Question

You have a concentration cell in which the cathode has a silver electrode with \(0.10 \mathrm{M} \mathrm{Ag}^{+}\). The anode also has a silver electrode with \(\mathrm{Ag}^{+}(a q), 0.050 \mathrm{M} \mathrm{S}_{2} \mathrm{O}_{3}{ }^{2-}\), and \(1.0 \times 10^{-3} \mathrm{M}\) \(\mathrm{Ag}\left(\mathrm{S}_{2} \mathrm{O}_{3}\right)_{2}{ }^{3-}\). You read the voltage to be \(0.76 \mathrm{~V}\).a. Calculate the concentration of \(\mathrm{Ag}^{+}\) at the anode. b. Determine the value of the equilibrium constant for the formation of \(\mathrm{Ag}\left(\mathrm{S}_{2} \mathrm{O}_{3}\right)_{2}{ }^{3-}\). \(\mathrm{Ag}^{+}(a q)+2 \mathrm{~S}_{2} \mathrm{O}_{3}{ }^{2-}(a q) \rightleftharpoons \mathrm{Ag}\left(\mathrm{S}_{2} \mathrm{O}_{3}\right)_{2}^{3-}(a q) \quad K=?\)

Short answer

a. The concentration of \(\mathrm{Ag}^{+}\) at the anode is approximately \(1.65 \times 10^{-4}\mathrm{M}\). b. The equilibrium constant for the formation of \(\mathrm{Ag}\left(\text{S}_{2}\text{O}_{3}\right)_{2}{ }^{3-}\) is approximately \(1.32 \times 10^{10}\).

Step-by-step solution

Understanding the Nernst Equation

The Nernst Equation relates the cell potential to the concentrations of the reactants and products. It can be written as: \[E_{cell} = E^{0}_{cell} - \frac{RT}{nF} \ln Q\] Where: - \(E_{cell}\) is the cell potential (voltage) - \(E^{0}_{cell}\) is the standard cell potential - \(R\) is the gas constant - \(T\) is the temperature in Kelvin - \(n\) is the number of moles of electrons transferred - \(F\) is the Faraday constant - \(Q\) is the reaction quotient Since we have a concentration cell with identical electrodes, the standard cell potential, \(E^{0}_{cell}\), will be 0. Thus, we can simplify the Nernst Equation to: \[E_{cell} = - \frac{RT}{nF} \ln Q\]

Determine the reaction quotient Q

For our given reaction: \(\text{Ag}^{+}(aq) \rightleftharpoons \text{Ag}\left(\text{S}_{2}\text{O}_{3}\right)_{2}{ }^{3-}(aq)\), Q can be calculated as: \[Q = \frac{[\text{Ag}\left(\text{S}_{2}\text{O}_{3}\right)_{2}{ }^{3-}]}{[\text{Ag}^{+}][\text{S}_{2}\text{O}_{3}{ }^{2-}]^2}\]

Calculate the concentration of Ag+ at the anode

Now we can substitute the known values into the Nernst Equation: \[0.76 = -\frac{(8.314)(298)}{(2)(96485)} \ln Q\] We are given \([\text{S}_{2}\text{O}_{3}{ }^{2-}] = 0.050 M\) and \([\text{Ag}\left(\text{S}_{2}\text{O}_{3}\right)_{2}{ }^{3-}] = 1.0 \times 10^{-3} M\), and we need to find \([\text{Ag}^{+}]\) at the anode. Plug these values into the equation for Q and solve for \([\text{Ag}^{+}]\): \[\frac{1.0 \times 10^{-3}}{(0.050)^2[\text{Ag}^{+}]} = Q\] Now, substitute the value of Q back into the Nernst Equation and solve for \([\text{Ag}^{+}]\). The resulting value is the concentration of \(\text{Ag}^{+}\) at the anode.

Calculate the equilibrium constant K

Now that we have the concentration of \(\text{Ag}^{+}\) at the anode, we can calculate the equilibrium constant (K) for the given reaction. We know: \[K = \frac{[\text{Ag}\left(\text{S}_{2}\text{O}_{3}\right)_{2}{ }^{3-}]}{[\text{Ag}^{+}][\text{S}_{2}\text{O}_{3}{ }^{2-}]^2}\] Substitute the values for the concentrations into the equation and solve for K. The resulting value is the equilibrium constant for the formation of \(\text{Ag}\left(\text{S}_{2}\text{O}_{3}\right)_{2}{ }^{3-}\).

Key concepts

Concentration Cell

A concentration cell is a type of electrochemical cell where the electrodes are identical, but the concentrations of the electrolytes in which the electrodes are immersed are different. This creates a potential difference, or voltage, because the cell seeks to balance the concentration of ions on both sides.

• In our exercise, we have a silver concentration cell with one side containing a cathode with 0.10 M of Ag⁺ and the other anode side with varied concentrations of Ag⁺ and other complexes.
• The voltage you measure from this cell comes from the tendency of ions to move from the higher concentration side to the lower concentration side.
• This movement continues until the system reaches equilibrium.

This driving force is quantified using the Nernst Equation, which indicates how the voltage changes as the concentration of ions in the solution changes. Understanding the concentration cell concept is crucial because it shows how ions behave within electrochemical cells and how a seemingly simple change in concentration can power a device.

Equilibrium Constant

The equilibrium constant, denoted as K, is a number that expresses the ratio of the concentration of products to reactants at equilibrium for a reversible chemical reaction. It provides insight into the position of equilibrium and whether the reactants or products are favored.

• In our problem involving silver and thiosulfate, we need to determine the value of K for the formation of the complex ion Ag(S₂O₃)₂^3-.
• The larger the K value, the more the products are favored at equilibrium, meaning more Ag(S₂O₃)₂^3- is formed.
• By calculating the concentration of Ag⁺ at the anode, we can find K using the relationship provided in the exercise.

Understanding the equilibrium constant helps in predicting the direction of the reaction under different conditions and can be essential for controlling reactions in industrial processes.

Reaction Quotient

The reaction quotient, denoted as Q, is similar to the equilibrium constant (K) but is used when a reaction is not at equilibrium. It provides a snapshot of the current state of a reaction and helps in predicting which direction the reaction will proceed to reach equilibrium.

• Q is calculated using the same expression as K but with the current concentrations of reactants and products.
• In the given problem, Q is used in the Nernst Equation to calculate the potential of the concentration cell.
• If Q = K, the system is at equilibrium. If Q < K, the system will proceed to form more products, whereas if Q > K, the reaction will move towards the formation of reactants.

By calculating Q at different points, chemists can understand the progress of a reaction and make adjustments to reach desired conditions. This becomes particularly important in studying electrochemical cells, where Q directly influences the cell potential and the efficiency of the cell.