Question

The relationship between decrease in Gibbs free energy and electrical energy is (a) \(-\Delta \mathrm{G}=n F E_{\text {cell }}\) (b) \(\Delta \mathrm{G}=n R T \ln E_{\text {cell }}\) (c) \(-\Delta G=-n F E_{\text {cell }}\) (d) \(\Delta G=-n R T \ln E_{\text {cell }}\)

Short answer

Statement (a) is the correct relationship between decrease in Gibbs free energy and cell potential.

Step-by-step solution

Evalutate Statement (a)

The statement \(-\Delta \mathrm{G}=n F E_{\text {cell }}\) is the correct relationship of a cell operating in spontaneous conditions. Here, \(-\Delta \mathrm{G}\) represents decrease in Gibbs free energy, \(n\) is the number of moles of electrons, \(F\) is the Faraday's constant and \(E_{\text {cell }}\) is the cell potential.

Evaluate Statement (b)

This expression \(\Delta \mathrm{G}=n R T \ln E_{\text {cell }}\) is incorrect. The Relationship between Gibbs free energy change and cell potential is not logarithmical. This modifies the Nernst equation's structure, which measures Gibbs free energy change through equilibrium.

Evaluate Statement (c)

This statement \(-\Delta G=-n F E_{\text {cell }}\) has a double negative on both sides of the equation and thus is wrong. This means it is describing a situation where an increase in Gibbs free energy corresponds to a decrease in cell potential, which isn't accurate as per the fundamentals of electrochemistry.

Evaluate Statement (d)

This statement \(\Delta G=-n R T \ln E_{\text {cell }}\) is also incorrect, for the same reason as explained in Statement (b). It is not giving a correct relationship between Gibbs free energy and cell potential.

Key concepts

electrochemistry

Electrochemistry is the branch of chemistry that deals with the relationship between electrical energy and chemical change. It involves the study of the movement of electrons or electric charge, which is crucial in both chemical reactions and processes. One of the key aspects of electrochemistry is the redox reaction, where oxidation and reduction reactions occur simultaneously. In these reactions:

• Oxidation involves the loss of electrons.
• Reduction involves the gain of electrons.

The understanding of electrochemical cells, which convert chemical energy into electrical energy (and vice versa), is pivotal for this field. They can be categorized into two types:

• Galvanic Cells: These cells generate electrical energy from spontaneous chemical reactions. This is the principle behind batteries.
• Electrolytic Cells: These cells use electrical energy to drive non-spontaneous chemical reactions. They are used in applications like electroplating.

The core principle behind these cells is the conversion and management of energy, making electrochemistry essential for industries involved in energy storage and conversion. Understanding electrochemistry is key to mastering concepts like cell potential and Gibbs free energy.

cell potential

Cell potential, often denoted as \( E_{\text{cell}} \), is a measure of the voltage difference between two electrodes in an electrochemical cell. It tells us how much work can be done by the electrons moving through the circuit. The cell potential is influenced by several factors:

• The type of materials used in the electrodes and electrolyte.
• The concentration of the ions involved in the reaction.
• The temperature of the cell.

The standard cell potential, \( E^0_{\text{cell}} \), is measured under standard conditions, which include:

• 1 M concentrations for all solutions.
• 1 atm pressure for all gases.
• A temperature of 25°C (298 K).

The cell potential is connected directly to the Gibbs free energy change of the reaction, using the equation \(-\Delta G = n F E_{\text{cell}}\), where:

• \(\Delta G\) is the change in Gibbs free energy.
• \(n\) is the number of moles of electrons transferred.
• \(F\) is Faraday's constant.

A positive cell potential indicates a spontaneous reaction, aligning with a negative \(\Delta G\). This relationship forms the backbone of energy calculations in electrochemical cells.

Faraday's constant

Faraday's constant, denoted as \( F \), is a fundamental constant in electrochemistry. It represents the charge carried by one mole of electrons. The value of Faraday's constant is approximately \( 96,485 \) coulombs per mole of electrons. This constant is crucial because:

• It connects the electric charge with the number of moles of electrons transferred in electrochemical reactions.
• It is used in calculations that involve converting between chemical energy and electrical energy.

In the context of Gibbs free energy and cell potential, Faraday's constant helps bridge the macroscopic world of measurable electric currents with the microscopic world of molecules and atoms. By knowing how much charge one mole of electrons carries, scientists can precisely calculate the work that can be done by electrochemical reactions. This constant is not only fundamental in electrochemical calculations but also pivotal in understanding how energy conversion occurs at a molecular level, thereby facilitating innovations in fields such as battery technology and electrolysis.

Nernst equation

The Nernst equation is a vital tool in electrochemistry for calculating the cell potential under non-standard conditions. It provides a way to modify the standard electrode potentials based on the actual concentrations of the reactants and products in a cell. The general form of the Nernst equation is:\[ E = E^0 - \frac{RT}{nF} \ln Q \]Where:

• \( E \) is the cell potential under non-standard conditions.
• \( E^0 \) 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 transferred in the reaction.
• \( F \) is Faraday's constant.
• \( Q \) is the reaction quotient, which is the ratio of the concentrations of products to reactants.

The Nernst equation demonstrates how the cell potential is affected by changes in the concentration of ions in solution. It is particularly useful for predicting the direction of the reaction and the equilibrium position in electrochemical cells. By accounting for these variables, the Nernst equation allows us to understand the dynamic nature of electrochemical systems, thus offering insights into manipulating these systems for practical applications, such as sensors and electrochemical cells.