Question

Explain the following relationships: \(\Delta G\) and \(w,\) cell potential and \(w,\) cell potential and \(\Delta G,\) cell potential and \(Q\) . Using these relationships, explain how you could make a cell in which both electrodes are the same metal and both solutions contain the same compound, but at different concentrations. Why does such a cell run spontaneously?

Short answer

The relationships between Gibbs Free Energy ($\Delta G$), work done by the cell ($w$), cell potential ($E$), and reaction quotient ($Q$) are as follows: 1. $\Delta G = -w$ - indicating that when the system does work, $\Delta G$ decreases. 2. $w = nFE$ - relating work to cell potential and the charge transferred during the reaction. 3. $\Delta G = -nFE$ - connecting Gibbs Free Energy to cell potential. 4. $E_{cell} = E^0_{cell} - \frac{RT}{nF} \ln Q$ - Nernst equation connecting cell potential to the reaction quotient. To create an electrochemical cell with the same metal electrodes and solutions containing the same compound at different concentrations, we can use a concentration cell. The driving force for the cell comes from the difference in concentrations, leading to electron transfer between half-cells. The spontaneity depends on the cell potential and Gibbs Free Energy. In a concentration cell, the positive cell potential and the negative Gibbs Free Energy indicate a spontaneous process. As the reaction progresses, the changing concentrations continue to drive the cell until reaching equilibrium.

Step-by-step solution

1. Relationship between Gibbs Free Energy (\(\Delta G\)) and Work (\(w\))

: The maximum non-expansion work that can be obtained from a reversible process at constant temperature and pressure is represented by Gibbs Free Energy (\(\Delta G\)). In an electrochemical cell, the non-expansion work is the electrical work (\(w\)) done by the cell. Therefore, the relationship between \(\Delta G\) and \(w\) can be expressed as: \[ \Delta G = -w \] Here, the negative sign indicates that when the system (cell) does work on the surroundings, the \(\Delta G\) of the system decreases.

2. Relationship between Cell Potential (\(E\)) and Work (\(w\))

: The amount of work done by an electrochemical cell can be determined by multiplying the cell potential (\(E\)) by the charge transferred during the redox reaction. This relationship can be written as: \[ w = nFE \] where \(n\) is the number of moles of electrons transferred, \(F\) is the Faraday constant (96,485 C/mol), and \(E\) is the cell potential.

3. Relationship between Cell Potential (\(E\)) and Gibbs Free Energy (\(\Delta G\))

: By combining the relationships between \(\Delta G\) and \(w\), and \(w\) and \(E\), we can derive the relationship between cell potential (\(E\)) and Gibbs Free Energy (\(\Delta G\)): \[ \Delta G = -nFE \]

4. Relationship between Cell Potential (\(E\)) and Reaction Quotient (\(Q\))

: The cell potential (\(E_{cell}\)) at any point can be calculated using the Nernst equation, which relates the cell potential to the reaction quotient (\(Q\)): \[ E_{cell} = E^0_{cell} - \frac{RT}{nF} \ln Q \] where \(E^0_{cell}\) is the standard cell potential, \(R\) is the gas constant (8.314 J/mol·K), \(T\) is the temperature in Kelvin, and \(Q\) is the reaction quotient.

5. Creating a Cell with Same Metal Electrodes and Different Concentrations

: To create an electrochemical cell with the same metal electrodes and solutions containing the same compound at different concentrations, a concentration cell can be used. In a concentration cell, the half-reactions are the same, but the concentrations of the metal ions in the two half-cells are different. The driving force for the cell comes from the difference in concentrations, resulting in the transfer of electrons from one half-cell to the other.

6. Spontaneity of the Cell

: The spontaneity of an electrochemical cell depends on the cell potential (\(E\)) and Gibbs Free Energy (\(\Delta G\)). A positive cell potential and a negative Gibbs Free Energy indicate a spontaneous process. In a concentration cell, since the concentrations are different, the cell potential will be positive as one half-cell will be more oxidized than the other. According to the relationship between \(E\) and \(\Delta G\) (\(\Delta G = -nFE\)), this positive cell potential results in a negative Gibbs Free Energy, making the reaction spontaneous. As the reaction progresses, the concentrations in both half-cells change and continue to drive the cell until reaching equilibrium.

Key concepts

Gibbs Free Energy

Gibbs Free Energy (\(\Delta G\)) is central to understanding chemical reactions and processes, as it tells us whether a process can occur spontaneously. In an electrochemical context, it signifies the energy that can be harnessed to do non-expansion work like electrical work. This contrasts with expansion work, such as pushing against atmospheric pressure. When a reaction occurs spontaneously in an electrochemical cell, energy is released, which can be used to drive an electrical current.

• The relationship between \(\Delta G\) and work (\(w\)) is crucial: \(\Delta G = -w\). A negative \(\Delta G\) indicates a spontaneous reaction, meaning energy is available for work.
• In practice, this translates to the cell's ability to produce electrical energy, where the system does work on the surroundings, resulting in decreased Gibbs Free Energy.

At a basic level, if a reaction within an electrochemical cell has a negative \(\Delta G\), it confirms that the process can naturally proceed, driving electrons through a circuit. This is why electrochemical cells, like batteries, can serve as effective power sources.

Cell Potential

Cell potential (\(E\)) is a measure of the energy per charge available from the redox reactions occurring in an electrochemical cell. It is vital for determining whether a reaction can perform work, such as generating electricity.The potential reflects the inherent energy differences between the oxidized and reduced forms of a substance. It can be calculated from the standard cell potential (\(E^0\)), but it also varies based on conditions such as reactant concentration.

• The mathematical connection between work (\(w\)) and cell potential is expressed as: \(w = nFE\).
• Here, \(n\) is the number of moles of electrons transferred, and \(F\) stands for the Faraday constant (\(96,485\, \text{C/mol}\)).
• This equation shows that the work done by a cell is directly related to the cell potential and the amount of charge moved.

Highlighting the direct relationship between cell potential and Gibbs Free Energy, the equation \(\Delta G = -nFE\) indicates that more positive cell potentials result in more negative \(\Delta G\) values, thus signifying spontaneous reactions.

Nernst Equation

The Nernst equation is a powerful tool in electrochemistry, enabling you to relate the cell potential to the concentrations of reactants and products in a cell. By adjusting for these concentration differences, it accounts for the deviation of actual cell conditions from standard conditions.The equation takes the following form:\[E_{cell} = E^0_{cell} - \frac{RT}{nF} \ln Q\]

• \(E^0_{cell}\) is the standard cell potential, reflecting idealized conditions.
• \(R\) is the universal gas constant (\(8.314 \, \text{J/mol}\cdot\text{K}\)), and \(T\) represents temperature in Kelvin.
• \(Q\), the reaction quotient, shows the ratio of product concentrations to reactant concentrations at any moment.

The Nernst equation shows how changes in concentration affect the cell potential, thus guiding predictions about reaction spontaneity and direction. For concentration cells, it is particularly insightful, as it discerns the potential driven solely by concentration differences.

Concentration Cell

In a concentration cell, both electrodes are made of the same material and the cell generates electrical energy solely from a difference in ion concentration between the two half-cells. This type of cell exploits the principle that electron flow is motivated by concentration imbalances. The steps involved in creating this cell entail:

• Using identical metal electrodes dipped into solutions of their respective ions, but at differing concentrations.
• The greater concentration functions as the cathode, while the lower concentration defines the anode, initiating electron flow from low to high concentration.

This concentration gradient is the cell's driving force. The cell's potential can be calculated with the Nernst equation, further validating the spontaneous direction of electron flow. A concentration cell naturally reaches equilibrium as concentrations equalize, driving the cell until it can no longer produce energy. Two primary reasons for its spontaneity include:

• The positive cell potential results from the higher energy state of the more concentrated solution.
• The negative Gibbs Free Energy aligns with spontaneous electron transfer and energy release, continuing until equilibrium is reached.

Thus, concentration cells are ingenious examples of chemical principles harnessed to create usable energy from simple concentration differences.