Problem 111 - Chemistry The Molecular Nature of Matter

Question

An \(\mathrm{Ag} / \mathrm{AgCl}\) electrode dipping into \(1.00 \mathrm{M} \mathrm{HCl}\) has a standard reduction potential of \(+0.2223 \mathrm{~V}\). The half reaction is $$ \mathrm{AgCl}(s)+e^{-} \rightleftharpoons \mathrm{Ag}(s)+\mathrm{Cl}^{-}(a q) $$ A second \(\mathrm{Ag} / \mathrm{AgCl}\) electrode is dipped into a solution containing \(\mathrm{Cl}\) at an unknown concentration. The cell generates a potential of \(0.0478 \mathrm{~V}\), with the electrode in the solution of unknown concentration having a negative charge. What is the molar concentration of \(\mathrm{Cl}\) in the unknown solution?

Step-by-Step Solution

Answer

The molar concentration of \(\mathrm{Cl}^{-}\) in the unknown solution is calculated using the Nernst equation, considering the reaction quotient as the unknown concentration of \(\mathrm{Cl}^{-}\), and by computing the antilog of the isolated variable.

Step 1: Identify Given Information

We are given the standard reduction potential of the Ag/AgCl electrode, which is +0.2223 V, and the potential generated when this electrode is in a solution with an unknown concentration of Cl-, which is 0.0478 V. The half-reaction of the electrode is: \[\begin{equation}\mathrm{AgCl}(s) + \mathrm{e}^{-} \rightleftharpoons \mathrm{Ag}(s) + \mathrm{Cl}^{-}(aq)\end{equation}\]

Step 2: Write the Nernst Equation

The Nernst equation for the half-reaction is: \[\begin{equation}E = E^\circ - \frac{0.05916}{n}\log Q\end{equation}\]where- \(E^\circ\) is the standard reduction potential- \(E\) is the measured cell potential- \(n\) is the number of electrons transferred in the half-reaction- \(Q\) is the reaction quotient

Step 3: Calculate the Number of Electrons Transferred (n)

The half-reaction shows that one electron is transferred:\[\begin{equation}\mathrm{AgCl}(s) + \mathrm{e}^{-} \rightleftharpoons \mathrm{Ag}(s) + \mathrm{Cl}^{-}(aq)\end{equation}\]Hence, \(n = 1\).

Step 4: Express the Reaction Quotient (Q)

The reaction quotient \(Q\) for the half-reaction is the concentration of the products over the reactants. Since the reactants include a solid and the products are in aqueous solution, \(Q\) is simply the concentration of \(\mathrm{Cl}^{-}\) in the solution:\[\begin{equation}Q = [\mathrm{Cl}^{-}]\end{equation}\]

Step 5: Plug into the Nernst Equation and Solve for Q

Substituting the values into the Nernst equation and solving for \(Q = [\mathrm{Cl}^{-}]\):\[\begin{equation}0.0478 = +0.2223 - \frac{0.05916}{1}\log [\mathrm{Cl}^{-}]\end{equation}\]Now, isolate the logarithmic term and solve for \([\mathrm{Cl}^{-}]\).

Step 6: Rearrange and Find the Logarithm

Rearrange the equation to isolate the logarithm:\[\begin{equation}\log [\mathrm{Cl}^{-}] = \frac{+0.2223 - 0.0478}{0.05916}\end{equation}\]Calculate the right-hand side of the equation.

Step 7: Take the Antilog to Find \([\mathrm{Cl}^{-}]\)

Compute the antilog (base 10) of the result from step 6 to find the molar concentration of \(\mathrm{Cl}^{-}\):\[\begin{equation}[\mathrm{Cl}^{-}] = 10^{\text{result from step 6}}\end{equation}\]This gives you the molar concentration of \(\mathrm{Cl}^{-}\) in the unknown solution.

Key Concepts

Electrochemical Cells

Electrochemical cells are the fundamental units of batteries and various types of electrochemical power sources. They operate by harnessing the chemical energy of a spontaneous redox reaction and converting it into electrical energy that can perform work. In its simplest form, an electrochemical cell consists of two different metal electrodes immersed in electrolyte solutions, each containing its respective metal ions.
An external circuit connects the electrodes, and as the redox reaction occurs, electrons flow through the circuit, creating an electric current. The electrode where oxidation occurs is called the anode, while the cathode is where reduction takes place. In the context of the given exercise, the Ag/AgCl electrode functions as either the anode or cathode, depending on whether we are charging or discharging the cell, with the direction of electron flow being determined by the electrode potentials.

Standard Reduction Potential

The standard reduction potential, denoted by the symbol E°, is a measure of the tendency of a chemical species to acquire electrons and thereby be reduced. This potential is measured under standard conditions, which is typically at a temperature of 298 K (25°C), a pressure of 1 atm, and with 1 M concentration of each ion participating in the reaction.
Values of standard reduction potentials are given with respect to the standard hydrogen electrode (SHE), which has an assigned potential of 0 volts. A positive standard reduction potential indicates a greater tendency to be reduced (gain electrons), while a negative value suggests a tendency to lose electrons. In the exercise, the Ag/AgCl electrode has a standard reduction potential of +0.2223 V, implying it tends to undergo reduction rather than oxidation when compared to the SHE.

Reaction Quotient

The reaction quotient, Q, is a calculation that expresses the relative concentration of products to reactants at any point during a chemical reaction, not just at equilibrium. For the electrode half-reaction AgCl(s) + e- ⇌ Ag(s) + Cl-(aq), Q is defined by the equation Q = [Cl-], where [Cl-] stands for the molar concentration of chloride ions in the solution.
In the scenario where solid AgCl and solid Ag are present, their activities are considered to be constant and equal to one, thereby not appearing in the reaction quotient expression. The reaction quotient is a critical aspect of the Nernst equation, allowing us to calculate the electrode potential at any given state of the reaction mixture, not just at standard conditions.

Electrode Potential Calculation

To calculate the electrode potential, E, in a cell that is not at standard conditions, we utilize the Nernst equation. This critical tool is expressed as follows:
\[E = E^\circ - \frac{0.05916}{n}\log Q\]
where E is the electrode potential under the current non-standard conditions, E° is the standard reduction potential, n is the number of electrons transferred in the half-reaction, and Q is the reaction quotient. Applying this equation in practice involves inserting known values and solving for the unknown. For the exercise, we used the Nernst equation to determine the concentration of Cl- ions in a solution by knowing the measured cell potential and the standard reduction potential of the Ag/AgCl electrode.