Question

The \(E^{\circ}\) at \(25^{\circ} \mathrm{C}\) for the following reaction at the indicated concentrations is \(1.50 \mathrm{~V}\). Calculate the \(\Delta G\) in \(\mathrm{kJ}\) at \(25^{\circ} \mathrm{C}\) : $$ \operatorname{Cr}(s)+3 \mathrm{Ag}^{+}(a q, 0.1 \mathrm{M}) \longrightarrow \mathrm{Ag}(s)+\mathrm{Cr}^{3+}(a q, 0.1 \mathrm{M}) $$ (a) \(-140.94\) (b) \(-295\) (c) \(-212\) (d) \(-422.83\)

Short answer

-422.83 kJ

Step-by-step solution

- Recall the relationship between free energy change and cell potential

The free energy change \(\Delta G\) of a reaction at constant temperature and pressure is related to the cell potential \(E^{\circ}\) by the equation \[\Delta G = -nFE^{\circ}\text{, where }n\text{ is the number of moles of electrons transferred, } F \text{ is the Faraday constant (}96485\text{ C/mol), and } E^{\circ} \text{ is the standard cell potential.}\]

- Determine the number of moles of electrons transferred

In the given reaction, each chromium atom transfers 3 electrons to 3 silver ions. Thus, \(n = 3\) moles of electrons are transferred per mole of \(\text{Cr}(s)\).

- Calculate the free energy change

Using the formula from Step 1, calculate the change in free energy: \[\Delta G = -nFE^{\circ} = -(3 \text{ mol})\times (96485 \text{ C/mol}) \times (1.50 \text{ V})\] Convert the result from joules to kilojoules by dividing by 1000 (since 1 kJ = 1000 J).

Key concepts

Electrochemistry

Electrochemistry is the branch of physical chemistry that studies chemical reactions which take place at the interface of an electrode, usually a solid metal or a semiconductor, and an ionic conductor, the electrolyte. These reactions involve electric charges moving between the electrode and the electrolyte, creating an electric current. Understanding electrochemistry is essential for many applications such as batteries, fuel cells, and electrolysis.

In the context of the given problem, we examine a redox reaction where chromium reacts with silver ions. The movement of electrons through an external circuit in such reactions can be harnessed to do work. This is the basis for galvanic cells, also known as voltaic cells, where a spontaneous reaction generates electricity. The overall potential of a cell, which drives the electron flow, is a critical aspect of electrochemistry that is quantified by the standard cell potential.

Gibbs Free Energy

Gibbs free energy, denoted as \(\Delta G\), is a thermodynamic quantity that represents the maximum amount of work that can be performed by a system at constant temperature and pressure. It's a measurement of the chemical potential energy available for work. When \(\Delta G\) is negative, the process or reaction is said to be spontaneous, meaning it can occur without the input of external energy.

In electrochemical reactions, the relationship between the change in Gibbs free energy and the cell potential is vitally important. As shown in the exercise, the cell potential helps determine if a redox reaction will occur spontaneously. High school and college students often encounter this concept in their chemistry classes when learning about reaction spontaneity and how it relates to electrochemical cell potential and the movement of electrons.

Standard Cell Potential

The standard cell potential, \(E^{\circ}\), is an indicator of how forcefully an electrochemical cell can push electrons through a circuit. This value is measured in volts (V) under standard conditions, which include all reactants and products at a concentration of 1 M, a pressure of 1 atmosphere, and at 25 degrees Celsius unless otherwise specified. The cell potential is determined by the difference in potential energy between the anode and cathode, or more fundamentally, between the oxidation and reduction half-reactions of the electrochemical cell.

For most problems involving electrochemistry, including the textbook exercise provided, the standard cell potential is critical in calculating other parameters such as the Gibbs free energy change. Knowing the standard cell potential for a reaction allows chemists and engineers to predict whether a reaction is thermodynamically favorable and to what extent it can produce or consume electrical energy.

Faraday's Constant

Faraday's constant, symbolized by \(F\), represents the total electric charge carried by one mole of electrons, amounting to approximately 96485 coulombs per mole (C/mol). It is named after the English scientist Michael Faraday, who made significant contributions to the understanding of electromagnetism. Faraday's constant plays a pivotal role in calculations involving electric charge in electrochemistry.

When calculating Gibbs free energy change in electrochemistry, as demonstrated in the textbook solution step, Faraday's constant is multiplied by the number of moles of electrons transferred during the reaction (\(n\)) and the standard cell potential (\(E^{\circ}\)). This formula quantitatively links the microscopic movement of electrons during a chemical reaction to the macroscopic observables that we measure, such as cell potential and energy change.