Question

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

Short answer

The \(K_a\) value for the weak acid HA in the given galvanic cell is approximately \(1.69 \times 10^{-4}\).

Step-by-step solution

Write down the Nernst equation

The Nernst equation is used to determine the cell potential of an electrochemical cell under non-standard conditions. It is given by: \[E_{cell} = E_{cell}^° - \frac{RT}{nF} \cdot \ln Q\] Where: \(E_{cell}\) = cell potential under non-standard conditions \(E_{cell}^°\) = standard cell potential \(R\) = gas constant (8.314 J/mol·K) \(T\) = temperature in Kelvin \(n\) = number of moles of electrons transferred \(F\) = Faraday constant (96,485 C/mol) \(Q\) = reaction quotient

Calculate the standard cell potential

We are given the standard potentials for each half-reaction. Add them to find the standard cell potential: \[E_{cell}^° = E_{Fe}^° + E_{H_2}^°\] \[E_{cell}^° = (-0.440 V) + (0.000 V)\] \[E_{cell}^° = -0.440 V\]

Calculate the reaction quotient (Q)

The balanced overall redox reaction is: \[\mathrm{Fe}^{2+}(aq) + \mathrm{H}_{2}(g) → \mathrm{Fe}(s) + 2\mathrm{H}^{+}(aq)\] The reaction quotient (Q) is given by: \[Q = \frac{[\mathrm{H}^{+}]^2}{[\mathrm{Fe}^{2+}]}\] Since the concentration of HA (weak acid) is 1.00 M and [\(\mathrm{H}^{+}\)] is initially assumed negligible compared to HA, we can say that: \[[\mathrm{H}^{+}] = x\] Therefore, \[Q = \frac{x^2}{1.00 \times 10^{-3}}\]

Obtain an expression for x and calculate Ka using the Nernst equation

Now, we can substitute the values in the Nernst equation: \[0.333 V = -0.440 V - \frac{(8.314 J/mol·K)(298 K)}{2(96,485 C/mol)} \cdot \ln{\frac{x^2}{1.00 \times 10^{-3}}}\] Rearrange the equation to solve for x: \[\ln{\frac{x^2}{1.00 \times 10^{-3}}} = \frac{2(96,485 C/mol)(0.333 V + 0.440 V)}{(8.314 J/mol·K)(298 K)}\] \[\ln{\frac{x^2}{1.00 \times 10^{-3}}} = 16.99\] Take the antilogarithm of both sides: \[\frac{x^2}{1.00 \times 10^{-3}} = e^{16.99}\] Now solve for x: \[x = \sqrt{1.00 \times 10^{-3} \cdot e^{16.99}}\] \[x \approx 1.30 \times 10^{-2} M\] The dissociation constant, \(K_a\), for the weak acid HA can be calculated by: \[K_a = \frac{x^2}{1 - x}\] \[K_a = \frac{(1.30 \times 10^{-2})^2}{(1 - 1.30 \times 10^{-2})}\] \[K_a \approx 1.69 \times 10^{-4}\] So, the \(K_a\) value for the weak acid HA is approximately \(1.69 \times 10^{-4}\).

Key concepts

Understanding the Nernst Equation

The Nernst equation is a fundamental principle in electrochemistry, providing insight into how the potential of an electrochemical cell changes with concentration. The equation is expressed as:
\[E_{cell} = E_{cell}^° - \frac{RT}{nF} \cdot \ln Q\]
Where:

• \(E_{cell}\) represents the cell potential under non-standard conditions.
• \(E_{cell}^°\) is the standard cell potential when the reactants and products are at standard conditions (1 M concentration for solutions, 1 atm pressure for gases).
• \(R\) is the universal gas constant, which has the value of 8.314 J/(mol·K).
• \(T\) is the temperature in Kelvin.
• \(n\) denotes the number of moles of electrons transferred in the cell reaction.
• \(F\) stands for the Faraday constant, approximately 96,485 C/mol, representing the charge of one mole of electrons.
• \(Q\) is the reaction quotient, which reflects the concentrations of the reactants and products at any given instant.

The Nernst equation is crucial as it allows us to calculate the real-world cell potential, taking into account the concentrations of the cell's components, rather than just the standard potential.

Standard Cell Potential

Standard cell potential, denoted as \(E_{cell}^°\), is a measure of the electromotive force (EMF) of a galvanic or voltaic cell under standard conditions. The standard conditions imply that the temperature is 298 K, all gases are at 1 atm pressure, and all aqueous species are at a 1 M concentration.
For a given galvanic cell, standard cell potential is calculated by summing up the standard potentials of the cathode and anode reactions:
\[E_{cell}^° = E_{cathode}^° - E_{anode}^°\]
It's important to remember that electrons flow from the anode to the cathode, and the standard cell potential is positive for spontaneous cell reactions. In the problem provided, a negative standard cell potential indicates a need for an external potential to drive the reaction.

Reaction Quotient (Q)

The reaction quotient, \(Q\), offers a snapshot of a reaction's quotient at any given moment. It's mathematically similar to the equilibrium constant but does not require that the system be at equilibrium. For a generic chemical reaction:
\[aA + bB \longrightarrow cC + dD\]
The reaction quotient is:
\[Q = \frac{[C]^c [D]^d}{[A]^a [B]^b}\]
where the concentrations of the products are raised to the power of their respective coefficients and divided by the concentrations of the reactants, also raised to the power of their respective coefficients. In an electrochemical cell, Q is used in the Nernst equation to adjust the standard cell potential and reflect how far the system is from equilibrium.

Acid Dissociation Constant (Ka)

The acid dissociation constant, represented by \(K_a\), quantifies the strength of an acid in solution. It is the equilibrium constant for the dissociation reaction of an acid, HA, into a hydrogen ion and its conjugate base, A-, as shown in the following equation:
\[HA \longleftrightarrow H^+ + A^-\]
Thus, the expression for the acid dissociation constant is:
\[K_a = \frac{[H^+][A^-]}{[HA]}\]
A higher \(K_a\) value suggests a stronger acid—that is, the acid more readily donates hydrogen ions in solution. Conversely, a lower \(K_a\) value indicates a weaker acid. In the exercise, the goal is to calculate the \(K_a\) of a weak acid from electrochemical data, which underscores the interconnection between electrochemistry and acid-base chemistry. Understanding these concepts not only aids in solving homework problems but also enhances comprehension of broader chemical behavior.