Problem 7 - Chemistry The Central Science In Si Units

Question

Consider a redox reaction for which \(E^{\circ}\) is a negative number. (a) What is the sign of \(\Delta G^{\circ}\) for the reaction? (b) Will the equilibrium constant for the reaction be larger or smaller than \(1 ?\) (c) Can an electrochemical cell based on this reaction accomplish work on its surroundings? [Section 20.5\(]\)

Step-by-Step Solution

Answer

(a) \(\Delta G^{\circ}\) is positive. (b) The equilibrium constant is less than 1. (c) The cell cannot do work on its surroundings.

Step 1: Understanding Standard Cell Potential

The standard cell potential, denoted as \(E^{\circ}\), is a measure of the driving force behind an electrochemical reaction. If \(E^{\circ}\) is negative, it indicates that the reaction is non-spontaneous under standard conditions.

Step 2: Determining the Sign of \(\Delta G^{\circ}\)

The relationship between \(\Delta G^{\circ}\) and \(E^{\circ}\) is given by the equation \( \Delta G^{\circ} = -nFE^{\circ} \), where \(n\) is the number of moles of electrons exchanged in the reaction, and \(F\) is Faraday's constant. Since \(E^{\circ}\) is negative, \(\Delta G^{\circ}\) will be positive, indicating the reaction is non-spontaneous.

Step 3: Analyzing the Equilibrium Constant

The relationship between \(\Delta G^{\circ}\) and the equilibrium constant \(K\) is given by \( \Delta G^{\circ} = -RT \ln K \). A positive \(\Delta G^{\circ}\) means \( \ln K \) is negative, implying \(K < 1\). Thus, the equilibrium constant will be less than 1, indicating reactants are favored.

Step 4: Evaluating the Ability to Do Work

A negative \(E^{\circ}\) indicates that the reaction cannot occur spontaneously, meaning an electrochemical cell based on it cannot do work on its surroundings. Cells do work when the reaction is spontaneous, i.e., when \(E^{\circ}\) is positive.

Key Concepts

Standard Cell Potential

The standard cell potential, represented as \(E^{\circ}\), is an essential concept in electrochemistry. It measures the ability of a chemical reaction within an electrochemical cell to be spontaneous when all reactants and products are in their standard states. To put it simply, it tells us how much push or pull a reaction has under standard conditions. When \(E^{\circ}\) is positive, the reaction is spontaneous. It means that the reaction can occur on its own and has the potential to do work. However, if \(E^{\circ}\) is negative, like in the exercise, it indicates a non-spontaneous reaction. This means the reaction needs external energy to proceed and cannot do work by itself.Remember:

• Positive \(E^{\circ}\): Reaction is spontaneous.
• Negative \(E^{\circ}\): Reaction is non-spontaneous.

Understanding this helps predict the feasibility of a given electrochemical process.

Gibbs Free Energy

Gibbs Free Energy, denoted as \(\Delta G^{\circ}\), is a concept that helps us quantify the energy available to do work in a system at constant temperature and pressure. It is closely related to the standard cell potential with the equation \( \Delta G^{\circ} = -nFE^{\circ} \). Here, \(n\) is the number of moles of electrons transferred, and \(F\) is Faraday's constant.When \(E^{\circ}\) is negative, as in the original exercise, \(\Delta G^{\circ}\) becomes positive. A positive \(\Delta G^{\circ}\) means the reaction is non-spontaneous. Essentially, it requires an input of energy to occur, rather than releasing energy to perform work.Key points to remember:

• If \(\Delta G^{\circ}\) is negative, the reaction is spontaneous.
• If \(\Delta G^{\circ}\) is positive, the reaction is non-spontaneous.

This relationship helps predict whether a reaction can occur under given conditions.

Equilibrium Constant

The equilibrium constant \(K\) tells us the ratio of products to reactants at equilibrium for a chemical reaction. For electrochemical reactions, the relationship between \(\Delta G^{\circ}\) and \(K\) is given by the equation \( \Delta G^{\circ} = -RT \ln K \), where \(R\) is the universal gas constant and \(T\) is the temperature in Kelvin.In the context of the exercise, a positive \(\Delta G^{\circ}\) means \( \ln K \) is negative, which leads to a \(K\) value less than 1. This tells us that at equilibrium, the concentration of reactants is higher than that of the products, meaning reactants are favored.It's important to understand that:

• When \(K > 1\), products are favored.
• When \(K < 1\), reactants are favored.

A low equilibrium constant indicates a non-favorable forward reaction under standard conditions.

Electrochemical Cells

Electrochemical cells are devices that convert chemical energy into electrical energy or, conversely, use electrical energy to provoke chemical change. They are the backbone of batteries, which power many of our daily devices.In a typical electrochemical cell, two reactions occur between two electrodes. One is reduction, and the other is oxidation – together known as redox reactions. These reactions lead to the flow of electrons through an external circuit resulting in electricity.However, when the standard cell potential \(E^{\circ}\) is negative, as in the given exercise, the reaction is non-spontaneous. This means the cell cannot generate an electric current or do work without external energy input. Essentially, such cells cannot act as self-sustaining batteries.Remember:

• Spontaneous reactions (positive \(E^{\circ}\)) can do work.
• Non-spontaneous reactions (negative \(E^{\circ}\)) need energy input.

This concept is crucial in designing batteries and understanding their efficiency.