Given are statements for certain thermodynamic variables,
(A) Internal energy, volume (V) and mass (M) are extensive variables.
(B) Pressure (P), temperature (T) and density ($\rho$) are intensive variables.
(C) Volume (V), temperature (T) and density ($\rho$) are intensive variables.
(D) Mass (M), temperature (T) and internal energy are extensive variables.
Choose the correct answer from the points given below :

Concepts of Extensive and Intensive Properties:

• Extensive variables depend on the amount of substance (they change when the system size changes). Examples: Mass (M), Volume (V), Internal Energy (U), Entropy (S), Enthalpy (H), etc.
• Intensive variables are independent of the amount of substance (they remain the same for any system size). Examples: Pressure (P), Temperature (T), Density (ρ), Specific Heat, etc.

Analysis of Each Statement:

(A) Internal energy, volume (V), and mass (M) are extensive variables. ✅ (Correct)

• Internal energy (U) is extensive because it depends on the amount of substance.
• Volume (V) is extensive since it changes when the system size changes.
• Mass (M) is extensive because it directly depends on the total quantity of the substance.
  ✅ This statement is correct.

(B) Pressure (P), temperature (T), and density (ρ) are intensive variables. ✅ (Correct)

• Pressure (P) is intensive because it is the same regardless of system size.
• Temperature (T) is intensive as it remains unchanged when the system is divided.
• Density (ρ) is intensive because it is a ratio ($\rho = M/V$), meaning it remains constant for any fraction of the system.
  ✅ This statement is correct.

(C) Volume (V), temperature (T), and density (ρ) are intensive variables. ❌ (Incorrect)

• Volume (V) is NOT an intensive variable; it depends on the size of the system, making it extensive.
  ❌ This statement is incorrect.

(D) Mass (M), temperature (T), and internal energy (U) are extensive variables. ❌ (Incorrect)

• Mass (M) is extensive (✅ Correct).
• Temperature (T) is NOT extensive; it is intensive (❌ Incorrect).
• Internal energy (U) is extensive (✅ Correct).
  ❌ Since temperature (T) is incorrectly classified, this statement is incorrect.