Questions: State Functions and Path Functions in Thermodynamics

Question 1

A fixed amount of gas is taken from state A (P₁, T₁) to state B (P₂, T₂) via two different processes: one isothermal and one adiabatic. Which statement correctly describes the relationship between the two processes?

• A. Both ΔU and Q will be identical for both processes because the endpoints are the same
• B. ΔU will be identical for both processes, but Q and W will each differ between them
• C. Q will be identical for both processes, but ΔU and W will differ
• D. W will be identical for both processes, but Q and ΔU will differ

Answer: (B)

Internal energy U is a state function — its change depends only on the initial and final thermodynamic states, not on how the process occurred. Since both processes connect the same states A and B, ΔU is identical. However, Q and W are path functions: the adiabatic process has Q = 0 by definition, while the isothermal process delivers nonzero heat. The first law (ΔU = Q − W) is satisfied in both cases with the same ΔU but different Q and W values. This is the fundamental operational consequence of the state/path distinction.

Question 2

Which of the following is a state function of a thermodynamic system?

• A. Heat transferred during a process
• B. Work done by the system during a process
• C. Enthalpy at a given temperature and pressure
• D. The amount of thermal energy stored in the system

Answer: (C)

Enthalpy H is a state function — its value is uniquely determined by the thermodynamic state (T, P, composition), which is why it can be tabulated in steam tables and refrigerant property charts. Options A and B are path functions: the same change of state can be accomplished with different amounts of heat and work depending on the process followed. Option D ('thermal energy stored') is not a valid thermodynamic concept — systems store internal energy U, not heat. Heat is energy in transit across a boundary, not something stored within the system.

Question 3

For any complete thermodynamic cycle, the net change in internal energy of the working fluid is zero, regardless of which processes make up the cycle.

• T. True
• F. False

Answer: True

Because internal energy is a state function, its value depends only on the current thermodynamic state. In a complete cycle, the working fluid returns exactly to its initial state, so U_final = U_initial and ΔU_cycle = 0. This holds regardless of which processes (isothermal, adiabatic, isobaric, etc.) make up the cycle. Applied to the first law: ΔU = Q_net − W_net = 0, therefore W_net = Q_net. Every cycle efficiency calculation depends on this identity — if U were a path function, the cycle analysis would collapse.

Question 4

A thermodynamic system contains a certain amount of 'heat' that can be measured and tabulated as a state property, just like internal energy.

• T. True
• F. False

Answer: False

Heat is not a substance stored in a system — it is energy in transit across a system boundary driven by a temperature difference. The phrase 'heat content of a system' is physically meaningless, a remnant of the discredited caloric theory. A system possesses internal energy U and enthalpy H, both of which can be tabulated because they are state functions. There is no 'stored heat' to measure. This is precisely why thermodynamicists use δQ notation (an inexact differential) rather than dQ — to emphasize that heat is a process quantity that cannot be integrated without knowing the path, not a state property.

Question 5

Why is it meaningful to look up the enthalpy of steam at a given temperature and pressure in a table, but meaningless to look up 'the heat stored in' that steam?

Model answer: Enthalpy H is a state function: its value is completely determined by the current thermodynamic state (temperature, pressure, composition). Every parcel of steam at 200°C and 1 MPa has exactly the same enthalpy, regardless of whether it was heated at constant pressure, flashed from high-pressure liquid, or generated by some other process. This uniqueness makes tabulation possible and useful — you can look it up once and use it for any problem involving that state. Heat Q is a path function: different processes connecting the same initial and final states deliver different amounts of heat. There is no single value of 'heat in the steam' to tabulate, because it depends on history, not on the current state.

The deeper point is that heat describes an interaction (energy crossing a boundary during a process) rather than a property (something the system possesses). Asking how much heat is stored in steam is like asking how much work is stored in a compressed spring — work describes a mode of energy transfer, not a stored quantity. The spring stores elastic potential energy; the steam stores internal energy. Heat and work are both boundary interactions that exist only while the process is occurring, not as attributes of the final equilibrium state. This is why the first law is written ΔU = Q − W rather than U = U_heat + U_work.