Question

Consider the following galvanic cell:Calculate the \(K_{\text {sp }}\) value for \(\mathrm{Ag}_{2} \mathrm{SO}_{4}(s)\). Note that to obtain silver ions in the right compartment (the cathode compartment), excess solid \(\mathrm{Ag}_{2} \mathrm{SO}_{4}\) was added and some of the salt dissolved.

Short answer

The \(K_{sp}\) value for \(\mathrm{Ag}_2\mathrm{SO}_4(s)\) in a galvanic cell can be calculated using the Nernst equation and the standard cell potential. After obtaining the balanced overall cell reaction and substituting the given values, the \(K_{sp}\) value is found to be approximately \(1.95 \times 10^{42}\).

Step-by-step solution

1. Write the balanced half-reactions occurring in the cell

Since there is no information given about the galvanic cell, we should assume that it is composed of two standard electrode potentials: one for the oxidation of silver ions to silver metal and another for the reduction of silver ions to zero valency. 1. Oxidation half-reaction: \( Ag_2SO_4 \rightarrow Ag^+ + SO_4^{2-} \) 2. Reduction half-reaction: \( Ag^+ (aq) + e^- \rightarrow Ag(s) \)

2. Write the balanced overall cell reaction

To write the overall cell reaction, first, the number of electrons exchanged should be equal. So, we have to multiply the oxidation half-reaction by 2 to get the same number of electrons as in the reduction half-reaction: \( 2 (Ag_2SO_4 \rightarrow Ag^+ + SO_4^{2-}) \) Now, we can combine the two half-reactions to get the overall cell reaction: \( 2 Ag_2SO_4 (s) + 2 e^- \rightarrow 3 Ag(s) + 2 SO_4^{2-}(aq) \)

3. Calculate the cell potential ( \(E_{cell}\) ) at standard conditions

At standard conditions, the cell potential is given by the difference in standard electrode potentials of cathode and anode: \( E_{cell}^0 = E_{cathode}^0 - E_{anode}^0 \) However, since we are not provided with any information about the anode potential, we must assume it to be zero: \( E_{cell}^0 = E_{cathode}^0 \) The standard electrode potential of the silver/silver ion half-cell is \(0.7996V\), thus the standard cell potential is also: \( E_{cell}^0 = 0.7996 V\)

4. Use the Nernst equation to relate the cell potential and reaction quotient (Q) to the standard cell potential

The Nernst equation relates the cell potential to the standard cell potential and the reaction quotient: \(E_{cell} = E_{cell}^0 - \frac{RT}{nF} \ln Q \) Since our cell is under standard conditions, the cell potential at standard conditions is equal to the standard cell potential: \( E_{cell}^0 = E_{cell}\) Now we can solve for the reaction quotient Q: \( Q = e^{\frac{2FE_{cell}}{RT}}\), where \(F\) is the Faraday constant (96485 C/mol), \(R\) is the gas constant (8.314 J/mol K) and \(T\) is the temperature in Kelvin (assume room temperature 298K)

5. Solve for Q to find the \(K_{sp}\) value

Substituting the values into the equation: \( Q = e^{\frac{2 \times 96485 C/mol \times 0.7996 V}{8.314 J/mol K \times 298 K}} \) \( Q = e^{96.64} \) Now, recall that the overall cell reaction is \( 2 Ag_2SO_4 (s) + 2 e^- \rightarrow 3 Ag(s) + 2 SO_4^{2-}(aq) \). The reaction quotient Q for this reaction is given by the products of the concentration of the products divided by the concentration of the reactants: \( Q = \frac{[Ag]^3 [SO_4^{2-}]^2}{[Ag_2SO_4]^2} \) Since the solubility product constant \(K_{sp}\) is defined as the equilibrium constant for the dissolution of a solid in a solution and only involves the concentrations of the dissolved species, we can put \(K_{sp}\) in place of Q for the dissolution reaction: \( K_{sp} = [Ag]^3 [SO_4^{2-}]^2 \) Now we have: \( K_{sp} = e^{96.64} \) Calculating \(K_{sp}\): \( K_{sp} = 1.95 \times 10^{42} \) Thus, the \(K_{sp}\) value of \(\mathrm{Ag}_2\mathrm{SO}_4(s)\) is approximately \(1.95 \times 10^{42}\).

Key concepts

Galvanic Cell

A galvanic cell, also known as a voltaic cell, is an electrochemical cell that derives electrical energy from spontaneous redox reactions taking place within the cell. It typically consists of two different metals connected by a salt bridge, or separated by a porous membrane in separate solutions. Each metal acts as an electrode in their respective solutions. One metal undergoes oxidation (loses electrons), while the other undergoes reduction (gains electrons).

The cell has two half-cells with the anode being the oxidation half-cell and the cathode being the reduction half-cell. The chemical reaction in a galvanic cell continues until one of the reactants has been consumed or the product of the reaction builds up to such a level that it suppresses the reaction.

Electrochemistry

Electrochemistry is the branch of chemistry that deals with the relationship between electricity and chemical changes. It includes the study of both spontaneous and non-spontaneous processes. The applications of electrochemistry are extensive and include electroplating, electrolysis, and batteries.

In the context of a galvanic cell, electrochemistry focuses on the spontaneous chemical reactions that produce electrical energy. These reactions are governed by the movement of electrons from the anode to the cathode through an external circuit, which generates an electric current that can be harnessed to do work.

Nernst Equation

The Nernst equation is a fundamental equation in electrochemistry. It provides a quantitative relationship between the concentration of species in an electrochemical cell and the cell's potential.

The equation is expressed as: \[E_{cell} = E_{cell}^0 - \frac{RT}{nF} \ln Q\]
where \(E_{cell}\) is the cell potential, \(E_{cell}^0\) is the standard cell potential, \(R\) is the ideal gas constant, \(T\) is the temperature in Kelvin, \(n\) is the number of moles of electrons transferred in the cell reaction, \(F\) is the Faraday constant, and \(Q\) is the reaction quotient. The reaction quotient \(Q\) reflects the ratio of the concentrations of the products to reactants at any point in time (not just at equilibrium).

Solubility Product Constant

The solubility product constant, denoted as \(K_{sp}\), is a special case of an equilibrium constant that applies to the dissolution of sparingly soluble ionic compounds. It is defined as the product of the molar concentrations of the constituent ions, each raised to the power of its stoichiometric coefficient in the dissolution equation.

The calculation of \(K_{sp}\) typically involves determining the saturation concentration of ions when a solid ionic compound is in equilibrium with a solution of its ions. It is key to predicting whether a precipitate will form when solutions of different ions are mixed. In the provided exercise, the calculation of the \(K_{sp}\) for silver sulfate \(\mathrm{Ag}_2\mathrm{SO}_4\) involves first determining the reaction quotient using the Nernst equation, and subsequently equating it with \(K_{sp}\) to find the solubility product in a saturated solution.