Problem 16 - Chemistry in 30 days

Question

Consider the cell: \(\mathrm{Cu} / \mathrm{Cu}^{2+}(0.01 \mathrm{M}) 11 \mathrm{Ag}^{+}(0.1 \mathrm{M}) \mid \mathrm{Ag}\). The e.m.f. of the cell can be increased by (a) Exchanging the concentration in the two half cells (b) Decreasing the concentrations to one tenth in each half cell (c) Making the concentrations \(1.0 \mathrm{M}\) for each side (d) Increasing the concentrations to ten times in each half cell

Step-by-Step Solution

Answer

Increasing concentrations tenfold in each half-cell (option d) increases the e.m.f. of the cell.

Step 1: Identify Nernst Equation for the Cell

The Nernst Equation can calculate the cell potential, considering concentration: \[ E_{cell} = E^0_{cell} - \frac{RT}{nF} \ln \left(\frac{[\text{products}]}{[\text{reactants}]}\right) \]. For a standard condition with one electron transfer, this simplifies to: \[ E_{cell} = E^0_{cell} - \frac{0.059}{n} \log \left(\frac{[ ext{Ag}^+]}{[ ext{Cu}^{2+}]}\right) \] where \(n\) is the number of electrons transferred.

Step 2: Evaluate Condition (a)

Exchanging concentrations in the two half cells would mean \([ ext{Cu}^{2+}] = 0.1 \, \text{M}\) and \([ ext{Ag}^+] = 0.01 \, \text{M}\). This would increase the reaction quotient \(\frac{[ ext{Ag}^+]}{[ ext{Cu}^{2+}]}\) and thus decrease the cell potential, not increase it.

Step 3: Evaluate Condition (b)

Decreasing both concentrations to one-tenth (\( [ ext{Cu}^{2+}] = 0.001 \, \text{M}, \, [ ext{Ag}^+] = 0.01 \, \text{M} \)). The reaction quotient \(\frac{[ ext{Ag}^+]}{[ ext{Cu}^{2+}]}\) becomes smaller, which might suggest an increase in potential, but reduction in concentration generally makes the cell closer to equilibrium, decreasing the potential.

Step 4: Evaluate Condition (c)

Making concentrations equal to 1 M for both half-cells simplifies the equation to standard conditions, where the concentration of each ion does not affect the cell potential. For this case, \(\log \left(\frac{1}{1}\right) = 0\), exercising no influence on cell potential.

Step 5: Evaluate Condition (d)

Increasing concentrations tenfold (\( [ ext{Cu}^{2+}] = 0.1 \, \text{M}, \, [ ext{Ag}^+] = 1.0 \, \text{M} \)). Here, the reaction quotient \(\frac{[ ext{Ag}^+]}{[ ext{Cu}^{2+}]}\) considerably increases, reducing the negative part of the equation and thus increasing the cell potential.

Key Concepts

Understanding Electrochemistry

Electrochemistry is the study of chemical processes that cause electrons to move. This movement of electrons is a current, which can be harnessed to perform work in electrochemical cells. A key part of electrochemistry involves oxidation-reduction reactions, where the transfer of electrons between species takes place. One species loses electrons (oxidation), and another gains electrons (reduction).
These reactions occur in separate compartments called half-cells. Each half-cell contains a different electrode in a solution with its ions. In these cells, electrons flow from the anode (where oxidation happens) to the cathode (where reduction takes place), generating a current that can be measured.

• The anode and cathode reactions are separated but connected via a salt bridge that allows ions to flow between the compartments.
• The potential difference between these two reactions is the cell potential.

Cell potential is a central concept in electrochemistry because it defines the voltage output of a cell. Electrochemistry provides the foundational knowledge necessary for creating batteries and understanding various chemical storage methods.

Exploring Cell Potential

Cell potential, also known as electromotive force (emf), is the measure of the potential energy difference between the two electrodes in an electrochemical cell. It represents the cell's ability to do work, usually in generating an electrical current. This potential difference drives the electron flow from the anode to the cathode. In the context of an electrochemical cell, cell potential is denoted by E, and it can change depending on the conditions of the cell.
There are several factors that affect cell potential:

• Nature of the electrodes: Different metals and their ions in solution can contribute various potentials due to their specific characteristics.
• Concentration of ionic solutions: Higher concentrations usually lead to increased cell potential due to enhanced reaction tendencies.

The Nernst Equation is critical for determining the cell potential under non-standard conditions:\[ E_{cell} = E^0_{cell} - \frac{RT}{nF} \ln \left(\frac{[\text{products}]}{[\text{reactants}]}\right) \]Here, \( E^0_{cell} \) is the standard potential when concentrations are at 1M, T is the temperature, R is the gas constant, F is Faraday's constant, and n is moles of electrons transferred. In simpler terms,\[ E_{cell} = E^0_{cell} - \frac{0.059}{n} \log \left(\frac{[\text{Ag}^+]}{[\text{Cu}^{2+}]}\right) \]The equation demonstrates that changes in concentration can significantly shift the cell potential.

Concentration Effects on Cell Potential

Changes in concentration can profoundly affect cell potential. This principle is crucial when dealing with non-standard conditions in electrochemical cells. The concentration of reactants and products can shift the position of equilibrium, thus altering the cell potential.
The Nernst Equation provides insights into how these changes impact the potential. When concentrations of reactants increase, the reaction quotient \( \left( \frac{[\text{products}]}{[\text{reactants}]} \right) \) decreases, often resulting in an increased cell potential. Conversely, if the concentration of products increases, this quotient increases, potentially decreasing cell potential.
Let's break down the scenarios:

• High concentration of reactants: These favor the forward reaction, thereby increasing cell potential.
• Equal concentrations at 1M: This establishes standard conditions, where the potential is not influenced by concentration as the quotient becomes 1.
• Decrease in concentrations: This shifts the quotient towards equilibrium, often reducing the potential.

These variations explain why electrochemical cells can be fine-tuned by adjusting the concentration of solutions, making them highly adaptable for specific uses such as in batteries and sensors.