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 \(0.10 \mathrm{M} \mathrm{NaOH}\) that is saturated with \(\mathrm{Cu}(\mathrm{OH})_{2}\), what is the cell potential at \(25^{\circ} \mathrm{C} ?\left[\right.\) For \(\mathrm{Cu}(\mathrm{OH})_{2}, K_{\text {sp }}=1.6 \times 10^{-19}\) ]

Short answer

The cell potential of the given electrochemical cell at \(25^{\circ}\mathrm{C}\) is approximately \(0.65 V\).

Step-by-step solution

Calculate the concentration of \(Cu^{2+}\)

First, the solubility product expression for the reaction \(Cu(OH)_2 \rightleftharpoons Cu^{2+} + 2OH^-\) can be written as: \[K_{sp}=[Cu^{2+}][OH^-]^2\] Since the solubility product constant, \(K_{sp} = 1.6 \times 10^{-19}\), and the concentration of OH- ions is given as \(0.10 M "\mathrm{NaOH}"\), we can solve for the concentration of \(Cu^{2+}\): \[1.6 \times 10^{-19} = [Cu^{2+}] (0.10)^2\] \[ [Cu^{2+}]= \frac{1.6\times10^{-19}}{0.01}\] \[ [Cu^{2+}]= 1.6\times10^{-17} M\]

Calculate the reduction potential of copper electrode using Nernst equation

For the redox reaction: \[Cu^{2+}(aq) + 2e^- → Cu(s)\] We can use the Nernst equation to find the cell potential. The Nernst equation is given as: \[E_{cell}=E_{cell}^0 - \frac{0.0592}{n}\log{Q}\] Our reaction has \(n = 2\) electrons, and the standard cell potential for the reduction of Cu2+ to Cu is \(E_{cell}^0 = 0.34 V\). The reaction quotient, Q, can be formulated as: \[Q=[Cu^{2+}]\] Now, we can substitute the values calculated earlier in the Nernst Equation: \[E_{cell} = 0.34 - \frac{0.0592}{2}\log{1.6\times10^{-17}}\] Evaluating the expression gives us: \[E_{cell} \approx 0.65 V\]

Determine the overall cell potential

Now we can determine the overall cell potential as follows: \[E_{overall} = E_{copper} - E_{hydrogen}\] Since the standard hydrogen electrode has a reduction potential of 0 V: \[E_{overall} = E_{copper}\] Therefore, the overall cell potential at \(25^{\circ}\mathrm{C}\) is: \[E_{overall} \approx 0.65 V\]

Key concepts

Standard Hydrogen Electrode

An electrochemical cell relies on two electrodes, each dipped in an electrolyte. One of these, the

• standard hydrogen electrode (SHE)

is often used as a reference point. It has a defined potential of precisely 0 volts. This standardization helps compare different electrode potentials.

In an SHE, hydrogen gas at 1 atm pressure bubbles over a platinum electrode immersed in 1 M HCl, where hydrogen ions are present. It provides a clear baseline when measuring the potential of other electrodes, such as copper in this exercise.

Copper Electrode

The other part of our electrochemical cell involves a copper electrode. Copper is a common choice due to its

• good electrical conductivity and stability.

When copper is placed in a solution with dissolved ions, it can participate in redox (reduction-oxidation) reactions.

In this setup, the copper undergoes a reduction reaction: \[Cu^{2+}(aq) + 2e^- \rightarrow Cu(s)\]The copper ions gain electrons to become solid copper again. In this case, as the copper electrode reacts with hydroxide ions in solution, it aids in calculating the cell potential using the Nernst equation.

Nernst Equation

Electrochemical cell reactions are dynamic, and conditions may vary from standard states. That's where the Nernst equation shines. It's a formula that allows the calculation of cell potential based on actual conditions. The equation is: \[E_{cell} = E^0_{cell} - \frac{0.0592}{n}\log{Q}\]Here,

• \(E^0_{cell}\) is the standard cell potential,
• \(n\) is the number of electrons transferred, and
• \(Q\) is the reaction quotient.

For the copper electrode in our exercise, the Nernst equation adjusts potential from 0.34 V under given conditions, yielding about 0.65 V. It's a tool for fine-tuning potential predictions based on actual ion concentrations, which can differ significantly from 1 M.

Solubility Product

Understanding how substances dissolve is a must in electrochemistry. The solubility product constant

• \(K_{sp}\)

reflects how much of a compound can dissolve in a solution. For copper hydroxide, the equilibrium can be written as: \[Cu(OH)_2 \rightleftharpoons Cu^{2+} + 2OH^-\]The calculation in the exercise involves \(K_{sp} = 1.6 \times 10^{-19}\). Given a solution saturated with NaOH, you find \([Cu^{2+}]\). The concentration intricacies here are vital for accurate Nernst equation usage.

Cell Potential

Cell potential, or electromotive force (emf), tells us the voltage an electrochemical cell can provide. It depends on the potential difference between the two electrodes in the setup.

In this example, the copper electrode's adjusted potential, thanks to the Nernst equation, adds key insight. Paired with the standard hydrogen electrode's potential, which is set at 0 V, this setup's resulting potential, \(E_{overall} \approx 0.65\) V, reflects the copper's contribution solely.

The overall cell potential thus embodies the practical applicability of theoretical concepts to real laboratory and industrial cells, indicating a cell's efficiency and capabilities.