Question

An electrochemical cell consists of a standard hydrogen electrode and a copper metal electrode. a. What is the potential of the cell at \(25^{\circ} \mathrm{C}\) if the copper electrode is placed in a solution in which \(\left[\mathrm{Cu}^{2+}\right]=2.5 \times 10^{-4} M ?\) b. The copper electrode is placed in a solution of unknown \(\left[\mathrm{Cu}^{2+}\right]\). The measured potential at \(25^{\circ} \mathrm{C}\) is \(0.195 \mathrm{~V}\). What is \(\left[\mathrm{Cu}^{2+}\right] ?\) (Assume \(\mathrm{Cu}^{2+}\) is reduced.)

Short answer

a. The potential of the cell at \(25^{\circ} \mathrm{C}\) with \(\left[\mathrm{Cu}^{2+}\right]=2.5 \times 10^{-4} \, \mathrm{M}\) is \(0.52 \, \mathrm{V}\). b. The concentration of \(\mathrm{Cu}^{2+}\) when the measured potential at \(25^{\circ} \mathrm{C}\) is \(0.195 \mathrm{~V}\) is \(8.57 \times 10^{-4} \, \mathrm{M}\).

Step-by-step solution

Part a: Calculate the cell potential at given concentration

. To calculate the cell potential at the given concentration of \(\mathrm{Cu}^{2+}\), we will use the Nernst equation: \(E_{cell} = E^{\circ}_{cell} - \frac{RT}{nF} \ln Q\), where \(E_{cell}\) = cell potential, \(E^{\circ}_{cell}\) = standard cell potential, \(R\) = gas constant = \(8.314 \, \mathrm{J /(mol \cdot K)}\), \(T\) = temperature in Kelvin, \(n\) = number of electrons exchanged, \(F\) = Faraday's constant = \(96485 \, \mathrm{C/mol}\), \(Q\) = reaction quotient. First, we need to write a balanced chemical equation for the given cell: \( \mathrm{2H^{+} + 2e^{-} \rightleftharpoons H_{2}}\) (Hydrogen half-cell reaction), \( \mathrm{Cu^{2+} + 2e^{-} \rightleftharpoons Cu}\) (Copper half-cell reaction). Now we can find the standard cell potential (\(E^{\circ}_{cell}\)) by considering the reduction potentials of each half-cell: For hydrogen, \(E^{\circ}(\mathrm{H}^{+}/\mathrm{H}_{2}) = 0 \, \mathrm{V}\), For copper, \(E^{\circ}(\mathrm{Cu}^{2+}/\mathrm{Cu}) = +0.34 \, \mathrm{V}\). Therefore, the standard cell potential (\(E^{\circ}_{cell}\)) is given by: \(E^{\circ}_{cell} = E^{\circ}(\mathrm{Cu}^{2+}/\mathrm{Cu}) - E^{\circ}(\mathrm{H}^{+}/\mathrm{H}_{2}) = 0.34 \, \mathrm{V}\) Now using the Nernst equation, we can obtain the cell potential (\(E_{cell}\)): \(E_{cell} = 0.34 - \frac{(8.314)(298)}{(2)(96485)} \ln \frac{[H^{+}]^{2}}{[Cu^{2+}]} \), We are given that the concentration of \([\mathrm{Cu}^{2+}]\) is \(2.5 \times 10^{-4} \, \mathrm{M}\) and we can assume the concentration of acidic protons \([\mathrm{H}^{+}]\) as 1 M since it's a standard hydrogen electrode: Now substitute the given values into the equation: \( \\ E_{cell} = 0.34 - \frac{(8.314)(298)}{(2)(96485)} \ln \frac{(1)^{2}}{2.5 \times 10^{-4}} \\ \\ E_{cell} = 0.34 - \frac{0.042929}{2} \ln \frac{1}{2.5 \times 10^{-4}} \\ \\ E_{cell} = 0.34 + 0.021465 \cdot 8.3 \\ \\ E_{cell} = 0.52 \, \mathrm{V} \) So, the potential of the cell at \(25^{\circ} \mathrm{C}\) is \(0.52 \, \mathrm{V}\).

Part b: Calculate the concentration of \(\mathrm{Cu}^{2+}\) at given potential

. Here, we are given the cell potential. We will use the Nernst equation again and solve for the concentration of \(\mathrm{Cu}^{2+}\). Given cell potential (\(E_{cell}\)) = 0.195 V We have the Nernst equation from the previous part: \(E_{cell} = 0.34 - \frac{(8.314)(298)}{(2)(96485)}\ln \frac{[H^{+}]^{2}}{[Cu^{2+}]}\), and substituting the cell potential: \(0.195 = 0.34 - \frac{(8.314)(298)}{(2)(96485)} \ln \frac{(1)^{2}}{[Cu^{2+}]}\), Now, moving forward, we have to solve for the concentration of \(\mathrm{Cu}^{2+}\): \[ \begin{align*} \frac{(8.314)(298)}{(2)(96485)} \ln \frac{1}{[Cu^{2+}]} &= 0.34 - 0.195 \\ -0.021465 \ln \frac{1}{[Cu^{2+}]} &= 0.145 \\ \ln \frac{1}{[Cu^{2+}]} &= -6.75 \\ \frac{1}{[Cu^{2+}]} &= e^{-6.75} \\ [Cu^{2+}] &= \frac{1}{e^{-6.75}} \end{align*} \] Now, calculating the concentration of \(\mathrm{Cu}^{2+}\): \([Cu^{2+}] = \frac{1}{e^{-6.75}} = 8.57 \times 10^{-4} \, \mathrm{M}\) Thus, the concentration of \(\mathrm{Cu}^{2+}\) in the solution is \(8.57 \times 10^{-4} \, \mathrm{M}\).

Key concepts

Understanding the Nernst Equation

The Nernst equation is a fundamental tool in electrochemistry for determining the potential of an electrochemical cell under non-standard conditions. Given a reaction at equilibrium, the Nernst equation directly relates the reduction potential of the electrochemical cell to the concentration of the participating species.

Its mathematical form is often expressed as: \[E_{cell} = E^{\text{o}}_{cell} - \frac{RT}{nF} \ln Q\], where \(E_{cell}\) is the cell potential, \(E^{\text{o}}_{cell}\) is the standard cell potential, \(R\) is the universal gas constant, \(T\) is the temperature in Kelvin, \(n\) is the number of moles of electrons transferred in the electrochemical reaction, \(F\) is Faraday's constant, and \(Q\) is the reaction quotient reflecting the concentrations of the reactants and products.

Please note that the logarithm used in the Nernst equation is the natural logarithm (ln), which may sometimes be expressed as log to the base e. When applying the Nernst equation, it is essential to handle the sign of the electrode potentials carefully. Understanding this formula helps students delve into understanding how the concentrations of ionic species and temperature affect cell potential, aiding in grasping more complex electrochemical concepts.

Standard Hydrogen Electrode Explained

The standard hydrogen electrode (SHE) is the benchmark by which all other electrode potentials are compared. It serves as the reference point and has a defined potential of 0 volts. The SHE is based on the redox reaction involving hydrogen gas \(H_2\) and hydrogen ions \(H^+\) in solution: \[2H^+ (aq) + 2e^- \rightleftharpoons H_2 (g)\].

In practice, it consists of a platinum electrode in contact with 1 M \(H^+\) and bathed by hydrogen gas at 1 atm pressure. The 'standard' conditions imply a temperature of \(25^\circ C\), which is also equivalent to 298 K. It's important to understand that the use of standard conditions allows chemists to compare different electrochemical cells objectively. In the context of textbook problems, students are often asked to use SHE as the reference electrode while calculating the standard cell potential or when using the Nernst equation to complete their exercises.

Copper Electrode Reduction Potential

Electrode reduction potential measures the tendency of a chemical element to acquire electrons and be reduced. Each half-cell has its reduction potential, and for the copper electrode, the standard reduction potential is given by the reaction: \[Cu^{2+} (aq) + 2e^- \rightleftharpoons Cu (s)\].

The standard reduction potential for the copper electrode (\(E^{\text{o}}\)) is \(+0.34 V\). This value represents the potential difference that would occur if the copper half-cell were connected to the SHE under standard conditions.

When a student is faced with a problem involving copper in an electrochemical cell, they must consider the copper's reduction potential to calculate the overall cell potential. Moreover, when conditions are not standard, as in the example provided in the textbook exercise, the Nernst equation incorporates the copper electrode's standard potential and adjusts it according to changes in concentration or temperature. Understanding the standard electrode potential concept is crucial for predicting the electrochemical behavior of elements in different scenarios and is widely applied in fields such as battery design, corrosion studies, and electroplating.