Question

Show that the oxidation of \(\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{2+}\) to \(\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}\) referred to on page 1098 should occur spontaneously in alkaline solution with \(\mathrm{H}_{2} \mathrm{O}_{2}\) as an oxidizing agent.

Short answer

The spontaneity of the oxidation of \(\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{2+}\) to \(\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}\) in an alkaline solution with \(\mathrm{H}_{2} \mathrm{O}_{2}\) as an oxidizing agent can be shown by writing the half-reactions, combining them into the redox reaction, and using the Nernst equation to calculate the redox potential. If the potential is positive, the reaction is spontaneous.

Step-by-step solution

Write the Half-reactions

Begin by writing the half-reactions. For the oxidation of \(\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{2+}\), the half-reaction is: \[ \left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{2+} → \left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+} + e^- \]. For the reduction of \(\mathrm{H}_{2} \mathrm{O}_{2}\), in alkaline conditions, the half-reaction is: \[ \mathrm{H}_{2} \mathrm{O}_{2} + 2 e^- → 2 OH^- \]

Combine the Half-reactions

Combine the two half-reactions to form the overall redox reaction. Ensure that the electrons are balanced. This yields: \[ \left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{2+} + \mathrm{H}_{2} \mathrm{O}_{2} → \left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+} + 2 OH^- \]

Use the Nernst Equation

The Nernst equation is used to calculate the redox potential of a half-cell in an electrochemical cell. It is given by: \[ E = E^\circ - \frac{RT}{nF} \ln Q \] where \(E\) is the cell potential, \(E^\circ\) 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 redox reaction, \(F\) is the Faraday's constant and \(Q\) is the reaction quotient. The values of \(E^\circ\) for the half-reactions can be obtained from standard redox potential tables. Calculate \(E\) for the reaction and if \(E>0\), the reaction is spontaneous.

Determine Spontaneity

If the calculated cell potential (\(E\)) is positive, the reaction is spontaneous. If \(E=0\), the reaction is in equilibrium. If \(E<0\), the reaction is non-spontaneous and will not occur under standard conditions.

Key concepts

Half-Reactions

When dealing with redox reactions, it's crucial to break down the overall process into half-reactions. Each half-reaction represents either the oxidation or the reduction process separately.
By doing so, we can easily track the movement of electrons and balance the charges. For the scenario where - * we are oxidizing \(\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{2+}\) - * to \(\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}\), we write this half-reaction as:\[ \left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{2+} \rightarrow \left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+} + e^- \]
**In the reduction half-reaction**, * we consider \(\mathrm{H}_{2} \mathrm{O}_{2}\) * which in an alkaline solution reduces, gaining electrons as shown:\[ \mathrm{H}_{2} \mathrm{O}_{2} + 2 e^- \rightarrow 2 OH^- \]Combining these half-reactions ensures that they are balanced not only in terms of mass but also in charge, leading to a comprehensive understanding of the complete redox equation.

Nernst Equation

To determine the spontaneity of a redox reaction, the Nernst equation is an essential tool. It's used to evaluate the cell potential under non-standard conditions.
Given by the equation:\[ E = E^\circ - \frac{RT}{nF} \ln Q \]We break down each part:- **\(E\)** represents the actual cell potential we're interested in.- **\(E^\circ\)** is the standard cell potential, often available in tables.- **\(R\)** stands for the universal gas constant ( 8.314 J (mol·K)^{-1} ).- **\(T\)** is the temperature (measured in Kelvin).- **\(n\)** indicates the moles of electrons transferred.- **\(F\)** corresponds to Faraday's constant ( 96485 C mol^{-1} ).- **\(Q\)** is the reaction quotient, derived from the concentrations of the reactants and products.The Nernst equation allows predicting how the cell potential changes with concentration changes, and thus deduces whether a given reaction under particular conditions will proceed spontaneously.

Reaction Spontaneity

The spontaneity of a chemical reaction, especially redox reactions, is an indicator of whether the reaction will occur on its own under given conditions. This is determined primarily by evaluating the cell potential \(E\).
Here's a simple breakdown:- **If \(E > 0\):**The reaction is spontaneous, meaning it can naturally proceed without added external energy.- **If \(E = 0\):**The reaction is at equilibrium; there is no net change as the forward and reverse reactions occur at the same rate.- **If \(E < 0\):**The reaction is non-spontaneous. To proceed, it would require an input of energy from an external source.In our context, the measure of spontaneity also corroborates with Le Châtelier's Principle, as any changes in concentration, pressure, or temperature will shift the reaction towards equilibrium, suggesting the direction of the spontaneous reaction.