Problem 99 - Chemistry The Central Science In Si Units

Question

The standard free energy of formation of solid maltose is \(-1334.42 \mathrm{~kJ} / \mathrm{mol}\), whereas that of solid glucose is \(-910.56 \mathrm{~kJ} / \mathrm{mol}\). What is \(\Delta G^{\circ}\) for the condensation of two glucose molecules to form maltose?

Step-by-Step Solution

Answer

The \(\Delta G^{\circ}\) for the reaction is 486.7 kJ/mol.

Step 1: Write the Balanced Chemical Equation

The chemical reaction for the condensation of two glucose molecules to form a maltose molecule can be represented as \[ 2 \text{ C}_6\text{H}_{12}\text{O}_6(s) \rightarrow \text{C}_{12}\text{H}_{22}\text{O}_{11}(s) + \text{H}_2\text{O}(l) \].

Step 2: Identify the Standard Free Energies of Formation

According to the problem, the standard free energy of formation for maltose, \( \Delta G_f^{\circ}(\text{maltose}) \), is \(-1334.42 \text{ kJ/mol} \), and for glucose, \( \Delta G_f^{\circ}(\text{glucose}) \), it is \(-910.56 \text{ kJ/mol} \).

Step 3: Calculate the Total Standard Free Energy of Reactants

For two moles of glucose, the total standard free energy of formation is: \[ 2 \times (-910.56) = -1821.12 \text{ kJ/mol} \].

Step 4: Construct the Equation for Standard Free Energy Change

The standard free energy change for the reaction, \( \Delta G^{\circ}_{reaction} \), is calculated by subtracting the standard free energy of formation of the reactants from that of the products. The equation is: \[ \Delta G^{\circ}_{reaction} = \Delta G_f^{\circ}(\text{products}) - \Delta G_f^{\circ}(\text{reactants}) \].

Step 5: Substitute Values into the Equation

Substitute the values into the equation to find \( \Delta G^{\circ}_{reaction} \): \[ \Delta G^{\circ}_{reaction} = [-1334.42] - [-1821.12] = 486.7 \text{ kJ/mol} \].

Key Concepts

Understanding Standard Free Energy

The standard free energy, often denoted as \( \Delta G^{\circ} \), is a crucial concept in thermodynamics. It is the measure of the maximum reversible work that can be performed by a thermodynamic system at constant temperature and pressure. Typically, the standard free energy is expressed in kilojoules per mole (kJ/mol) and provides insight into the spontaneity of a chemical reaction.

In general, a negative \( \Delta G^{\circ} \) indicates that a reaction is spontaneous under standard conditions, while a positive \( \Delta G^{\circ} \) suggests non-spontaneity, meaning energy needs to be added for the reaction to proceed.

Standard conditions imply that the reactants and products are in their standard states (pure solids, liquids, and gases at 1 atmosphere pressure and a specified temperature, usually 298 K). Knowing the standard free energy allows us to predict how a reaction will behave without actually carrying it out.

The Role of Chemical Reactions in Thermodynamics

Chemical reactions are the heart of chemistry and can be described in terms of energy changes. When a chemical reaction occurs, bonds between atoms are broken and formed, leading to a change in energy. This energy change is crucial because it affects reaction spontaneity and rate.

In the provided exercise, the chemical reaction under consideration is the condensation of two glucose molecules to form maltose and water. We begin with a balanced chemical equation, which in this case is:

• 2 \( \text{C}_6\text{H}_{12}\text{O}_6(s) \rightarrow \text{C}_{12}\text{H}_{22}\text{O}_{11}(s) + \text{H}_2\text{O}(l) \)

This equation shows how reactants (two glucose molecules) combine to form products (maltose and water). The balanced equation is essential for computing the energies involved since it informs us about the stoichiometry or the proportions of reactants and products.

By understanding the energy changes involved in the reaction, one can determine the standard free energy change (\( \Delta G^{\circ}_{reaction} \)), which in this exercise is calculated to predict the reaction's tendency to occur under standard conditions.

The Concept of Free Energy of Formation

The free energy of formation, denoted as \( \Delta G_f^{\circ} \), is a thermodynamic quantity that represents the free energy change when 1 mole of a compound forms from its elements in their standard states. This concept is pivotal in calculating the standard free energy change for a reaction.

For the exercise provided, the free energy of formation is given for solid glucose and maltose.

• \(\Delta G_f^{\circ}(\text{glucose}) = -910.56 \text{ kJ/mol} \)
• \(\Delta G_f^{\circ}(\text{maltose}) = -1334.42 \text{ kJ/mol} \)

The values indicate the energy needed to form these substances from their most basic elements, such as carbon, hydrogen, and oxygen.

Understanding free energy of formation helps us calculate the total energy involved for all reactants and products, which leads to determining the overall energy change of a reaction. In this specific problem, calculating the total free energy for two moles of glucose allows us to use the equation:

• \( \Delta G^{\circ}_{reaction} = \Delta G_f^{\circ}(\text{products}) - \Delta G_f^{\circ}(\text{reactants}) \)

This formula is key to finding out whether the reaction releases or requires energy. By plugging in the given values, we found that the reaction has a \( \Delta G^{\circ} \) of 486.7 kJ/mol, indicating that it requires energy input to proceed spontaneously under standard conditions.