Question

A concentration cell is constructed of two hydrogen electrodes: one immersed in a solution with \(\left[\mathrm{H}^{+}\right]=1.0 \mathrm{M}\) and the other in \(0.65 \mathrm{M} \mathrm{KOH}\) (a) Determine \(E_{\text {cell for the reaction that occurs. }}\) (b) Compare this value of \(E_{\text {cell }}\) with \(E^{\circ}\) for the reduction of \(\mathrm{H}_{2} \mathrm{O}\) to \(\mathrm{H}_{2}(\mathrm{g})\) in basic solution, and explain the relationship between them.

Short answer

The calculated \(E_{\text {cell}}\) and \(E^{\circ}\) are equal as they represent the same electrochemical reaction. The sign of \(E_{\text {cell}}\) is negative, suggesting that the reaction is not spontaneous under the given conditions.

Step-by-step solution

Determine the concentration of \(OH^{-}\) in \(0.65 \mathrm{M} \mathrm{KOH}\)

Since KOH is a strong base and completely dissociates in water, the concentration of \([OH^{-}]=0.65 M\).

Calculate the concentration of \(H^{+}\) ions using the ionic product of water

Considering neutral solution at 25 °C, the ionic product of water KW is given by \(KW=[H^{+}][OH^{-}]=1.0×10^{−14}\). Therefore, the concentration of \(H^{+}\) can be found using this equation.

Determine the cell potential \(E_{\text {cell}}\)

Apply the Nernst equation: \(E_{\text {cell}}=E^0-\frac{0.0592}{n} \log Q\), where \(E^0\) is the standard cell potential, n is the number of moles of electrons transferred in the half-reaction, and Q is the reaction quotient. Since it's a hydrogen concentration cell, \(E^0=0\) and n=2. Therefore, our equation simplifies as \(E_{\text {cell}}=-\frac{0.0592}{2} \log Q\).

Evaluate the reaction quotient Q

Here, Q can be defined as the ratio of the concentration of products over reactants. Because the equation is \(2H^{+} + 2e^{-} → H_{2}\), Q would be \(Q=[H^{+}_{product}]/[H^{+}_{reactant}]\). Replace and calculate Q.

Calculate \(E_{\text {cell}}\) using the Nernst equation

Replace Q in the Nernst equation with the calculated value and solve for \(E_{\text {cell}}\).

Compare \(E_{\text {cell}}\) and \(E^{\circ}\)

Since it's a hydrogen electrode, the procedure for the reduction of \(H_{2}O\) to \(H_{2}(g)\) in basic solution is the same as for this cell, thus, \(E^{\circ} = 0\). Discuss the relationship between \(E_{\text {cell}}\) and \(E^{\circ}\) based on the comparison.

Key concepts

Nernst Equation

The Nernst equation is instrumental in electrochemistry as it allows us to calculate the voltage of an electrochemical cell under non-standard conditions. It relates the measurable cell potential, temperature, and the reaction quotient (Q), which indicates concentrations of ions at a given moment.

For a general redox reaction occurring in a cell, the Nernst equation is given by: \[ E_{\text{cell}} = E^{\circ}_{\text{cell}} - \frac{0.0592}{n} \log Q \]
where \( E_{\text{cell}} \) is the cell potential under non-standard conditions, \( E^{\circ}_{\text{cell}} \) is the standard cell potential, \( n \) is the number of moles of electrons transferred in the half-reaction, and \( Q \) is the reaction quotient.
To apply the Nernst equation effectively, one must understand how to determine the reaction quotient and factor in the appropriate number of electron transfers occurring in the half-reactions. In concentration cells, such as a hydrogen cell with differing concentrations, the equation simplifies since the standard cell potential, \( E^{\circ}_{\text{cell}} \), is zero.

Standard Cell Potential

The standard cell potential (\( E^{\circ}_{\text{cell}} \)) is a measure of the inherent voltage of an electrochemical cell under standard conditions, which typically means solutes at 1 M concentration, gases at 1 atm pressure, and a temperature of 25°C (298 K). It is determined by the difference in the standard potentials of the cathode and the anode.

In other words, \( E^{\circ}_{\text{cell}} \) reflects the cell's ability to drive an electric current when the reactants and products are in their standard states. For a concentration cell comprised of identical electrodes with different ion concentrations, the \( E^{\circ}_{\text{cell}} \) is zero because under standard conditions, the cells would have equal concentration, and therefore, no potential difference would exist.

Ionic Product of Water

The ionic product of water (\( K_w \)) is a fundamental concept in acid-base chemistry. It is the product of the molar concentrations of hydrogen ions (\( H^+ \)) and hydroxide ions (\( OH^- \)) in water at a given temperature. At 25°C, this value is \( 1.0 \times 10^{-14} \).

In a solution, the product of \( [H^+] \) and \( [OH^-] \) must always equal \( K_w \), which implies that if we know the concentration of one ion, we can calculate the other. This principle is crucial when solving problems involving concentration cells, as it helps us determine the concentration of either ion when the other is available, thus contributing to the calculation of the reaction quotient (Q) for the Nernst equation.

Hydrogen Electrode

The hydrogen electrode serves as a reference electrode in electrochemical measurements. It consists of a platinum wire or sheet in contact with hydrogen gas and immersed in an acidic solution where the concentration of hydrogen ions is well-defined.

The hydrogen electrode works on the simple principle that hydrogen gas can be reduced to form hydrogen ions or hydrogen ions can be oxidized to form hydrogen gas. It plays a pivotal role in setting up concentration cells as it establishes a half-cell with a potential determined by the hydrogen ion concentration. When constructing a concentration cell with hydrogen electrodes, different concentrations of \( H^+ \) ions produce different potentials at each electrode, leading to a measurable voltage for the cell powered by the tendency of the system to reach equilibrium.