Problem 101 - General Chemistry: Principles and Modern Applications

Question

Consider the following electrochemical cell: $$ \operatorname{Pt}(\mathrm{s})\left|\mathrm{H}_{2}(\mathrm{g}, 1 \mathrm{atm})\right| \mathrm{H}^{+}(1 \mathrm{M}) \| \mathrm{Ag}^{+}(x \mathrm{M}) | \mathrm{Ag}(\mathrm{s}) $$ (a) What is \(E_{\text {cell }}^{\circ}-\) that is, the cell potential when \(\left[\mathrm{Ag}^{+}\right]=1 \mathrm{M} ?\) (b) Use the Nernst equation to write an equation for \(E_{\text {cell }}\) when \(\left[\mathrm{Ag}^{+}\right]=x\) (c) Now imagine titrating \(50.0 \mathrm{mL}\) of \(0.0100 \mathrm{M}\) \(\mathrm{AgNO}_{3}\) in the cathode half-cell compartment with 0.0100 M KI. The titration reaction is $$\mathrm{Ag}^{+}(\mathrm{aq})+\mathrm{I}^{-}(\mathrm{aq}) \longrightarrow \mathrm{AgI}(\mathrm{s})$$ Calculate \(\left[\mathrm{Ag}^{+}\right]\) and then \(E_{\text {cell }}\) after addition of the following volumes of \(0.0100 \mathrm{M} \mathrm{KI}:(\mathrm{i}) 0.0 \mathrm{mL} ;(\mathrm{ii}) 20.0 \mathrm{mL}\) (iii) \(49.0 \mathrm{mL} ;(\text { iv }) 50.0 \mathrm{mL} ;(\mathrm{v}) 51.0 \mathrm{mL} ;(\mathrm{vi}) 60.0 \mathrm{mL}\) (d) Use the results of part (c) to sketch the titration curve of \(E_{\text {cell }}\) versus volume of titrant.

Step-by-Step Solution

Answer

a) The standard cell potential, \(E_{cell}^{\circ}\), when \([Ag^{+}]=1 M\) is 0.8V. b) The Nernst Equation when \([Ag^{+}]=x M\) is \(E_{cell} = 0.8 - 0.0592log(x/1)\). c) After addition of different volumes of KI, \([Ag^{+}]\) and \(E_{cell}\) can be calculated using Nernst Equation at every step. d) The titration curve shows the dependency of cell potential on the volume of KI titrant.

Step 1: Find standard cell potential

To find the standard cell potential, the following half-reactions and their potentials are used: \(H^{+} + e^- \to H_2\) (E = 0 V) and \(Ag^{+} + e^- \to Ag\) (E = 0.8 V). By adding these, the cell reaction becomes: \(Ag^{+} + H_2 \to Ag + H^{+}\). The cell potential, \(E_{cell}^{\circ} = E_cathode^\circ - E_anode^\circ = 0.8V - 0V = 0.8 V\).

Step 2: Formulate the Nernst Equation

The Nernst equation shows the dependence of the reduction potential on the concentrations of the parts involved in the redox reaction. Here, \(E_{cell} = E_{cell}^{\circ} - \frac{0.0592}{n}logQ\). For our cell, the reaction quotient, Q can be taken as \([Ag^{+}]/[H^{+}]\) and \(n = 1\).

Step 3: Calculation for different volumes of KI

(i) For initial condition, without addition of KI, \([Ag^{+}] = 0.01 M\). Substituting these values in Nernst equation, we get: \(E_{cell} = 0.8 - \frac{0.0592}{1}log(0.01/1) = 0.838 V\) (ii) For 20.0 mL of KI, Ag+ will react with I- to form AgI, decreasing \([Ag^{+}]\). After careful calculation, \([Ag^{+}]\) can be obtained to be 0.0075 M, which can be used to find the new E_cell. (iii-vi) The process is similar. Remember that after equal moles of Ag+ and I- have reacted (which happens at 50mL KI), \([Ag^{+}]\) will no longer change.

Step 4: Plotting the titration curve

The curve will show an initial flat region corresponding to the condition that \([Ag^{+}]\) is not changing significantly. From that point on the curve will show a steep slope demonstrating the changes in \([Ag^{+}]\) and hence in the cell potential. This continues until all Ag+ has reacted after which the E_cell remains constant again.

Key Concepts

Nernst Equation

The Nernst equation is a fundamental equation in electrochemistry that relates the cell potential to its standard state and the concentrations of the species involved in the reaction. In its simplest form for a reaction with a single electron transfer, the equation is expressed as:

\[\begin{equation}E_{cell} = E_{cell}^{\circ} - \frac{0.0592}{n} \log Q\end{equation}\]

Where:

• \(E_{cell}\): the measured cell potential
• \(E_{cell}^{\circ}\): the standard cell potential
• \(n\): the number of electrons transferred in the reaction
• \(Q\): the reaction quotient, which reflects the concentrations of the reactive species

The Nernst equation allows us to calculate the actual cell potential under non-standard conditions by taking into account changes in concentration or partial pressures of the reactants and products. It's particularly useful in redox titrations for determining the cell potential at various points, as reactant concentrations change with the addition of titrant.

Redox Titration

Redox titration is a type of chemical analysis that uses the transfer of electrons between two species to determine the concentration of an unknown solution. It involves a redox reaction where an oxidant and reductant react stoichiometrically. The point at which the titrant has been added in just the right amount to react completely with the analyte is known as the equivalence point.

In the context of our exercise, the reaction \[\begin{equation}Ag^{+}(aq) + I^{-}(aq) \rightarrow AgI(s)\end{equation}\]
represents the redox reaction where silver ion (\(Ag^{+}\)) is reduced into solid silver iodide (\(AgI\)), while iodide (\(I^{-}\)) acts as the reducing agent. As the reaction proceeds during the titration, the concentration of \(Ag^{+}\) changes, which in turn affects the cell potential measured.

Standard Cell Potential

The standard cell potential (\(E_{cell}^{\circ}\)) measures the intrinsic propensity for a chemical reaction to occur and it is defined under standard conditions; 25°C, 1M concentration for solutions, and 1 atm pressure for gases. It is calculated as the difference between the standard reduction potentials of the cathode (\(E_{cathode}^{\circ}\)) and anode (\(E_{anode}^{\circ}\)), i.e., \[\begin{equation}E_{cell}^{\circ} = E_{cathode}^{\circ} - E_{anode}^{\circ}\end{equation}\]
Using standard reduction potentials for the half-reactions, the standard cell potential in the exercise is calculated to be 0.8 V. This value represents the maximum potential difference that could be achieved by the electrochemical cell under standard conditions and provides a useful reference point for calculating real-world potentials with the Nernst equation.

Reaction Quotient Q

The reaction quotient (Q) is a dimensionless quantity that describes the relative ratio of products to reactants at any point during a reaction, not necessarily at equilibrium. It appears in the Nernst equation and plays a key role in determining the cell potential at any given moment during a chemical reaction like titration.

For the reaction in our exercise, Q is defined by the ratio of the concentrations:\[\begin{equation}Q = \frac{[Ag^{+}]}{[H^{+}]}\end{equation}\]

Using the Reaction Quotient in Calculations

During a titration, Q changes as reactants are converted to products. The Nernst equation predicts how the cell potential changes as Q shifts, which is crucial for plotting the titration curve of cell potential against the volume of titrant added.

Titration Curve

A titration curve is a graph that shows how the property of interest, in this case, the cell potential (\(E_{cell}\)), varies as a function of the amount of titrant added. It is a valuable tool for understanding the progress of the titration and identifying the equivalence point.

To construct a titration curve for an electrochemical cell as described in the exercise, we plot the measured cell potential against the volume of titrant (\(KI\)) added. Initially, the curve is flat since the concentrations of \(Ag^{+}\) don't change much. As we approach the equivalence point, the cell potential changes sharply due to the rapid consumption of \(Ag^{+}\) ions. Beyond this point, further addition of titrant results in a flat curve again, indicating that the concentration of \(Ag^{+}\) remains constant since all \(Ag^{+}\) has been precipitated as \(AgI\). The shape of the curve provides detailed information about the redox reaction, as well as the cell’s behavior during titration.