Question

The thermodynamic criterion/criteria for a spontaneous process is/are

• A. \( \Delta U>0 \) at constant \( S \) and \( V \)
• B. \( \Delta S>0 \) at constant \( U \) and \( V \)
• C. \( \Delta(H - TS)>0 \) at constant \( T \) and \( P \)
• D. \( \Delta(U - TS)<0 \) at constant \( T \) and \( V \)

Hint:

Remember the free energy criteria for spontaneity: - \( \Delta G<0 \) at constant \( T, P \) - \( \Delta A<0 \) at constant \( T, V \) Also, \( \Delta S>0 \) at constant \( U, V \) implies spontaneity.

Answer 1

The Correct Option is B, D

The correct thermodynamic criteria for spontaneity under different constraints are:

• (A) \( \Delta U > 0 \) at constant \( S \) and \( V \):
  This is incorrect. At constant entropy and volume, a spontaneous process is characterized by minimum internal energy, so \( \Delta U < 0 \) is required. WRONG
• (B) \( \Delta S > 0 \) at constant \( U \) and \( V \):
  This is correct. At constant internal energy and volume, spontaneity is driven by an increase in entropy. CORRECT
• (C) \( \Delta(H - TS) > 0 \) at constant \( T \) and \( P \):
  This corresponds to \( \Delta G > 0 \), which means non-spontaneity. For spontaneity, \( \Delta G < 0 \). WRONG
• (D) \( \Delta(U - TS) < 0 \) at constant \( T \) and \( V \):
  This expression represents the Helmholtz free energy, \( A = U - TS \), and for spontaneity at constant \( T \) and \( V \), \( \Delta A < 0 \) is the correct criterion. CORRECT

\[ \boxed{\text{Correct options: (B) and (D)}} \]