Question

An electrochemical cell consists of a standard hydrogen electrode and a copper metal electrode. If the copper electrode is placed in a solution of 0.10\(M \mathrm{NaOH}\) that is saturated with \(\mathrm{Cu}(\mathrm{OH})_{2},\) what is the cell potential at \(25^{\circ} \mathrm{C} ?\left[\text { For } \mathrm{Cu}(\mathrm{OH})_{2}\right.\) \(K_{\mathrm{sp}}=1.6 \times 10^{-19} . ]\)

Short answer

The cell potential of the electrochemical cell consisting of a standard hydrogen electrode and a copper metal electrode at 25°C is approximately 0.3685 V.

Step-by-step solution

Write down the dissolution equation for Copper(II) hydroxide

The dissolution equation for Copper(II) hydroxide in water can be written as: \[ \mathrm{Cu}(\mathrm{OH})_{2}(s) \rightleftarrows \mathrm{Cu}^{2+}(aq) + 2 \mathrm{OH}^{-}(aq) \]

Find the concentration of Cu2+ ions in equilibrium

Using the given \(K_{sp}=1.6 \times 10^{-19}\), we can set up the following solubility equilibrium expression: \[K_{sp} = [\mathrm{Cu}^{2+}][\mathrm{OH}^{-}]^{2}\] Since the solution contains 0.10 M of NaOH, we know the initial concentration of OH- ions is 0.10 M. Let \(x\) be the concentration of \(\mathrm{Cu}^{2+}\) ions in equilibrium, the concentration of \(\mathrm{OH}^{-}\) will then be \(0.10+x\). So, the equilibrium expression becomes: \[1.6 \times 10^{-19} = x(0.10+x)^2\]

Solve for x (Concentration of Cu2+ ions)

We can approximate that \(x << 0.10\), and thus neglect \(x\) in the term \(0.10+x\). Therefore, we have: \[1.6 \times 10^{-19} = x(0.10)^2\] \[x = \frac{1.6 \times 10^{-19}}{(0.10)^2}\] \[x = 1.6 \times 10^{-17} \] So, the concentration of \(\mathrm{Cu}^{2+}\) ions in equilibrium is approximately \(1.6 \times 10^{-17} \mathrm{M}\).

Write down the half-cell reactions

The two half-cell reactions are: Cathode (Reduction): Cu2+ + 2e- → Cu (E° = +0.34 V) Anode (Oxidation): 2H+ + 2e- → H2 (Standard Hydrogen Electrode, E° = 0.00 V)

Calculate the cell potential using the Nernst equation

Now we can use the Nernst equation to determine the cell potential at 25°C (298.15 K): \[E_{cell} = E_{cathode}° - E_{anode}° - \frac{RT}{nF} \ln\frac{[\mathrm{H}^{+}]^{2}}{[\mathrm{Cu}^{2+}]}\] where R = 8.314 J/mol·K (gas constant), T = 298.15 K (temperature), n = 2 (moles of electrons exchanged), and F = 96485 C/mol (Faraday's constant). Remember that \(E_{anode}° = 0.00 \, \mathrm{V}\) since the anode is the standard hydrogen electrode. We are given \(E_{cathode}° = +0.34 \, \mathrm{V}\), and we know that pH of a 0.10 M NaOH solution is \(pH = 13\), so the concentration of \(\mathrm{H}^{+}\) ions can be calculated as \([\mathrm{H}^{+}] = 10^{-pH} = 10^{-13}\,\mathrm{M}\). Inserting these values into the Nernst equation and solving for \(E_{cell}\): \[E_{cell} = 0.34 - \frac{8.314 \cdot 298.15}{2 \cdot 96485} \ln{\frac{(10^{-13})^2}{1.6 \times 10^{-17}}}\] \[E_{cell} = 0.34 + 0.0285\] \[E_{cell} = 0.3685 \, \mathrm{V}\] Therefore, the cell potential of the electrochemical cell is approximately 0.3685 V at 25°C.

Key concepts

Nernst equation

The Nernst equation is essential in electrochemistry. It allows you to determine the potential of an electrochemical cell based on the concentrations of the reactants and products involved. The general form of the Nernst equation is:
\[E = E^0 - \frac{RT}{nF} \ln Q\]Where:

• \(E\) is the cell potential under non-standard conditions.
• \(E^0\) is the standard cell potential.
• \(R\) is the universal gas constant (8.314 J/mol·K).
• \(T\) is the temperature in Kelvin.
• \(n\) is the number of moles of electrons transferred in the redox reaction.
• \(F\) is Faraday's constant (96485 C/mol).
• \(Q\) is the reaction quotient.

In the context of an electrochemical cell, using the Nernst equation helps in calculating the cell potential by considering the concentrations of ions involved in both half-reactions. This can be crucial for reactions occurring outside the standard conditions.
Using the values provided, you can adjust the potential you calculate using the Nernst equation to find the true value of cell potential for different conditions.

Standard hydrogen electrode

At the heart of many electrochemical calculations is the standard hydrogen electrode (SHE), which serves as a universal reference. It's defined under standard conditions, where its potential is universally taken to be 0.00 V. This electrode involved the half-reaction:
\[2H^+ + 2e^- \rightarrow H_2(g)\]The SHE is crucial because it offers a stable baseline for measuring electrode potentials. By coupling it with other electrodes, like the copper electrode in this exercise, you can measure the electrode potentials confidently and standardize measurements across different conditions.
Using the SHE allows scientists and engineers to compare the potentials of different half-reactions accurately. In the exercise, it's paired with the copper electrode to establish an understandable comparison for the potential calculation.

Dissolution equilibrium

Dissolution equilibrium involves the balance between a solute dissolving into a solvent and its recrystallization. In electrochemistry, this comes into play when dealing with a saturated solution and a sparingly soluble compound, like copper(II) hydroxide.
This equilibrium can be described using the solubility product constant \(K_{sp}\), shown in the reaction:
\[\mathrm{Cu}(\mathrm{OH})_2(s) \rightleftarrows \mathrm{Cu}^{2+}(aq) + 2 \mathrm{OH}^{-}(aq)\]The \(K_{sp}\) helps establish the concentrations of the ions at saturation, a critical aspect when applying principles like the Nernst equation.
Given copper(II) hydroxide's low \(K_{sp}\), the number of dissolved ions is minimal. In our scenarios, this impacts the concentration of \(\mathrm{Cu}^{2+}\) ions directly, which is a crucial step for computing the cell potential.

Cell potential calculation

Estimating an electrochemical cell's potential involves considering multiple factors, including the electrodes involved and ionic concentrations. After understanding the dissolution equilibrium and the standard potentials, you use the Nernst equation to compute the precise cell potential based on current conditions.
In the exercise, the cell contains a standard hydrogen electrode and a copper electrode. Each electrode has specific standard potentials: 0.00 V for the hydrogen electrode and specific values relevant to copper. Given the ionic concentrations from the dissolution equilibrium, you can adopt these into the Nernst equation:
\[ \large E_{cell} = E_{cathode}^0 - E_{anode}^0 - \frac{RT}{nF} \ln \frac{[H^+]^2}{[Cu^{2+}]} \]
Upon substituting the known values and conducting the calculations, the exact cell potential at 25°C in this instance is approximately 0.3685 V.
This accurate approach allows prediction of electrochemical behavior under variable conditions, providing thorough insights into how the system will perform.