Step 1: Understanding the Concept:
Thermodynamic properties are strictly classified as either state functions (independent of the path taken) or path functions (dependent on the specific sequence of steps).
Step 2: Key Formula or Approach:
Analyze each statement against fundamental thermodynamic definitions, such as conditions for isothermal/adiabatic processes, and identify state vs. path variables.
Step 3: Detailed Explanation:
(A) In an isothermal process involving an ideal gas, temperature remains constant. Since the internal energy ($U$) of an ideal gas depends solely on temperature, internal energy does {not} change ($\Delta U = 0$). Therefore, it doesn't change in all processes. (Incorrect)
(B) Internal energy ($U$) and Entropy ($S$) depend exclusively on the current equilibrium state of the system (characterized by P, V, T), not on how that state was reached. They are state functions. (Correct)
(C) Work ($w$) and Heat ($q$) are heavily dependent on the specific path (e.g., reversible vs. irreversible expansion). They are path functions, not state functions. (Incorrect)
(D) In an adiabatic process, heat exchange is zero ($q = 0$). According to the First Law ($\Delta U = q + w$), the change in internal energy equals the work done ($\Delta U = w_{ad}$). Work is not zero unless it's free expansion. (Incorrect)
Step 4: Final Answer:
Statement (B) is the correct statement.