Problem 51 - Chemistry: An Atoms-Focused Approach

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

Answer: The potential of a hydrogen electrode at pH 7 is approximately -0.414 V.

Step 1: Reminder of the Nernst Equation

The Nernst equation allows us to calculate the potential of an electrode under non-standard conditions. It can be written as: $$ E = E^{0} - \dfrac{RT}{nF} \ln{Q} $$ where \(E\) is the electrode potential, \(E^{0}\) is the standard electrode potential, \(R\) is the gas constant, \(T\) is the temperature, \(n\) is the number of electrons transferred, \(F\) is the Faraday's constant, and \(Q\) is the reaction quotient. In our case, since we have the reaction: $$ 2 \mathrm{H}^{+}(a q)+2 \mathrm{e}^{-} \rightarrow \mathrm{H}_{2}(g) $$ The Nernst equation can be simplified to: $$ E = E^{0} - \dfrac{RT}{2F} \ln{\dfrac{[\mathrm{H}^{+}]^{2}}{P(\mathrm{H}_{2})}} $$

Step 2: Establish the initial conditions

At the initial pH 0, the electrode potential is 0 volts (\(E_1 = 0.000\,\mathrm{V}\)), and the hydroxide ion concentration is \([\mathrm{H}^{+}] = 10^{-0} = 1\,\mathrm{M}\). Also, the pressure is given as 1 atmosphere (P(H\(_2\)) = 1 atm). Therefore, we can write: $$ E_1 = E^{0} - \dfrac{RT}{2F} \ln{\dfrac{1^{2}}{1}} $$

Step 3: Establish the final conditions

At the final pH of 7, we need to calculate the electrode potential (\(E_2\)). The hydroxide ion concentration at the final pH is \([\mathrm{H}^{+}] = 10^{-7}\,\mathrm{M}\). We will still assume the pressure to be 1 atmosphere (P(H\(_2\)) = 1 atm). So we can write: $$ E_2 = E^{0} - \dfrac{RT}{2F} \ln{\dfrac{(10^{-7})^{2}}{1}} $$

Step 4: Calculate the potential difference

From the equations in steps 2 and 3, we can find the potential difference by subtracting the two Nernst equations: $$ E_2 - E_1 = - \dfrac{RT}{2F} \ln{\dfrac{(10^{-7})^{2}}{1}} + \dfrac{RT}{2F} \ln{(1)^{2}} $$ Since \(E_1 = 0\,\mathrm{V}\) and \(\ln{1}=0\), the equation becomes: $$ E_2 = - \dfrac{RT}{2F} \ln{(10^{-7})^{2}} $$

Step 5: Calculate the final potential

To calculate the final potential, we need the value of R, T, and F. At room temperature (25°C), we have: - R = 8.314 J/(mol·K) - T = 25°C + 273.15 = 298.15 K - F = 96485 C/mol Substitute these values into the equation: $$ E_2 = - \dfrac{8.314\,\mathrm{J/mol\cdot K} \times 298.15\,\mathrm{K}}{2 \times 96485\,\mathrm{C/mol}} \ln{(10^{-7})^{2}} $$ Solving the equation for \(E_2\) gives: $$ E_2 = -0.414\,\mathrm{V} $$ This means that the potential of the hydrogen electrode at pH 7 is approximately \(-0.414\,\mathrm{V}\).

Key Concepts

Nernst equation

The Nernst Equation is a fundamental tool in chemistry that helps us understand and calculate the electrode potential of a cell under non-standard conditions. We have seen that the equation is given by:\[\mathit{E} = E^{0} - \frac{RT}{nF} \ln{Q}\]Here’s a breakdown of its components:

• \(E\) is the electrode potential, the voltage across the electrode.
• \(E^{0}\) is the standard electrode potential under standard conditions, such as 1 molar concentration.
• \(R\) is the universal gas constant, approximately \(8.314 \text{ J/mol·K}\).
• \(T\) is the absolute temperature in Kelvin.
• \(n\) is the number of electrons transferred in the half-reaction.
• \(F\) is Faraday's constant, approximately \(96485 \text{ C/mol}\).
• \(Q\) is the reaction quotient, representing the relative concentrations of products and reactants at any given time.

By adjusting these parameters, the Nernst Equation allows us to calculate the potential of an electrochemical cell, giving insight into how conditions like concentration and temperature affect cell voltage.
It reveals the importance of chemical concentrations in determining how much voltage an electrode can provide.

pH level

pH is an essential concept in chemistry since it helps determine the acidity or basicity of a solution, influencing many reactions and processes. It is a scale ranging from 0 to 14, with 7 being neutral.

• A pH less than 7 indicates an acidic solution, containing a higher concentration of \([\mathrm{H}^+]\) ions.
• Conversely, a pH higher than 7 indicates a basic solution, with more \([\mathrm{OH}^-]\) ions.

The formula for pH is:\[\text{pH} = -\log_{10}[\mathrm{H}^+]\]This relationship shows how the concentration of hydrogen ions determines the pH of a solution. In the hydrogen electrode exercise, we used the equation to understand how a change from pH 0 to pH 7 affects the electrode potential.
Knowing that at pH 7, \([\mathrm{H}^+] = 10^{-7} \, \text{M}\), the Nernst equation helps us calculate the electrode potential, demonstrating the impact of pH changes on electrochemical reactions.

hydrogen electrode

A hydrogen electrode is a type of reference electrode used in electrochemistry to determine the electrode potentials of other cells.It is based on the reversible reaction:\[2 \mathrm{H}^{+}(aq) + 2 \mathrm{e}^{-} \rightarrow \mathrm{H}_{2}(g)\]What makes it essential in this context is its standard potential:

• At standard conditions (pH 0, 1 M concentration, and 1 atm pressure), this potential is set to \(0.000 \text{ V}\), serving as a baseline for measuring other electrode reactions.

In our exercise, we examined how changing the pH from 0 to 7 influenced the potential of the same hydrogen electrode.
This theory allows scientists to investigate chemical reactions in cells by making comparisons to a well-defined standard.

reaction quotient

The reaction quotient \(Q\), is a ratio that compares the concentrations of the products to the reactants for a particular reaction at any point in time. For the general reaction:\[\text{aA} + \text{bB} \rightarrow \text{cC} + \text{dD}\]
\(Q\) is given by:\[Q = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]Understanding \(Q\) is crucial because it helps determine the direction a reaction must proceed to reach equilibrium.

• If \(Q < K\) (equilibrium constant), the reaction moves forward, producing more products.
• If \(Q > K\), the reaction pushes back, generating more reactants.

In the context of the Nernst equation, \(Q\) helps us calculate cell potentials by considering real-time concentrations.
This aspect emphasizes the dynamic nature of chemical reactions, as they strive to reach equilibrium, impacting the voltages seen in electrochemical cells.