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 $\left[{\mathrm{Cu}}^{2+}\right]=$ $2.5\times {10}^{-4}\mathrm{M}?$ 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

The cell potential in part a is approximately $0.308V$, and the unknown concentration of copper ions in the solution for part b is approximately $80M$.

Step-by-step solution

Gather given information and convert temperature to Kelvin

Given values: - Standard hydrogen electrode - Copper metal electrode - Temperature: ${25}^{\circ }C=298K$ For part a: - $[{\mathrm{Cu}}^{2+}]=2.5\times {10}^{-4}M$ For part b: - Measured potential: $0.195V$ - Unknown $[{\mathrm{Cu}}^{2+}]$

Write down the Nernst equation

The Nernst equation is as follows: $$E=E^{\circ }-\frac{RT}{nF}\times \ln Q$$ where - E: cell potential - $E^{\circ }$: standard cell potential - R: universal gas constant ($8.314J{K}^{-1}mo{l}^{-1}$) - T: temperature in Kelvin - n: number of moles of electrons transferred - F: Faraday's constant ($96,485Cmo{l}^{-1}$) - Q: reaction quotient

Calculate E for part a, given the Cu(2+) concentration

Using the standard reduction potentials, we can determine the standard cell potential for the given cell configuration. Standard reduction potential for hydrogen: $E_{H+}^{\circ }=0.000V$ Standard reduction potential for copper: $E_{C{u}^{2+}/Cu}^{\circ }=0.34V$ $E_{cell}^{\circ }=E_{C{u}^{2+}/Cu}^{\circ }-E_{H+}^{\circ }=0.34V$ For the cell reaction, the reduction half-cell reaction involves two electrons: $${\mathrm{Cu}}^{2+}+2e^{-}\to \mathrm{Cu}$$ So, $n=2$, and we have a concentration of $[{\mathrm{Cu}}^{2+}]=2.5\times {10}^{-4}M$ Now, let's plug all these values into the Nernst equation: $$E_{cell}=0.34-\frac{(8.314)(298)}{(2)(96485)}\times \ln (2.5\times {10}^{-4})$$

Solve for the cell potential E for part a

$$E_{cell}=0.34-\frac{(8.314)(298)}{(2)(96485)}\times \ln (2.5\times {10}^{-4})\approx 0.308V$$ So, the cell potential in part a is $0.308V$.

Setup Nernst equation for part b

We already know the measured potential for part b (it's given as $0.195V$), so E can be replaced with this value. $$0.195=0.34-\frac{(8.314)(298)}{(2)(96485)}\times \ln [{\mathrm{Cu}}^{2+}]$$ Then, isolate the unknown concentration term $[{\mathrm{Cu}}^{2+}]$ to solve for it: $$\ln [{\mathrm{Cu}}^{2+}]=\frac{(0.34-0.195)(2)(96485)}{(8.314)(298)}$$

Solve for the unknown Cu(2+) concentration in part b

First, compute the value of the logarithm: $$\ln [{\mathrm{Cu}}^{2+}]=\frac{(0.34-0.195)(2)(96485)}{(8.314)(298)}\approx 4.38$$ Next, exponentiate both sides to find the concentration: $$[{\mathrm{Cu}}^{2+}]=e^{4.38}\approx 80M$$ Thus, the unknown concentration of copper ions in the solution for part b is approximately $80M$.

Key concepts

Nernst Equation

The Nernst Equation is a fundamental principle used to calculate the cell potential under non-standard conditions in electrochemical cells. This equation relates the cell potential at any given conditions to the standard cell potential, temperature, and the activities or concentrations of the chemical species involved. It is expressed as follows:

$$E=E^{\circ }-\frac{RT}{nF}\times \ln Q$$

In this equation:

• $E$ is the cell potential.
• $E^{\circ }$ is the standard cell potential, determined under standard conditions (1 atm, 1 M, 25°C).
• $R$ is the universal gas constant (8.314 J K^-1 mol^-1).
• $T$ is the temperature in Kelvin.
• $n$ is the number of moles of electrons transferred in the balanced equation for the cell reaction.
• $F$ is Faraday's constant (96,485 C mol^-1).
• $Q$ is the reaction quotient, which reflects the ratio of concentrations of products to reactants.

Using the Nernst Equation allows us to adjust the cell potential for real-world applications where conditions often deviate from standards.

Standard Hydrogen Electrode

The Standard Hydrogen Electrode (SHE) plays a vital role as a reference point in electrochemistry. It is used to set a baseline for measuring the electrode potentials of other substances. The SHE consists of a platinum electrode in contact with 1M H₊ ions and hydrogen gas at a pressure of 1 atmosphere.

This electrode sets the potential of the hydrogen half-reaction:

• $2H^{+}+2e^{-}\to H_2$, $E^{\circ }=0.000V$.

This zero value makes it an ideal benchmark to compare with other half-cell potentials.For example, in the given exercise, the SHE was paired with a copper electrode to evaluate the potential of the electrochemical cell. By knowing the potentials of the reactions involved, we can calculate the cell potential using the Nernst Equation.

Concentration Calculation

Calculating the concentration of ions in a solution is crucial for determining the cell potential using the Nernst Equation. Concentrations directly affect the reaction quotient $Q$, which in turn influences the cell's potential.

For part (b) of the original exercise, given a measured potential, we utilized the Nernst Equation:
$$0.195=0.34-\frac{(8.314)(298)}{(2)(96485)}\times \ln [C{u}^{2+}]$$
This involves transferring the terms to isolate $[C{u}^{2+}]$:

Rearrange the equation to find $\ln [C{u}^{2+}]$:
$$\ln [C{u}^{2+}]=\frac{(0.34-0.195)(2)(96485)}{(8.314)(298)}$$

Once we solve for the logarithm, we exponentiate to get the concentration:
$$[C{u}^{2+}]=e^{4.38}\approx 80M$$.

This demonstrates how potential measurements can assist in finding unknown concentrations in electrochemical systems.

Standard Reduction Potential

Standard Reduction Potential, often denoted as $E^{\circ }$, is a measure of the tendency of a chemical species to gain electrons and thereby be reduced. It is determined under standard conditions (1M concentration, 1 atm pressure, and 25°C temperature).

This value is crucial in calculating the overall cell potential, as the standard reduction potential for each half-cell reaction is combined to form the standard cell potential $E_{cell}^{\circ }$.

• For instance, in our exercise, the standard reduction potential for copper: $E_{C{u}^{2+}/Cu}^{\circ }=0.34V$.
  This value was used along with the potential for the SHE: $E_{H+}^{\circ }=0.000V$ to calculate the standard cell potential:
  $$E_{cell}^{\circ }=E_{C{u}^{2+}/Cu}^{\circ }-E_{H+}^{\circ }=0.34V$$.

Recognizing these values enables the use of the Nernst Equation to determine potentials under various conditions, enhancing our understanding of electrochemical cells.