Question

The isothermal expansion of one mole of an ideal gas from \( V_i \) to \( V_f \) at temperature \( T \) occurs in two ways:
Path I: a reversible isothermal expansion;
Path II: free expansion against zero external pressure.
The CORRECT option for the values of \( \Delta U \), \( q \), and \( w \) for Path I and Path II is:

• A. Path I: \( \Delta U = 0,\, q>0,\, w<0 \) 
  Path II: \( \Delta U = 0,\, q = 0,\, w = 0 \)
• B. Path I: \( \Delta U = 0,\, q>0,\, w<0 \) 
  Path II: \( \Delta U>0,\, q>0,\, w = 0 \)
• C. Path I: \( \Delta U = 0,\, q<0,\, w>0 \) 
  Path II: \( \Delta U = 0,\, q>0,\, w<0 \)
• D. Path I: \( \Delta U = 0,\, q<0,\, w>0 \) 
  Path II: \( \Delta U<0,\, q = 0,\, w = 0 \)

Hint:

In free expansion, there is no work done because external pressure is zero, and no heat is exchanged with surroundings.

Answer 1

The Correct Option is A

Step 1: For an isothermal process, \( \Delta U = 0 \) since internal energy of an ideal gas depends only on temperature, which is constant.
Step 2: In Path I (reversible isothermal expansion), the gas does work (\( w<0 \)) and to maintain constant internal energy, heat must be absorbed (\( q>0 \)) such that:
\[ \Delta U = q + w \Rightarrow 0 = q + w \Rightarrow q = -w \] Step 3: In Path II (free expansion), the gas expands against zero external pressure:
- No work is done \( (w = 0) \)
- No heat is exchanged \( (q = 0) \)
- Therefore, \( \Delta U = 0 \)
Conclusion: The correct values are:
Path I: \( \Delta U = 0,\, q>0,\, w<0 \)
Path II: \( \Delta U = 0,\, q = 0,\, w = 0 \)