A galvanic cell is based on the following half-reactions: $$ \begin{aligned} \mathrm{Fe}^{2+}+2 \mathrm{e}^{-} \longrightarrow \mathrm{Fe}(s) & \mathscr{E}^{\circ} &=-0.440 \mathrm{~V} \\ 2 \mathrm{H}^{+}+2 \mathrm{e}^{-} \longrightarrow \mathrm{H}_{2}(g) & \mathscr{E}^{\circ} &=0.000 \mathrm{~V} \end{aligned} $$ where the iron compartment contains an iron electrode and \(\left[\mathrm{Fe}^{2+}\right]=1.00 \times 10^{-3} M\) and the hydrogen compartmentcontains a platinum electrode, \(P_{\mathrm{H}_{2}}=1.00 \mathrm{~atm}\), and a weak acid, HA, at an initial concentration of \(1.00 M .\) If the observed cell potential is \(0.333 \mathrm{~V}\) at \(25^{\circ} \mathrm{C}\), calculate the \(K_{\mathrm{a}}\) value for the weak acid HA.

One of the most fundamental equations in electrochemistry is the Nernst equation. It establishes a connection between the cell potential under non-standard conditions and the concentrations of the electroactive species. The general form of the Nernst equation is as follows:
\[\begin{equation}\ E_{cell} = E^{\circ}_{cell} - \frac{RT}{nF}\ln\left(\frac{\text{Products}}{\text{Reactants}}\right)\end{equation}\]
Where:

• \(E_{cell}\) is the measured cell potential.
• \(E^{\circ}_{cell}\) is the standard cell potential.
• R is the universal gas constant (8.314 J/(mol\textbullet K)).
• T is the temperature in Kelvin.
• n is the number of moles of electrons transferred in the reaction.
• F is the Faraday constant (96485 C/mol).

In the case of our galvanic cell example involving iron and hydrogen, the Nernst equation allows us to calculate the cell's potential by considering the concentrations of iron ions (Fe2+) and hydrogen ions (H+) as well as the pressure of hydrogen gas (H2). The equation takes into account the temperature and the number of electrons involved in the electrode processes, providing an accurate way to determine the behavior of the cell under the given conditions.
For students to better understand how to apply the Nernst equation, it's helpful to show step-by-step calculations and remind them to ensure the reaction quotient is correctly formulated—considering products over reactants—and that they correctly identify the reduced and oxidized forms of the species involved in the half-reactions.

Standard reduction potential, often denoted as \(\mathscr{E}^{\circ}\), is a measure of the tendency of a chemical species to be reduced and thus to gain electrons. Measured in volts under standard conditions (all reactants and products at 1M concentration, gas pressures at 1 atm, and at a temperature of 25°C or 298K), these potentials are key in determining the direction and spontaneity of redox reactions.

• The more positive the standard reduction potential, the greater the species' affinity for electrons (i.e., it is a stronger oxidizing agent).
• The more negative the standard reduction potential, the less likely the species is to gain electrons (i.e., it is a stronger reducing agent).

In our example, the standard reduction potential for the iron half-reaction is \(\mathscr{E}^{\circ} = -0.440 V\), indicating its tendency to be reduced, and for the hydrogen half-reaction it is \(\mathscr{E}^{\circ} = 0.000 V\), representing the standard potential for hydrogen's reduction reference. By comparing these values, it is possible to predict that hydrogen gas, having a higher reduction potential, would act as an oxidizing agent, while iron(II) ions, with a lower reduction potential, would act as a reducing agent. These potentials are crucial for calculating the standard cell potential and for understanding the electrochemical series which orders substances by their electrode potentials.

The acid dissociation constant (Ka) is a quantitative measure of the strength of an acid in solution. It is the equilibrium constant for the dissociation of a weak acid into its conjugate base and hydrogen ions in water. The stronger the acid, the larger the Ka value and the more the acid is dissociated. The general expression for the Ka of a weak acid \(HA\) dissociating into \(A^-\) and \(H^+\) can be written as:\[\begin{equation}\ K_a = \frac{[A^-][H^+]}{[HA]}\end{equation}\]
When working with weak acids in the context of electrochemistry, you might encounter situations where you have to calculate the Ka value, as seen in the exercise involving the galvanic cell and the weak acid HA. Students should understand that to find Ka, they need to know the equilibrium concentrations of the acid itself and its dissociated ions. By experimentally determining the concentration of hydrogen ions in solution, which can be approached using the Nernst equation as shown in the exercise, we are able to solve for Ka.
It's also vital for learners to grasp that for a weak acid, the concentration of undissociated acid remains almost the same as its initial concentration because the dissociation is minimal. Moreover, given the concentrations of the products and the remaining reactant, students should be careful in their calculations taking into account the slight change in the concentration of the weak acid due to dissociation. Through this understanding, students can appreciate the relationship between the electrochemical behavior of a cell and the underlying chemical equilibria.