Question

A Cu electrode is immersed in a solution that is \(1.00 \mathrm{M} \mathrm{NH}_{3}\) and \(1.00 \mathrm{M}\) in \(\left[\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}\right]^{2+} .\) If a standard hydrogen electrode is the cathode, \(E_{\text {cell }}\) is \(+0.08 \mathrm{V} .\) What is the value obtained by this method for the formation constant, \(K_{f},\) of \(\left[\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}\right]^{2+} ?\)

Short answer

The formation constant (\(K_f\)) for the complex \(\left[\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}\right]^{2+}\) is \(3.16 * 10^{17}\).

Step-by-step solution

Understanding the Nernst Equation

First, understanding and writing down the Nernst equation, which gives us the relation between the Gibbs free energy (\(ΔG\)) and the cell potential (\(E_{\text {cell}}\)). It can be written as: \[ ΔG = -nFE_{\text {cell}} \] where \(n\) is the number of moles of electons transferred (2 in this case as we are dealing with a copper (II) complex) and \(F\) is the Faraday's constant (\(96485 \, C \, mol^{-1}\)).

Calculating Gibbs free energy

Plug the given cell potential (\(E_{\text {cell}}\)) and the values for \(n\) and \(F\) into the Nernst Equation to calculate \(ΔG\). This comes out to be: \[ ΔG = -2 * 96485 * 0.08 \, J = -15437.6 \, J = -15.44 \, kJ\]

Using the Gibbs free energy change in the formation constant equation

The equation for the formation constant (\(K_f\)) in terms of \(ΔG\) is \(ΔG = -RTlnK_f\), where \(R\) is the gas constant (\(8.314 J K^{-1} mol^{-1}\)). Rearranging this equation, we get \(K_f = e^{-ΔG/RT}\). Substituting the given values, we obtain: \[ K_f = e^{-(-15.44 \, kJ)/(8.314 * 298)} = 3.16 * 10^{17}\]

Key concepts

Nernst Equation

The Nernst Equation is the cornerstone of electrochemistry, which expresses the relationship between the reduction potential of a chemical reaction and its concentration gradient. It's incredibly useful for predicting the direction of redox reactions and calculating cell potentials under non-standard conditions. Essentially, the equation links thermodynamic quantities at the micro level with observable electrochemical phenomena.

In our example, we use the Nernst Equation to correlate the cell potential, the number of electrons transferred, and the Gibbs free energy, thus providing a quantifiable expression of electrochemical cell efficiency. This quantification is essential for students to understand how changes at the molecular level can directly influence a measurable voltage in a system.

In practical application, the equation is traditionally written as: \[ E = E^0 - \frac{RT}{nF} \ln\frac{\text{Products}}{\text{Reactants}} \] where:

• \(E\) is the cell potential,
• \(E^0\) 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 exchanged in the redox reaction, and
• \(F\) is Faraday's constant.

By understanding this equation, students can determine how potential energy within a reaction varies with concentration, which is pivotal when dealing with any electrochemical system.

Gibbs Free Energy

When addressing problems in electrochemistry, the concept of Gibbs Free Energy (\(ΔG\)) is inseparable from the discussion. It represents the maximum amount of work that a thermodynamic system can perform at constant temperature and pressure, and essentially dictates the spontaneity of reactions; negative \(ΔG\) values suggest that a reaction is favorable and will occur without external input.

Our textbook problem demonstrates the practical use of \(ΔG\) when calculating the energy change associated with an electrochemical cell. By determining this value, students can infer whether energy must be supplied for a reaction to proceed or if it will release energy. In fact, \(ΔG\) can be directly related to the cell potential through the Nernst Equation:
\[ ΔG = -nFE_{\text{cell}} \]
This equation implies that energy change is proportional to both the number of electrons transferred and the cell potential, which is particularly helpful when reinforcing the connection between electrochemistry and thermodynamics in educational settings.

Formation Constant

The formation constant (\(K_f\)), which appears in our exercise, reflects the stability of a complex in solution, shedding light on the affinity between the metal ion and its ligands. It is essentially a special type of equilibrium constant used for complex ions. A higher \(K_f\) value indicates a more stable or favored complex formation.

The way to calculate the formation constant from Gibbs free energy is via the equation:\[ ΔG = -RT\ln{K_f} \]
When \(ΔG\) is calculated from electrochemical data, as we've seen with the Nernst Equation, we can rework this relationship to isolate \(K_f\). The exercise illustrates that by knowing the Gibbs free energy, we can determine the equilibrium properties of a system, such as the formation constant.\[ K_f = e^{-ΔG/RT} \]
This approach makes it tangible for students to understand how macroscopic measurements, such as cell potential, translate to intrinsic properties of chemical species, like the formation constant of a complex ion. It encapsulates the predictive power of electrochemistry and its applications in fields ranging from metallurgy to environmental science.