An ideal gas is taken from state A (300 K, 1 atm) to state B (600 K, 2 atm) via two different paths: Path 1 is an isothermal compression followed by constant-volume heating; Path 2 is an isobaric heating followed by isothermal compression. Which statement about Q, W, and ΔU for these two paths is correct?
AQ, W, and ΔU are all the same for both paths because the initial and final states are identical
BQ and W differ between paths, but ΔU is the same for both paths
CΔU differs between paths, but Q + W is the same
DQ is the same for both paths but W differs
ΔU is a state function — it depends only on the initial and final states (A and B), not the route. For an ideal gas, ΔU = nCvΔT, which depends only on the temperature change from 300 K to 600 K, giving the same value regardless of path. Q and W are path functions: the heat absorbed and work done vary dramatically between the isothermal-then-constant-volume path and the isobaric-then-isothermal path. The first law ΔU = Q − W holds for both paths, but the individual values of Q and W differ even though their combination ΔU is fixed.
Question 2 Multiple Choice
Which of the following is a state function?
AWork done by the gas during an expansion
BHeat absorbed by the system during a heating process
CEnthalpy H = U + PV
DThe heat exchanged during a reversible isothermal process
Enthalpy H = U + PV is a state function because it is defined entirely in terms of state variables (U, P, V). Its change ΔH depends only on initial and final states. Work (options A and D) and heat (option B) are path functions — they depend on the specific process. Note that option D is tempting because reversible isothermal processes have a specific, calculable heat exchange, but the heat still depends on the path (e.g., reversible isothermal gives W = Q = nRT ln(V₂/V₁), while irreversible isothermal gives different values), not just the endpoints.
Question 3 True / False
A system has a definite 'heat content' at any given thermodynamic state, just as it has a definite internal energy.
TTrue
FFalse
Answer: False
Heat is a path function, not a property of a state. You cannot say 'the system contains 500 J of heat' the way you can say 'the system has internal energy U = 500 J.' Heat is energy in transit — it only exists during a process, and the amount depends on how the process is carried out. The common mistake of treating heat as a stored quantity (like saying 'this object has more heat than that one') confuses heat with internal energy or enthalpy. Internal energy U is a state function with a definite value at each state; heat Q has no meaning except in reference to a specific path.
Question 4 True / False
If a system undergoes a complete thermodynamic cycle (ending in the same state it started), then ΔU = 0 regardless of the processes that made up the cycle.
TTrue
FFalse
Answer: True
Since internal energy U is a state function, its change ΔU depends only on the initial and final states. In a complete cycle, the initial and final states are identical, so ΔU = 0. This does NOT mean Q = 0 or W = 0 individually — they can both be nonzero and equal (Q = W for a cycle since ΔU = 0). This is the basis for analyzing heat engines: the net work output of a cycle equals the net heat input, and the efficiency is determined by how much heat must be rejected to a cold reservoir.
Question 5 Short Answer
Explain why you can talk about a system's 'internal energy content' but not its 'heat content.' What is the fundamental difference between internal energy and heat?
Think about your answer, then reveal below.
Model answer: Internal energy U is a state function — a property of the system's thermodynamic state. At any given state (defined by temperature, pressure, and composition), U has a unique, definite value. It makes sense to say 'the system has internal energy U' just as it makes sense to say 'the system has temperature T.' Heat Q, by contrast, is a path function — it describes energy transfer across the system boundary during a process. Heat only exists during a process; it is not stored in the system. Once the process ends, you cannot identify which part of U 'came from heat' because U is simply what it is, independent of how the system arrived there.
This distinction resolves one of the most persistent confusions in thermodynamics. The misconception of 'heat content' treats heat as if it were stored in a system like a fluid (caloric theory), which was the pre-19th-century understanding. The modern view recognizes that heat and work are modes of energy transfer, not properties of a system's state. Only state functions like U, H, S, and G can be 'contained' in a system.