Question

The following voltaic cell registers an \(E_{\text {cell }}=0.108 \mathrm{V}\) What is the pH of the unknown solution? $$\operatorname{Pt}\left|\mathrm{H}_{2}(\mathrm{g}, 1 \mathrm{atm})\right| \mathrm{H}^{+}(x \mathrm{M}) \| \mathrm{H}^{+}(1.00 \mathrm{M}) |$$ $$\mathrm{H}_{2}(\mathrm{g}, 1 \mathrm{atm}) | \mathrm{Pt}$$

Short answer

The pH of the unknown solution can be calculated by firstly using the Nernst equation to find the concentration of H+ ions (x M) in the solution, then use the pH formula, pH = -log[H+]. After calculating, substitute the concentration (x) into the pH equation to obtain the final answer.

Step-by-step solution

Understand the Chemical Reaction

Firstly, try to understand the redox reaction which is happening. In this cell, Hydrogen gas at the anode is being converted to H+ ions, while at the cathode, H+ ions are being reduced to Hydrogen gas. Essentially, the same half-reaction is occurring twice, but with differing concentrations of H+ ions. The overall cell reaction would thus be: \[2H^{+}(1M) + 2e^- \rightarrow 2H^{+}(xM) + 2e^-\]

Calculate the Cell Potential

Next, write down the Nernst equation, which is used to calculate the cell potential based on the concentrations of reactants and products: \[E = E^0 - \frac{0.05916}{n}logQ\] In this case, n equals 2 for the transferred electrons and Q is the reaction quotient which would be the ratio of the concentration of the hydrogen ion of the unknown solution to that of the standard hydrogen electrode solution. So, the Nernst equation in this case would be: \[E = E^0 - \frac{0.05916}{2}log \frac{[H^{+}](xM)}{[H^{+}](1M)}\]

Substitute Known Values

The E^0 for the reaction would be 0 because it's a standard hydrogen electrode. Substitute E^0, E and Q values into the Nernst equation. Therefore, \[0.108 = 0 - \frac{0.0592}{2}logx\]

Calculate the Value of x

Now solve for x: \[logx = - \frac{2*0.108}{0.0592}\] \[x = 10^{- \frac{2*0.108}{0.0592}}\]

Calculate pH

Finally, calculate the pH of the solution with the equation: \[pH = -log[H^{+}]\] \[pH = -log(x)\]

Final Calculation

Plug the previously calculated value of x into the equation to obtain the pH. Thus giving the final pH of the solution.

Key concepts

Voltaic Cell

A voltaic cell, also known as a galvanic cell, plays a fundamental role in electrochemistry, as it converts chemical energy into electrical energy through spontaneous redox reactions. Structurally, this device is composed of two different metals connected by a wire and a salt bridge. In an educational setting, you're likely to encounter a simplified version involving two half-cells, each with a metal electrode and an electrolyte solution, where oxidation occurs at the anode and reduction at the cathode.

It's fascinating to watch how chemical reactions can produce electricity, and understanding the working of a voltaic cell is crucial for grasping concepts in redox chemistry. For example, in a hydrogen-based cell, pure hydrogen gas is oxidized, releasing electrons that travel through the external circuit to the other half-cell, generating an electromotive force (EMF).

Cell Potential

Cell potential, represented as E_cell, measures the driving force behind the electrical current in a voltaic cell. It is the difference in potential between two half-cells and indicates how willing electrons are to flow from one half-reaction to another. The greater the difference in potential, the greater the cell's ability to do work through electrical energy.

In the context of pH calculations, the cell potential can be calculated using the Nernst equation. This involves knowledge of the standard cell potential (E⁰), the charge passed in the reaction (number of electrons, n), and the reaction quotient, Q, which tells us the ratio of product to reactant concentrations at a given moment. All these factors together determine the current potential of the cell as compared to its standard-state potential.

Redox Reaction

A redox reaction, or oxidation-reduction reaction, is a chemical reaction involving the transfer of electrons between two species. It's composed of two half-reactions: oxidation, where electrons are lost, and reduction, where electrons are gained. In many educational resources, you'll find that balancing redox reactions is crucial for understanding and solving electrochemical problems, such as those involving voltaic cells.

In our exercise, the redox process involves hydrogen gas (H₂) at the anode giving up electrons to become H⁺ ions, indicating oxidation, while the reverse occurs at the cathode. The ability to recognize and understand these reactions is fundamental in electrochemistry, particularly in computing cell potentials by linking chemical behavior with electrical energy changes.

Standard Hydrogen Electrode

The standard hydrogen electrode (SHE) is the universal reference for measuring electrode potentials. It's set at zero volts by definition, and any other electrode's potential is measured against it. The SHE consists of a platinum electrode with hydrogen gas at 1 atm bubbled through a solution of 1 M H⁺ ions.

In our given example, the SHE is used in conjunction with a different concentration of H⁺, creating a potential difference, which we calculate to find the pH of the unknown solution. Understanding the role of the SHE is essential for comparing different electrodes and calculating cell potentials accurately.

Reaction Quotient

The reaction quotient, Q, is a measure of the relative amounts of products and reactants present during a reaction at a particular point in time. It is used to predict the direction of the reaction and is instrumental in the Nernst equation for determining cell potential. Q is a ratio that compares the concentrations of ionic species in a reaction at any given moment to those at equilibrium.

In the context of our exercise, Q is used to calculate the E_cell for a hydrogen voltaic cell. Recognizing how reaction quotients change and affect cell potential is a significant part of understanding chemical reactions, especially their dynamic nature and how they move towards equilibrium.

pH Calculation

pH calculation is a fundamental aspect of acid-base chemistry, involving the measure of the acidity or alkalinity of a solution. In essence, pH is the negative log of the hydrogen ion (H⁺) concentration in a solution. The pH scale generally ranges from 0 to 14, with lower values representing acidic solutions, higher values representing basic solutions, and 7 indicating neutrality.

Relating this to electrochemistry, we often measure pH by calculating H⁺ ion concentration from the cell potential in a voltaic cell, which then allows us to take the negative logarithm to find the pH, as was done in the given exercise. The process intimately ties together the understanding of cell potential and acid-base equilibrium.

Chemical Equilibrium

Chemical equilibrium refers to the state of a reaction where the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the amounts of reactants and products. At equilibrium, the reaction quotient, Q, becomes the equilibrium constant, K. It's pivotal in understanding the behavior of chemical systems and the conditions under which they operate.

Although the reaction in the exercise hasn't reached equilibrium, understanding this concept is crucial for interpreting the cell potential and predicting the direction of the reaction. Equilibrium concepts underpin much of the chemistry involved in electrochemical processes like those seen in voltaic cells.