Problem 20 - Chemistry The Molecular Nature of Matter

Question

Write the equation that relates the standard cell potential to the standard free energy change for a reaction.

Step-by-Step Solution

Answer

The standard cell potential is related to the standard free energy change by the equation \( \Delta G^\circ = -nFE^\circ \).

Step 1: Identify the Relevant Equation

The equation that relates the standard cell potential, \( E^\circ \), to the standard free energy change, \( \Delta G^\circ \), for a reaction is known as the Gibbs free energy equation for electrochemical cells. This equation is derived from the principles of thermodynamics and electrochemistry.

Step 2: Write the Gibbs Free Energy Equation

The equation that directly relates the standard cell potential to the standard free energy change is given by \[ \Delta G^\circ = -nFE^\circ \] where \( \Delta G^\circ \) is the standard free energy change, \( n \) is the number of moles of electrons transferred in the reaction, \( F \) is the Faraday's constant (approximately 96485 C/mol), and \( E^\circ \) is the standard cell potential.

Step 3: Understanding the Components

The standard cell potential \( E^\circ \) is a measure of how much potential energy is available from the electrochemical reaction when all reactants and products are in their standard states. The standard free energy change \( \Delta G^\circ \) represents the maximum amount of work that the reaction can perform under standard conditions, and it's expressed in joules per mole. The Faraday's constant \( F \) converts the charge transferred in the reaction (in moles of electrons) to coulombs.

Key Concepts

Gibbs Free Energy Equation

The Gibbs free energy equation is fundamental to understanding the relationship between chemical spontaneity and the direction of a reaction. It is written as ewline \[ ewline \text{ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline} Delta G = -nFE \] where ewline \( ewline \text{ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline} G \) is the Gibbs free energy change, \(n\) stands for the number of moles of electrons transferred in the electronic reaction, \(F\) is Faraday's constant, measuring the electric charge per mole of electrons, and \(E\) represents the cell potential or electromotive force. The minus sign indicates that a spontaneous process (with a positive standard cell potential) will have a negative Gibbs free energy, meaning the process can do work on the surroundings.

Thermodynamics in Electrochemistry

Thermodynamics plays a crucial role in electrochemical reactions, dictating whether a reaction is energetically feasible.

Electrochemical cells convert chemical energy into electrical energy through redox reactions. Here, the Gibbs free energy change provides insight into the reaction's spontaneity. A negative ewline \(Delta G^circ circ \) indicates a spontaneous reaction, thus capable of producing an electric current.

Cells with a positive standard potential (\(E^circ \) > 0) can do work on the surroundings, as they have enough energy to move electrons against an external force, like in a battery. Conversely, cells with a negative standard potential require work, indicating a non-spontaneous reaction that needs an external voltage to proceed, such as in electrolysis.

Faraday's Constant

Faraday's constant (\(F\)) is a bridge between the macroscopic world of chemical reactions and the microscopic world of charge in physics.

It is approximately 96485 Coulombs per mole (C/mol), representing the total charge carried by one mole of electrons. This constant not only allows us to equate the chemical aspect (\(n\)) of our equations to the physical aspect (\(E\)), but it also scales the electron transfer in reactions to the amount of current produced or consumed. Without Faraday's constant, we would not be able to use electrochemical concepts to quantify real-world applications like the operation of batteries or the amount of substance produced in electrolysis.