Problem 75 - Chemistry: Principles and Reactions

Question

Given the following data: $$ \begin{array}{cc} \mathrm{PtCl}_{4}{\underline{\phantom{xx}}}^{2-}(a q)+2 e^{-} \longrightarrow \mathrm{Pt}(s)+4 \mathrm{Cl}^{-}(a q) & E^{o}=+0.73 \mathrm{~V} \\ \mathrm{Pt}^{2+}(a q)+4 \mathrm{Cl}^{-}(a q) \longrightarrow \mathrm{PtCl}_{4}^{2-}(a q) & K_{f}=1 \times 10^{16} \end{array} $$ find \(E^{\circ}\) for the half-cell $$ \mathrm{Pt}^{2+}(a q)+2 e^{-} \longrightarrow \operatorname{Pt}(s) $$

Step-by-Step Solution

Answer

Answer: The standard reduction potential of Pt²⁺ in the half-cell reaction is approximately 1.18 V.

Step 1: 1. Overall Reaction

Write down the overall reaction by combining the given reactions: $$ \begin{array}{c} \mathrm{PtCl}_{4}^{2-}(a q)+2 e^{-} \longrightarrow \mathrm{Pt}(s)+4\mathrm{Cl}^{-}(a q)\\ \hline \mathrm{Pt}^{2+}(a q)+4 \mathrm{Cl}^{-}(a q) \longrightarrow \mathrm{PtCl}_{4}^{2-}(a q) \end{array} $$ Adding the above two reactions: $$ \mathrm{Pt}^{2+}(a q)+2 e^{-} \longrightarrow \operatorname{Pt}(s) $$ This is the half-cell reaction we need to find the standard reduction potential for.

Step 2: 2. Calculate the equilibrium constant for reduction half-cell reaction

Since we know the formation constant of the overall reaction, we can calculate the equilibrium constant (K_c) for the half-cell reaction using the formula: $$ K_c = \frac{K_f}{E^\circ} $$ Where \(K_f\) is the formation constant, and \(E^\circ\) is the standard reduction potential of the given half-cell reaction. For the half-cell reaction: $$ \mathrm{PtCl}_{4}^{2-}(a q)+2 e^{-} \longrightarrow \mathrm{Pt}(s)+4\mathrm{Cl}^{-}(a q) $$ Using the given values: $$ K_c = \frac{1 \times 10^{16}}{0.73} $$ $$ K_c \approx 1.37 \times 10^{16} $$

Step 3: 3. Use the Nernst Equation to find the standard potential

Now, we will use the Nernst Equation which relates the standard potential (Eº) of a half-cell reaction, the number of electrons involved (n), the equilibrium constant (K_c), and the standard reduction potential (Eº) of the half-cell reaction we want to find. The equation is: $$ 0 = E^°_{Pt^{2+}/Pt} - \frac{0.0592}{n} \log{K_c} $$ We already have the value of K_c, and in our half-cell reaction, 2 electrons are transferred (n=2). Inserting the values into the equation: $$ 0 = E^°_{Pt^{2+}/Pt} - \frac{0.0592}{2} \log{(1.37 \times 10^{16})} $$ Now solve for \(E^°_{Pt^{2+}/Pt}\): $$ E^°_{Pt^{2+}/Pt} = \frac{0.0592}{2} \log{(1.37 \times 10^{16})} $$ $$ E^°_{Pt^{2+}/Pt} \approx 1.18~V $$ So the standard reduction potential for the half-cell reaction: $$ \mathrm{Pt}^{2+}(a q)+2 e^{-} \longrightarrow \operatorname{Pt}(s) $$ is \(E^°_{Pt^{2+}/Pt} \approx 1.18 V\).

Key Concepts

Standard Reduction Potential

In electrochemistry, the standard reduction potential of a half-cell is a crucial concept. It reflects the inherent tendency of a substance to gain electrons, i.e., to be reduced, in an electrochemical reaction. The standard reduction potential is measured under standard conditions, which are generally set at 25°C, 1 M concentration for all aqueous solutes, and a pressure of 1 atm for any gases involved. This measurement is taken relative to the standard hydrogen electrode (SHE), which is defined as having a potential of exactly 0.00 volts.

Reduction potentials are fundamentally important because they allow us to predict the direction of electron flow in electrochemical cells. For example, if a half-cell has a higher reduction potential than another, it will have a greater ability to attract or gain electrons.

• A positive standard reduction potential indicates a strong tendency to gain electrons and be reduced.
• The more positive the potential, the greater the substance’s affinity for electrons.

In the exercise provided, the goal was to find the standard reduction potential for a specific half-cell reaction, which eventually led to a calculated value of approximately 1.18 V. This indicates a strong tendency for the Pt²⁺ ions in the solution to gain electrons and be reduced to Pt metal.

Nernst Equation

The Nernst Equation is a fundamental equation in electrochemistry that relates the cell potential to the standard reduction potential, the temperature, the concentration of reactants, and the number of electrons transferred in the half-cell reactions. The equation is usually stated as:

\[ E = E^° - \frac{0.0592}{n} \log{Q} \]

Where:

• \(E\) is the cell potential under non-standard conditions.
• \(E^°\) is the standard reduction potential.
• \(n\) is the number of moles of electrons transferred.
• \(Q\) is the reaction quotient, which is essentially the ratio of the concentrations of products over reactants.

The Nernst Equation helps to calculate the actual cell potential when conditions are not standard. It essentially adjusts the potential based on concentration changes from the standard 1 M, taking into account the effect of temperature and pressure.

In the solution of the exercise, the Nernst equation was used to link the known equilibrium constant with the unknown standard reduction potential. By rearranging the formula, the standard reduction potential for the Pt(²⁺)/Pt(s) reduction was determined. The Nernst Equation allows us to understand how electrochemical cells perform in real-life conditions outside the standard laboratory environment.

Equilibrium Constant

The equilibrium constant, \(K\), is a dimensionless number that gives insight into the position of equilibrium in a chemical reaction. It is computed as the ratio of the concentrations of the products to the reactants, each raised to the power of their respective coefficients in the balanced equation. The larger the equilibrium constant, the more the reaction favors the formation of products.

• If \(K\) is much greater than 1, the reaction is product-favored, indicating that, at equilibrium, the concentration of products is substantially higher than that of reactants.
• Conversely, if \(K\) is much less than 1, the reaction is reactant-favored, with reactants being present in a higher concentration at equilibrium.

In electrochemical reactions, the equilibrium constant directly connects to the cell potential through the Nernst Equation. For instance, in this exercise, the given formation constant \(K_f = 1 \times 10^{16}\) was used to calculate the equilibrium constant for the electrochemical reaction.

By deciphering \(K\) in conjunction with the Nernst Equation, we arrived at discovering the required standard reduction potential. A high \(K_f\) signified a very product-favored reaction, confirming that the conversion of Pt²⁺ to Pt is favorable under standard conditions.