Question

We have seen that complex formation can stabilize oxidation states. An important illustration of this fact is the oxidation of water in acidic solutions by \(\mathrm{Co}^{3+}(\) aq) but not by \(\left[\mathrm{Co}(\mathrm{en})_{3}\right]^{3+} .\) Use the following data. \(\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}+\mathrm{e}^{-} \longrightarrow\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}\) $$ E^{\circ}=1.82 \mathrm{V} $$ \(\left[\operatorname{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}+3 \mathrm{en} \longrightarrow\left[\mathrm{Co}(\mathrm{en})_{3}\right]^{2+}+6 \mathrm{H}_{2} \mathrm{O}(1)\) $$ \log \beta_{3}=12.18 $$ \(\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}+3 \mathrm{en} \longrightarrow\left[\mathrm{Co}(\mathrm{en})_{3}\right]^{3+}+6 \mathrm{H}_{2} \mathrm{O}(1)\) $$ \log \beta_{3}=47.30 $$ Calculate \(E^{\circ}\) for the reaction $$ \left[\mathrm{Co}(\mathrm{en})_{3}\right]^{3+}+\mathrm{e}^{-} \longrightarrow\left[\mathrm{Co}(\mathrm{en})_{3}\right]^{2+} $$ Show that \(\left[\mathrm{Co}(\mathrm{en})_{3}\right]^{3+}\) is stable in water but \(\mathrm{Co}^{3+}(\mathrm{aq})\) is not.

Short answer

After computation, we obtain the standard potential \(E^{\circ}\) for the complex reduction reaction \(\left[\mathrm{Co}(\mathrm{en})_{3}\right]^{3+}\rightarrow \(\left[\mathrm{Co}(\mathrm{en})_{3}\right]^{2+}\). Comparison of this value with the provided value of \(E^{\circ}\) for the \(\mathrm{Co}^{3+}\) ion reveals that the \(\left[\mathrm{Co}(\mathrm{en})_{3}\right]^{3+}\) ion is more stable in water, while \(\mathrm{Co}^{3+}\) is not.

Step-by-step solution

Calculate \(E^{\circ}\) for the complex compound reduction reaction

Use the Nernst equation for calculating \(E^{\circ}\) values. This equation \(\Delta G = -nFE^{\circ}\) links the Gibbs free energy change \(\Delta G\) to the cell potential \(E^{\circ}\). Using \(\Delta G = -RT\ln(K)\) equation, where \(K\) is the equilibrium constant. In this case, the equilibrium constant is given by \(10^{\log \beta_3}\). Hence, \(E^{\circ} = -\frac{RT}{nF}\ln(K)\), substituting for \(R =8.314\ JK^{-1}mol^{-1}\), \(T = 298\ K\), \(F = 96485\ Cmol^{-1}\), \(n=1\) and \(\log \beta_3 = 47.30\), you can calculate the value of \(E^{\circ}\).

Compare the stability of cobalt III complexes in water

The difference in oxidation stability for the \(\mathrm{Co}^{3+}\) and \(\left[\mathrm{Co}(\mathrm{en})_{3}\right]^{3+}\) ions relies on the comparison of their \(E^{\circ}\) values. A more positive \(E^{\circ}\) indicates a greater tendency for reduction to occur, hence making the ion less stable in its oxidized state. Comparing the \(E^{\circ}\) values, it can be shown how \(\left[\mathrm{Co}(\mathrm{en})_{3}\right]^{3+}\) is more stable than \(\mathrm{Co}^{3+}\).

Key concepts

Oxidation States

Oxidation states are essential in determining how atoms exchange electrons during chemical reactions. Often depicted as the charge of an ion, an oxidation state reflects the number of electrons an atom has lost or gained relative to its elemental state. For instance, in the exercise we are considering cobalt ions: \(\mathrm{Co}^{2+}\) and \(\mathrm{Co}^{3+}\). The oxidation states here indicate a loss of two and three electrons, respectively. Understanding these states helps us predict how a given substance will interact with others in chemical reactions. When cobalt forms complexes, the oxidation state can be stabilized by these complexes. This is evident when comparing how \(\mathrm{Co}^{3+}(aq)\) and \(\left[\mathrm{Co}(\mathrm{en})_{3}\right]^{3+}\) behave in water. The stable configuration of the electron orbitals due to coordination with ligands such as ethylenediamine (en) plays a pivotal role here. In summary, oxidation states offer insight into the electron transfer potential, which is crucial for understanding complex formation and stability.

Nernst Equation

The Nernst Equation is a powerful tool for calculating cell potentials and understanding electrochemical systems. It describes the relationship between the cell potential \(E\), the standard cell potential \(E^{\circ}\), temperature, and the reaction quotient. The equation in its simplified form reads: \[ E = E^{\circ} - \frac{RT}{nF} \ln Q \]Where:

• \(E^{\circ}\) is the standard electrode potential.
• \(R\) is the universal gas constant.
• \(T\) is the temperature in Kelvin.
• \(n\) is the number of moles of electrons transferred in the reaction.
• \(F\) is Faraday's constant.
• \(Q\) is the reaction quotient or equilibrium constant.

In the given exercise, this equation helps us compute \(E^{\circ}\) for the reaction involving \(\left[\mathrm{Co}(\mathrm{en})_{3}\right]^{3+}\) to \(\left[\mathrm{Co}(\mathrm{en})_{3}\right]^{2+}\). We substitute known values like \(\log \beta_3\) into the equation to find the needed standard potential. This step is necessary to compare the stability of different cobalt complexes in aqueous environments.

Stability of Cobalt Complexes

Cobalt complexes often show varied stability depending on the ligands coordinating the cobalt metal. This principle is at play when observing the stability of \(\left[\mathrm{Co}(\mathrm{en})_{3}\right]^{3+}\) versus \(\mathrm{Co}^{3+}(aq)\). The complex with ethylenediamine (en) ligands forms a tightly-bound structure that stabilizes the higher oxidation state of cobalt. The stability of a metal complex in solution is often determined by the equilibrium constants such as \(\log \beta_n\) values. These values represent the affinity between metal ions and ligands, where a higher number means a more stable complex. In our case, the \(\log \beta_3\) for the \(\left[\mathrm{Co}(\mathrm{en})_{3}\right]^{3+}\) complex is significantly higher than the simple \(\mathrm{Co}^{3+}(aq)\), indicating greater stability.

• Greater the \(\log \beta\) value, the more stable the complex.
• Complex formation can alter the redox potential, as evidenced in the given exercise.

The stability speaks to why certain cobalt species resist reduction or oxidation in different environmental conditions. These concepts are foundational for understanding transition metal chemistry and how different ligands impact metal ion behavior.