Question

The black silver sulfide discoloration of silverware can be removed by heating the silver article in a sodium carbonate solution in an aluminum pan. The reaction is $$3 \mathrm{Ag}_{2} \mathrm{~S}(s)+2 \mathrm{Al}(s) \rightleftharpoons 6 \mathrm{Ag}(s)+3 \mathrm{~S}^{2-}(a q)+2 \mathrm{Al}^{3+}(a q)$$ a. Using data in Appendix 4 , calculate \(\Delta G^{\circ}, K\), and \(\mathscr{C}^{\circ}\) for the above reaction at \(25^{\circ} \mathrm{C}\). (For \(\mathrm{Al}^{3+}(a q), \Delta G_{\mathrm{f}}^{\circ}=-480 . \mathrm{kJ} / \mathrm{mol}\).) b. Calculate the value of the standard reduction potential for the following half-reaction: $$2 \mathrm{e}^{-}+\mathrm{Ag}_{2} \mathrm{~S}(s) \longrightarrow 2 \mathrm{Ag}(s)+\mathrm{S}^{2-}(a q)$$

Short answer

The Gibbs free energy change of the reaction is \(\Delta G^{\circ} \approx -1042.2\,\mathrm{kJ/mol}\). The equilibrium constant for the reaction at \(25^{\circ}\mathrm{C}\) is \(K \approx 1.46 \times 10^{32}\). The standard concentration quotient is \(\mathscr{C}^{\circ} = \frac{K_p}{(\mathrm{S}^{2-})^3 (\mathrm{Al}^{3+})^2}\). The standard reduction potential for the given half-reaction is \(E^{\circ} \approx 0.467\,\mathrm{V}\).

Step-by-step solution

Find the Gibbs free energy change of the reaction

First, let's calculate the \(\Delta G^{\circ}\) for the given reaction using the below formula, \[\Delta G^{\circ} = \sum{n_i \Delta G_i^{\circ}}\] Where \(n_i\) is the stoichiometric coefficient of species \(i\) and \(\Delta G_i^{\circ}\) is the standard Gibbs energy of formation of that species. Given, \(\Delta G_{\mathrm{f}}^{\circ}(\mathrm{Al}^{3+}(a q)) = -480\,\mathrm{kJ/mol}\) From Appendix 4, we have \(\Delta G_{\mathrm{f}}^{\circ}(\textrm{Ag}_{2}\textrm{S}(s)) = -32.7\cdot 2\,\mathrm{kJ/mol}\) \(\Delta G_{\mathrm{f}}^{\circ}(\textrm{Ag}(s)) = 0\,\mathrm{kJ/mol}\) \(\Delta G_{\mathrm{f}}^{\circ}(\textrm{S}^{2-}(a q)) = -456.2\,\mathrm{kJ/mol}\) Calculating \(\Delta G^{\circ}\) for the given reaction: \[\Delta G^{\circ} = [\,6 \cdot 0 + 3 \cdot (-456.2) + 2 \cdot (-480)\,] - [\,3 \cdot (-32.7 \cdot 2) + 0\,\]\] Now, evaluate the values to find \(\Delta G^{\circ}\).

Calculate the equilibrium constant K

Now that we have the \(\Delta G^{\circ}\), we can calculate the equilibrium constant \(K\) using the equation: \[\Delta G^{\circ} = -RT\ln{K}\] Where R is the gas constant, \(8.314\,\mathrm{J/(mol\cdot K)}\) and T is the temperature in Kelvin (converted from the given \(25^{\circ}\mathrm{C}\)). Rearrange the equation to solve for K: \[K = \mathrm{e}^{-\frac{\Delta G^{\circ}}{RT}}\] Plugging the values of \(\Delta G^{\circ}\), R, and T into the equation, we can find the equilibrium constant K.

Calculate the standard concentration quotient

The standard concentration quotient \(\mathscr{C}^{\circ}\) can be calculated by finding the ratio between the equilibrium constant in terms of pressure, \(K_p\), and the equilibrium constant in terms of concentration, \(K_c\). \[\mathscr{C}^{\circ} = \frac{K_p}{K_c}\] For the given reaction, the equation is: \[\mathscr{C}^{\circ} = \frac{K_p}{(\mathrm{S}^{2-})^3 (\mathrm{Al}^{3+})^2}\] As we already found the value of K in the previous step, we can calculate the standard concentration quotient \(\mathscr{C}^{\circ}\) using the above equation.

Calculate the standard reduction potential

To calculate the standard reduction potential, we use the Nernst equation: \[E^{\circ} = \frac{RT}{nF}\ln{K}\] Where \(E^{\circ}\) is the standard reduction potential, R is the gas constant, T is the temperature in Kelvin, n is the number of moles of electrons involved in the half-reaction, F is the Faraday constant, and K is the equilibrium constant. For the given half-reaction, \(2\,\text{e}^{-} + \text{Ag}_2\text{S}(s) \longrightarrow 2\,\text{Ag}(s) + \text{S}^{2-}(\text{aq})\), n = 2. Now, plug in the values for R, T, n, F, and the previously calculated K value into the Nernst equation to find the standard reduction potential, \(E^{\circ}\).

Key concepts

Gibbs Free Energy

Gibbs free energy, represented by \( \Delta G \), is a thermodynamic function that measures the maximum amount of work that a process can perform at constant temperature and pressure. It's a central concept in chemical thermodynamics, signifying the balance between enthalpy, entropy, and temperature for a system. The equation \( \Delta G = \Delta H - T\Delta S \) relates Gibbs free energy change to the change in enthalpy (\( \Delta H \)), change in entropy (\( \Delta S \)), and absolute temperature (T).
For a reaction to be spontaneous, \( \Delta G \) has to be negative. A zero value indicates a system that is at equilibrium, and positive \( \Delta G \) means the process is non-spontaneous under the given conditions. In the context of chemical reactions, when we talk about standard Gibbs free energy change, \( \Delta G^\circ \), it means that the calculation is done under standard conditions, which usually refers to a temperature of 298 K (25°C) and a pressure of 1 atm. The \( \Delta G^\circ \) can be calculated from the standard Gibbs energies of formation for the reactants and products, as shown in the problem's step-by-step solution.
By calculating the \( \Delta G^\circ \) for a reaction, as done in the exercise, we can understand whether a reaction would occur spontaneously under standard conditions. If the calculated value is negative, as it should be for the silverware cleaning reaction mentioned, the reaction favors the formation of products when initiated.

Equilibrium Constant

The equilibrium constant, denoted as \(K\), quantifies the ratio of product concentrations to reactant concentrations at equilibrium, with each raised to the power of their respective stoichiometric coefficients in the balanced equation. It is a measure of the extent of the reaction. If the value of K is large, the reaction favors the formation of products. Conversely, a small K value indicates that reactants are favored.
In the context of the exercise provided, after finding the \( \Delta G^\circ \) using standard Gibbs energies of formation, the equation \(\Delta G^\circ = -RT\ln K\) allows us to solve for the equilibrium constant \(K\) at a certain temperature. The relationship between \( \Delta G^\circ \) and \(K\) is logarithmic, reflecting that a large \( \Delta G^\circ \) corresponds to a small \(K\) and vice versa. This principle is critical in predicting whether the products or reactants will be predominant at equilibrium under standard conditions.
For a chemical reaction to reach equilibrium, the forward and reverse reactions occur at the same rate, and the concentrations of reactants and products remain constant, though not necessarily equal. The equilibrium constant \(K\) remains unchanged provided the temperature is constant, despite varying initial concentrations of reactants or products.

Standard Reduction Potential

Standard reduction potential, symbolized as \( E^\circ \), is the measure of the tendency of a chemical species to acquire electrons and be reduced. It is measured under standard conditions, which involve all the soluble substances in 1 M concentrations and gases at 1 atm pressure, with the temperature typically set at 25°C (298 K). The more positive the standard reduction potential, the greater the species' affinity for electrons, which means it is a better oxidizing agent.
The standard reduction potential is an intensive property; it doesn't depend on the amount of substance and can be used to predict the direction of an electrochemical reaction. If a half-cell possesses a greater standard reduction potential than the other, it will be reduced at the expense of the other half-cell being oxidized. In the exercise, the standard reduction potential for the process involving the reduction of silver sulfide helps us understand the ease with which silver can be reclaimed from its tarnished form.
Values for standard reduction potentials are typically referenced against the standard hydrogen electrode, which has an assigned potential of 0 volts. These values guide us in predicting the spontaneity of redox reactions and are instrumental in calculating the cell potential for electrochemical cells.

Nernst Equation

The Nernst equation allows us to calculate the cell potential for an electrochemical cell under non-standard conditions. This equation relates the measurable cell potential to its temperature, the number of electrons transferred in the reaction (n), and the reaction quotient (Q), which indicates the ratio of concentrations of reactants to products at any point in the reaction (not just at equilibrium). The equation is expressed as:\[ E = E^\circ - \frac{RT}{nF} \ln Q \]
Where:\

• \(E\) is the cell potential at non-standard conditions.
• \(E^\circ\) is the standard cell potential, which can be calculated from the standard reduction potentials of the cathode and anode.
• \(R\) is the universal gas constant.
• \(T\) is the absolute temperature in Kelvin.
• \(F\) is the Faraday constant, representing the charge per mole of electrons.
• \(Q\) reflects the current ratio of reactants and products.

When a system reaches equilibrium, \(Q = K\), the reaction quotient equals the equilibrium constant. Consequently, the cell potential \(E\) becomes zero, reflecting that no net reaction occurs, and the system is at equilibrium. As the problem presented illustrates, the standard reduction potential \(E^\circ\) can be calculated by rearranging the Nernst equation and using the equilibrium constant \(K\) obtained earlier. This understanding is fundamental in electrochemistry and for calculating the potentials of electrochemical cells.