Consider a twofold expansion of 1 mol of an ideal gas at 25 °C in the isolated system shown in Figure 18.1. (a) What are the values of ∆H, ∆S, and ∆G for the process?

Verified step by step guidance

1. Identify the type of process: Since the system is isolated, it is an adiabatic process, meaning no heat is exchanged with the surroundings.
2. Determine \( \Delta H \): In an adiabatic process for an ideal gas, the enthalpy change \( \Delta H \) is zero because there is no heat exchange.
3. Determine \( \Delta S \): For an isolated system, the entropy change \( \Delta S \) is positive because the process is spontaneous and the entropy of the universe increases.
4. Determine \( \Delta G \): The Gibbs free energy change \( \Delta G \) is zero for a spontaneous process in an isolated system because there is no exchange of energy with the surroundings.
5. Summarize the results: \( \Delta H = 0 \), \( \Delta S > 0 \), and \( \Delta G = 0 \) for the adiabatic expansion of an ideal gas in an isolated system.

Key Concepts

Enthalpy (∆H)

Enthalpy is a thermodynamic property that represents the total heat content of a system. It is defined as the sum of the internal energy and the product of pressure and volume (H = U + PV). In the context of an ideal gas undergoing expansion, the change in enthalpy (∆H) can indicate whether the process is endothermic or exothermic, which is crucial for understanding energy transfer during the expansion.

Entropy (∆S)

Entropy is a measure of the disorder or randomness in a system. It quantifies the number of ways a system can be arranged, reflecting the degree of uncertainty or energy dispersal. For an ideal gas expanding in an isolated system, the change in entropy (∆S) is positive, indicating an increase in disorder as the gas molecules occupy a larger volume, which is a key factor in determining the spontaneity of the process.

Gibbs Free Energy (∆G)

Gibbs Free Energy is a thermodynamic potential that measures the maximum reversible work obtainable from a thermodynamic system at constant temperature and pressure. The change in Gibbs Free Energy (∆G) is used to predict the spontaneity of a process: if ∆G is negative, the process is spontaneous; if positive, it is non-spontaneous. In the context of the ideal gas expansion, calculating ∆G helps assess the feasibility of the expansion under the given conditions.