Question

For a system to be in equilibrium at a given temperature and pressure ______

• A. the entropy must be a minimum
• B. the enthalpy must be a minimum
• C. the internal energy must be a minimum
• D. the Gibbs free energy must be a minimum

Hint:

Remember the thermodynamic equilibrium criteria based on independent variables:
1. Constant \( (U, V) \rightarrow S \) is maximized.
2. Constant \( (T, V) \rightarrow A \) is minimized.
3. Constant \( (T, P) \rightarrow G \) is minimized.
Most practical chemical processes are constant \( (T, P) \), making Gibbs free energy the key criterion.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks for the criterion of thermodynamic equilibrium of a system operating under constraints of constant temperature and pressure.
This is a core topic in chemical engineering thermodynamics, related to thermodynamic potentials and stability criteria.

Step 2: Key Formula or Approach:
The fundamental thermodynamic relation combines the first and second laws of thermodynamics.
For a system under constant temperature (\( T \)) and pressure (\( P \)), the change in Gibbs free energy (\( G \)) is related to spontaneity and equilibrium by:
\[ dG \le 0 \]
For a closed system at constant \( T \) and \( P \), the system reaches its stable equilibrium when its Gibbs free energy cannot decrease any further.

Step 3: Detailed Explanation:

• Spontaneous Processes: Any spontaneous process occurring at constant temperature and pressure must lead to a decrease in the Gibbs free energy of the system (\( dG \lt 0 \)).

• Equilibrium State: Once the system reaches state of equilibrium, no further spontaneous change can occur.
At this point, the Gibbs free energy reaches its lowest possible value under those conditions.
Therefore, the mathematical condition for equilibrium at constant \( T \) and \( P \) is:
\[ dG = 0 \quad \text{and} \quad G = \text{minimum} \]

• Other Potentials:
Entropy \( S \) must be maximized for an isolated system (constant internal energy \( U \) and volume \( V \)).
Helmholtz free energy \( A \) must be minimized for a system at constant temperature \( T \) and volume \( V \).
Internal energy \( U \) must be minimized at constant entropy \( S \) and volume \( V \).


Step 4: Final Answer:
For constant temperature and pressure, the Gibbs free energy must be a minimum at equilibrium.