Problem 57 - Chemistry The Science in Context

Question

If the potential of a hydrogen electrode based on the half-reaction $$2 \mathrm{H}^{+}(a q)+2 \mathrm{e}^{-} \rightarrow \mathrm{H}_{2}(g)$$ is \(0.000 \mathrm{V}\) at \(\mathrm{pH}=0.00,\) what is the potential of the same electrode at \(\mathrm{pH}=7.00 ?\)

Step-by-Step Solution

Answer

Using the Nernst equation, the potential of the hydrogen electrode at pH 7.00 can be calculated as approximately -0.414 V.

Step 1: Nernst Equation

The Nernst equation is given by: $$E = E^\circ - \dfrac{RT}{nF} \ln Q$$ Where: \(E\) - potential of the electrode \(E^\circ\) - standard potential of the electrode \(R\) - gas constant (8.314 J/mol K) \(T\) - temperature in Kelvin (assume 298K, standard temperature) \(n\) - number of electrons transferred in the reaction (2 in this case) \(F\) - Faraday's constant (96485 C/mol) \(Q\) - reaction quotient (has the same form as the equilibrium constant K, but with non-equilibrium concentrations)

Step 2: Calculate Reaction Quotient \(Q\)

From the half-reaction equation: $$2 \mathrm{H}^{+}(a q)+2 \mathrm{e}^{-} \rightarrow \mathrm{H}_{2}(g)$$ We can write the reaction quotient Q as: $$Q = \dfrac{1}{[\mathrm{H}^{+}]^2}$$ (assuming 1 atm for H2 gas and 1 M for H+ ions as standards)

Step 3: Find the standard potential \(E^\circ\)

Since we are given that the potential of the hydrogen electrode at \(\mathrm{pH}=0.00\) is \(0.000 \mathrm{V}\), we know that \([\mathrm{H}^{+}] = 1 \,\mathrm{M}\) (as \(\mathrm{pH}=-\log_{10}[\mathrm{H}^{+}]\)). Substituting this information into the Nernst equation, we can find the standard potential, \(E^\circ.\) $$0.000 \mathrm{V} = E^\circ - \dfrac{8.314 * 298}{2 * 96485} \ln \left(\dfrac{1}{1^2} \right)$$ As \(\ln 1 = 0,\) the equation becomes $$0.000 \mathrm{V} = E^\circ$$

Step 4: Find the H+ concentration at pH 7.00

The \(\mathrm{pH}\) of a solution is related to the concentration of H+ ions as: $$\mathrm{pH} = -\log_{10}[\mathrm{H}^{+}]$$ So for \(\mathrm{pH}=7.00\), the concentration of H+ ions is: $$[\mathrm{H}^{+}] = 10^{-7} \,\mathrm{M}$$

Step 5: Calculate the potential at pH 7.00

Now we have all the information needed to find the electrode potential at \(\mathrm{pH}=7.00\). Substituting the values into the Nernst equation: $$E = 0.000 \mathrm{V} - \dfrac{8.314 * 298}{2 * 96485} \ln \left(\dfrac{1}{(10^{-7})^2} \right)$$ $$E \approx -0.414 \mathrm{V}$$ The potential of the hydrogen electrode at \(\mathrm{pH}=7.00\) is approximately \(-0.414 \mathrm{V}\).

Key Concepts

Electrode Potential

The concept of electrode potential is pivotal in understanding electrochemistry. Electrode potential refers to the ability of an electrode in an electrochemical cell to drive an electric current. It essentially tells us the voltage necessary to move electrons from one side of the cell to the other.

• In a standard setup, the hydrogen electrode is set at 0 volts, serving as a reference point.
• The potential of other electrodes is measured relative to this baseline.
• This allows scientists to determine whether a given electrode can easily give up or gain electrons in a reaction.

Electrode potential is fundamentally connected to the Gibbs free energy of a reaction. This connection gives insight into how spontaneous a given reaction is. Using the Nernst equation, the potential of an electrode can change due to various factors such as concentrations, temperature, and pressure. Unlike standard potential, which is fixed, actual electrode potential depends on real conditions.

pH Scale

The pH scale is a measure of the acidity or basicity of an aqueous solution. It ranges from 0 to 14, with 7 being neutral. The pH value is directly related to the concentration of hydrogen ions ( H^+ ) in the solution.

• A low pH value (below 7) indicates a high concentration of hydrogen ions, making the solution acidic.
• A high pH value (above 7) signals a low concentration, resulting in a basic solution.
• pH affects the behavior of various chemical reactions and is a critical factor in the Nernst equation when calculating electrode potential.

In the context of the Nernst equation, changing the pH of a solution, alters the concentration of hydrogen ions. As demonstrated in the original exercise, moving from pH 0 to pH 7 involves changing the H^+ concentration from 1 M to 10^{-7} M. This shift substantially impacts the calculated electrode potential, highlighting the importance of pH in electrochemical calculations.

Reaction Quotient

The reaction quotient, denoted by \( Q \), plays a crucial role in predicting the direction of a chemical reaction. It is analogous to the equilibrium constant, \( K \), but is calculated under non-equilibrium conditions.

• \( Q \) helps determine how far a reaction is from reaching equilibrium.
• If \( Q = K \), the system is at equilibrium, and no net reaction occurs.
• If \( Q eq K \), the reaction will proceed in a specific direction to restore balance.

The value of \( Q \) is computed using the same formula as \( K \), but focusing on the current concentrations of the reactants and products. For the hydrogen electrode half-reaction, the reaction quotient is given by:\[ Q = \frac{1}{[\mathrm{H}^+]^2} \]
This illustrates how the concentrations of species influence the system's potential. A shift in hydrogen ion concentration profoundly affects \( Q \), guiding the magnitude of potential shift, as demonstrated in the exercise.