm{C} . )\)

Short answer

Combining the half-cell reactions and using the Nernst equation, the cell potential after the change in concentration of \(\mathrm{Zn^{2+}}\) by 0.20 mol/L is approximately \(1.53 V\).

Step-by-step solution

Write down the half-cell reactions

The half cell reactions of the cell are given below: Zn → Zn²⁺ + 2 e⁻ (oxidation half-cell) Ag⁺ + e⁻ → Ag (reduction half-cell)

Determine the overall cell reaction

Add the half-cell reactions to get the overall cell reaction: Zn + 2Ag⁺ → Zn²⁺ + 2Ag

Calculate the initial cell potential

We use the standard electrode potentials to determine the initial cell potential. The standard electrode potentials for both half-cells are: E₀(Zn / Zn²⁺) = -0.76 V E₀(Ag⁺ / Ag) = 0.80 V The initial cell potential, E₀_cell, is determined by subtracting the oxidation potential from the reduction potential: E₀_cell = E₀(Ag⁺ / Ag) - E₀(Zn / Zn²⁺) = 0.80 V - (-0.76V) = \(1.56 V\)

Apply the Nernst equation

The Nernst equation allows us to calculate the cell potential when the concentrations of species involved in the reaction are not at their standard states: E_cell = E₀_cell - \(\frac{RT}{nF}\) ln(Q) Where E_cell is the cell potential, R is the gas constant \(8.314 J/(mol \times K)\), T is the temperature in Kelvin, n is the number of moles of electrons transferred in the cell reaction, F is the Faraday constant \(96485 C/mol\), and Q is the reaction quotient. Given the information in the problem, the reaction quotient Q is: Q = \(\frac{[Zn^{2+}]}{[Ag^{+}]^2}\)

Calculate the change in concentration

The problem states that the concentration of Zn²⁺ ions has changed by 0.20 mol/L. We need to find the new concentrations of Zn²⁺ and Ag⁺: New [Zn²⁺] = Initial [Zn²⁺] + 0.20 mol/L = 1.00 M + 0.20 M = 1.20 M New [Ag⁺] = Initial [Ag⁺] - 0.20 mol/L = 1.00 M - 0.20 M = 0.80 M Using these values, we calculate the reaction quotient, Q: Q = \(\frac{1.20}{0.80^2} = \frac{1.20}{0.64} = 1.875\)

Calculate T(K) and the cell potential

Convert the temperature from Celsius to Kelvin: T = 25°C + 273.15 K = 298.15 K Now, we can plug everything into the Nernst equation and calculate the cell potential after the change in concentration: E_cell = E₀_cell - \(\frac{8.314 J/(mol \times K) \times 298.15 K}{2 \times 96485 C/mol}\) ln(1.875) E_cell = \(1.56 V - \frac{8.314 \times 298.15}{2 \times 96485} \times \ln(1.875)\) E_cell ≈ \(1.56 V - 0.026 V \approx 1.53 V\) After the reaction has operated long enough for the [Zn²⁺] to have changed by 0.20 mol/L, the cell potential is approximately \(1.53 V\).

Key concepts

Cell Potential

In electrochemistry, the cell potential is a measure of the potential difference between two electrodes in an electrochemical cell. This potential difference drives the redox reactions in the cell, causing electrons to flow from one electrode to another. The cell potential, often denoted as \(E_{\text{cell}}\), can be measured in volts and gives an idea about the cell's ability to perform work.

The initial cell potential is calculated under standard conditions, where the concentrations of all ionic species are 1 M, and the temperature is 25°C (298 K). However, in practical situations, the concentrations may differ due to ongoing reactions, which requires adjusting the calculated potential using the Nernst equation.

A positive cell potential indicates a spontaneous reaction in the forward direction, meaning the reaction will occur without any external energy applied. In our example, the initial cell potential was calculated to be 1.56 V using standard electrode potentials, but it changed to a final value of 1.53 V due to the change in ion concentrations.

Nernst Equation

The Nernst equation is essential in electrochemistry as it allows us to calculate the cell potential when the reaction conditions deviate from standard states. It incorporates factors such as changing ion concentrations, temperature, and pressure, giving a real-time potential of the cell.

The Nernst equation is written as:\[E_{\text{cell}} = E^0_{\text{cell}} - \left(\frac{RT}{nF}\right) \ln(Q)\] Where:

• \(E_{\text{cell}}\) is the actual cell potential under current conditions.
• \(E^0_{\text{cell}}\) is the standard cell potential.
• \(R\) is the universal gas constant, \(8.314 \, \text{J/(mol\,K)}\).
• \(T\) is the temperature in Kelvin.
• \(n\) is the number of moles of electrons exchanged in the reaction.
• \(F\) is the Faraday constant, \(96485 \, \text{C/mol}\).
• \(Q\) is the reaction quotient, which is the ratio of the products to the reactants, raised to the power of their stoichiometric coefficients.

This equation highlights how even a small change in concentration can impact the cell potential, reflecting on the overall cell performance.

Redox Reactions

Redox reactions are fundamental to the functioning of electrochemical cells. They involve the transfer of electrons between species; one species undergoes oxidation (loss of electrons) and the other undergoes reduction (gain of electrons). These processes happen in tandem.

In the example provided, the redox reactions include the oxidation of zinc: \( \text{Zn} \rightarrow \text{Zn}^{2+} + 2e^- \), which occurs at the anode, and the reduction of silver ions: \( \text{Ag}^+ + e^- \rightarrow \text{Ag} \), which occurs at the cathode. When combined, the overall redox reaction can be written as:\( \text{Zn} + 2 \text{Ag}^+ \rightarrow \text{Zn}^{2+} + 2\text{Ag} \).

Redox reactions are the driving force behind many applications, including batteries, corrosion, and electroplating. They are orchestrated in electrochemical cells to convert chemical energy into electrical energy efficiently.

Standard Electrode Potentials

Standard electrode potentials are tabulated values that indicate the propensity of a species to gain or lose electrons under standard conditions (1M concentration, 25°C, 1 atmosphere pressure). These potentials are measured in volts and are crucial for predicting the direction of redox reactions.

For the cell discussed, the standard electrode potentials are:

• \( E^0(\text{Zn}^{2+}/\text{Zn}) = -0.76 \, V \)
• \( E^0(\text{Ag}^+/\text{Ag}) = 0.80 \, V \)

These values suggest that silver ions are more inclined to gain electrons and undergo reduction than zinc ions, which prefer to lose electrons and undergo oxidation.

The overall standard cell potential, \(E^0_{\text{cell}}\), is calculated by taking the difference between the cathode and anode potentials (i.e., \(E^0_{\text{cathode}} - E^0_{\text{anode}}\)). These potentials guide the design and application of various electrochemical cells in different technological innovations.