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 various thermodynamic aspects of an electrochemical cell are as follows: (1) \(\Delta G = -w \) shows that the negative Gibbs free energy change indicates a spontaneous process with positive maximum work; (2) \(w = E_{cell} \cdot |q|\) relates work done by a cell to its potential and charge transferred; (3) \(\Delta G = -nFE_{cell}\) allows us to determine spontaneity based on cell potential; and (4) \(E_{cell} = E_{cell}^0 - \frac{RT}{nF} \ln Q\), the Nernst equation, relates cell potential to reaction quotient and concentrations. A cell with same metal electrodes and different concentrations, called a concentration cell, can be created, which runs spontaneously as it moves toward equilibrium to equalize the concentrations at both electrodes. This spontaneity is a result of the concentration difference driving ion flow and electron transfer between half-cells.

Step-by-step solution

Relationship 1: \(\Delta G\) and \(w\)

Gibbs free energy change (\(\Delta G\)) is the maximum amount of non-expansion work (\(w\)) that can be done by a system as it moves from an initial state to a final state. For an electrochemical cell, this relationship can be expressed as: \[\Delta G = -w \] This implies that a negative \(\Delta G\) (spontaneous process) means that the maximum possible work extracted from the cell is positive.

Relationship 2: Cell potential and \(w\)

The work done by an electrochemical cell (\(w\)) can be calculated in terms of cell potential. The work done is equal to the product of the cell potential and the charge transferred across the cell. Thus, the relationship between cell potential (\(E_{cell}\)) and work is given by: \[w = E_{cell} \cdot |q|\] Where \(|q|\) is the absolute value of the charge transferred.

Relationship 3: Cell potential and \(\Delta G\)

The relationship between the Gibbs free energy change (\(\Delta G\)) and the cell potential (\(E_{cell}\)) is given by the following equation: \[\Delta G = -nFE_{cell}\] Where \(n\) is the number of moles of electrons transferred and \(F\) is Faraday's constant, which is approximately 96,485 C/mol. This equation relates the change in Gibbs free energy to the cell potential, which allows us to determine whether a reaction will be spontaneous (negative ΔG) or non-spontaneous (positive ΔG) based on the cell potential.

Relationship 4: Cell potential and \(Q\)

The cell potential (\(E_{cell}\)) is related to the reaction quotient (\(Q\)) by the Nernst equation. The Nernst equation expresses the dependence of the cell potential on the concentration of the species involved in the reaction at both electrodes: \[E_{cell} = E_{cell}^0 - \frac{RT}{nF} \ln Q\] Where \(E_{cell}^0\) is the standard cell potential, \(R\) is the gas constant, \(T\) is the absolute temperature, \(n\) is the number of moles of electrons transferred per mole of product formed, and \(F\) is Faraday's constant.

Creating a cell with same metal electrodes and different concentrations

To create a cell with the same metal electrodes and different concentrations of a compound, we can set up a concentration cell. A concentration cell has the same metal electrodes but different concentrations of the electrolyte containing the metal ion. The cell spontaneously runs since the flow of electrons from one electrode to another creates a concentration gradient that wants to reach equilibrium. For example, if we have a copper concentration cell, we'll use two Cu electrodes, one dipped in a Cu(NO3)2 solution with a concentration of 1 M and the other dipped in a Cu(NO3)2 solution with a concentration of 0.1 M. The cell potential will be generated due to the concentration difference of Cu2+ ions in the half-cells.

Spontaneity of the concentration cell

A concentration cell runs spontaneously as the reactions move toward equilibrium to equalize the concentration of the species at both electrodes. The driving force behind the spontaneity of a concentration cell is the difference in concentration which causes the flow of ions and the transfer of electrons between the two half-cells. According to the Nernst equation, the cell potential is determined by the concentration of the species involved in the cell. As the cell operates, the concentration of the species in the two half-cells approaches equilibrium, and the cell potential decreases, eventually reaching zero when equilibrium is achieved. At this point, the concentration cell ceases to operate spontaneously as no further work can be done.

Key concepts

Cell Potential

Understanding the concept of cell potential is essential for determining the electrical potential of an electrochemical cell. Cell potential, denoted as \(E_{cell}\), is a measure of the voltage produced by the two half-reactions in an electrochemical cell. It is the difference between the electrode potentials of the cathode and anode.

When a chemical reaction occurs in an electrochemical cell, it results in the flow of electrons from the anode to the cathode. The potential difference created by this flow is what we measure as the cell potential. The greater the potential difference, the stronger the driving force for the electrons, and the greater the ability of the cell to do work.

To calculate the cell potential under standard conditions (where the solutes are at 1 M concentration and gases at 1 atm pressure), we add the reduction potentials of both half-reactions, taking into account their signs (reduction potential for the cathode and oxidation potential for the anode). This is easily remembered as the following equation:
\[E_{cell}^0 = E_{cathode}^0 - E_{anode}^0\]
The cell potential can be used to predict the spontaneity of a reaction, with a positive cell potential indicating a spontaneous reaction under standard conditions. However, in non-standard conditions, the cell potential must be calculated by accounting for the actual concentrations of reactants and products, which is where the Nernst equation comes into play.

Nernst Equation

The Nernst equation is pivotal in electrochemistry as it relates cell potential to the concentration of the reactants and products involved in the reaction.

The equation is written as:
\[E_{cell} = E_{cell}^0 - \frac{RT}{nF} \ln Q\]
where \(E_{cell}^0\) is the standard cell potential, \(R\) is the gas constant, \(T\) is the absolute temperature in Kelvin, \(n\) is the number of moles of electrons transferred in the reaction, \(F\) is Faraday's constant, and \(Q\) is the reaction quotient which represents the ratio of product activities to reactant activities.

The Nernst equation explains why cell potential decreases as the concentrations of products increase or reactants decrease, moving the system towards equilibrium. It is a fundamental equation for analyzing non-standard conditions in electrochemical cells and for determining how concentration differences impact cell potential.

Concentration Cell

Concentration cells are a special type of electrochemical cell where the electrodes are made of the same material and the electrochemical reactions involve the same species but differ in concentrations.

The driving force of a concentration cell is the tendency of the system to reach equilibrium. That means the chemical reaction will naturally shift to balance out the concentration differences between the two compartments. The cell operates due to the movement of ions from higher to lower concentration until the concentrations become equal.

To visualize a concentration cell, imagine two containers, each with a copper electrode, but one container has a higher concentration of \(Cu^{2+}\) ions than the other. The electrode in the higher concentration solution will act as the anode, oxidizing \(Cu\) to \(Cu^{2+}\), while the electrode in the lower concentration solution will act as the cathode, reducing \(Cu^{2+}\) back to \(Cu\). Electrons flow from the anode to the cathode, generating an electric current until the ion concentrations in both containers are the same and the cell potential drops to zero.

Spontaneity in Electrochemical Cells

The spontaneity of electrochemical cells is fundamentally associated with Gibbs free energy (\(\Delta G\)) and the cell potential. A key relationship to remember is that a spontaneous reaction in an electrochemical cell is one where the Gibbs free energy change is negative:
\[\Delta G < 0\]
This condition translates to positive cell potential, as proved by the equation:
\[\Delta G = -nFE_{cell}\]
The negative sign in the equation shows that a positive cell potential will result in a negative Gibbs free energy change, indicating spontaneity.

The concentration cell spontaneously runs because of natural tendencies toward equilibrium, which corresponds to a decrease in free energy. In the context of a concentration cell, the reaction moves spontaneously to even out the differing ion concentrations, driving the flow of electrons, and producing electrical work until equilibrium is reached. Understanding this equilibrium and the associated thermodynamics helps students to predict and explain why certain electrochemical cells can do work and why they stop working when equilibrium is attained.