Problem 8 - Chemistry An Atoms First Approach

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?

Step-by-Step Solution

Answer

The relationships between the variables in a galvanic cell are as follows: ΔG=-w_max, w=q*E_cell, ΔG=-n*F*E_cell, and E_cell=E°_cell-(RT/nF)*lnQ. A concentration cell can be created with the same metal electrodes and the same compound in different concentrations, generating a spontaneous reaction due to the natural tendency of systems to achieve equilibrium. Ions flow, causing a redox reaction and producing electrical energy until the concentrations in both half-cells are equal.

Step 1: Relationship 1: ΔG and w

In a galvanic cell, the maximum work that can be done is equal to the change in Gibbs free energy (ΔG). The relationship between ΔG and w is given as: ΔG = -w_max ΔG is negative for a spontaneous reaction, which indicates that the maximum work done by the system is positive.

Step 2: Relationship 2: Cell potential and w

Cell potential (E_cell) is the measure of the voltage generated by a galvanic cell. The work done per unit charge (w/q) is equal to the cell potential. The relationship between cell potential and w is given as: w = q * E_cell

Step 3: Relationship 3: Cell potential and ΔG

As we derived the relationships between ΔG and w and between cell potential and w, we can now link ΔG and cell potential. The relationship between cell potential and ΔG is given by: ΔG = -n * F * E_cell where n is the number of moles of electrons being transferred in the reaction, and F is the Faraday's constant (96485 C/mol).

Step 4: Relationship 4: Cell potential and Q

The cell potential at any point during the electrochemical reaction is given by the Nernst equation: E_cell = E°_cell - (RT/nF) * ln Q where E°_cell is the standard cell potential, R is the universal gas constant (8.314 J/(mol K)), T is the temperature in Kelvin, n is the number of moles of electrons being transferred in the reaction, F is Faraday's constant, and Q is the reaction quotient.

Step 5: Designing a cell with same metal electrodes and same compound in different concentrations

To create a galvanic cell where both electrodes are the same metal and both solutions contain the same compound in different concentrations, we need to use the concept of concentration cells. A concentration cell is an electrochemical cell where the emf (electromotive force) is generated solely due to the difference in the concentration of the same ionic species in both half-cells. An example of such a cell could involve two half-cells, each containing a copper electrode immersed in copper sulfate (CuSO4) solution. However, the concentrations of CuSO4 solutions in the two half-cells will be different, forming a concentration gradient. This concentration gradient will lead to a potential difference, hence generating a spontaneous reaction.

Step 6: Why does such a cell run spontaneously?

A concentration cell spontaneously generates an electrochemical reaction because of the natural tendency of systems to achieve equilibrium. When there is a difference in concentrations between the two half-cells, ions will start to flow and the redox reaction will take place. This process will continue until the concentrations of CuSO4 in both half-cells are equal, and in the process, electrical energy will be produced by the cell.

Key Concepts

Gibbs Free Energy (G)

Understanding Gibbs free energy (G) is pivotal to grasping the thermodynamics of a galvanic cell. Gibbs free energy, a measure of the total energy available to do work during a chemical process, is a thermodynamic quantity that helps predict the direction of a chemical reaction. For spontaneous reactions, G must be negative, signifying that the system is releasing energy.

This concept ties directly to the maximum work (G = -w_max) that can be done by the electrochemical cell. This maximum work is equivalent to the electrical energy obtained in the external circuit when electrons flow through it. Such spontaneous reactions in galvanic cells are fundamental in generating electrical energy from chemical reactions.

Cell Potential (E_cell)

Cell potential, often represented as (E_cell), is the driving force of a galvanic cell — essentially it's the voltage or electromotive force the cell can produce. The cell potential indicates how much voltage is available from the redox reaction occurring in the cell, and it's directly related to the work (w) that can be extracted from the cell ( w = q * E_cell), where q is the charge transferred.

This value is influenced by numerous factors such as the types of electrodes, the temperature, and the concentration of the solutions in the cell. In textbook problems, understanding how changes to these factors can affect the cell potential is key to solving for variables like work done or Gibbs free energy.

Nernst Equation

The Nernst equation is a formula used to calculate the cell potential at any given concentration, temperature, and pressure, not just under standard conditions. The equation is given by:

E_cell = E°_cell - (RT/nF) * ln Q

Here, E°_cell represents the standard cell potential, R is the universal gas constant, T is the temperature in Kelvin, n is the number of moles of electrons involved in the redox reaction, F is the Faraday's constant, and Q is the reaction quotient. The Nernst equation shows how cell potential changes with concentration ( Q), enabling us to explain concentration cells and predict the potential at non-standard conditions.

Concentration Cells

Concentration cells are a special type of galvanic cell where the electrodes are made of the same material and the electrolyte solutions are the same but have different concentrations. They operate on the principle that cells will attempt to reach equilibrium.

In a concentration cell, because of the concentration gradient, chemical species will move from areas of high concentration to low concentration, which creates a potential difference and hence generates an electromotive force (emf). This is what drives the reaction to be spontaneous, as the system seeks equilibrium.

Electromotive Force (emf)

The term electromotive force (emf) refers to the potential difference that drives electrons through an external circuit. It's essentially the 'pressure' that pushes the electrons, and in a galvanic cell, it results from the chemical reaction occurring within. The emf is measured in volts and is what is measured when we refer to the cell potential (E_cell).

A greater emf corresponds to a greater capacity to do electrical work, making it directly proportional to the cell's ability to do work and reflective of the various cell conditions, such as concentration, as elucidated by the Nernst equation.