2.5.2 · D5Thermodynamics (Chemical)
Question bank — State functions vs path functions
Before you start, keep three anchor ideas in hand, because every answer leans on them:
Recall The three anchors these questions test
Anchor 1 — State function = "where you are." Depends only on current . Its change ignores the route. Written with (exact differential). See 4.1.02-Exact-and-Inexact-Differentials. Anchor 2 — Path function = "how you got there." Depends on the process. Written with or đ (inexact). No such thing as "the of a state." Anchor 3 — First Law glue. : two path quantities combine into one path-independent quantity. See 2.5.01-First-Law-of-Thermodynamics.
True or false — justify
, , , , , , are all state functions.
True — each is fully fixed once you fix the current state of the system; none of them remembers the route taken to reach that state.
Heat is a state function because a hot object "contains heat."
False — a body contains internal energy, not heat; heat is energy in transit during a process, so it is a path function and has no value for a resting state.
For any cyclic process, .
True — a cycle returns to the initial state, so every state function's net change is zero; this is the defining test of a state function via 2.5.10-Cyclic-Processes-and-State-Functions.
For any cyclic process, .
False — over a full cycle the heat exchanged is generally nonzero (a steam engine absorbs heat and returns to its start each cycle); only holds.
If between two states, then for that process.
False — , so only means ; both can be large and equal, as in reversible isothermal expansion where .
and prove that heat is secretly a state function.
False — these say heat equals a state function's change only under a fixed constraint (constant or constant ); remove the constraint and again depends on the path.
The differential can always be integrated to give a "work content" of the state.
False — is inexact, so there is no function whose difference gives ; you can only integrate it along a specified path, not between endpoints alone.
Enthalpy is a state function because it is built from state functions , , .
True — is an algebraic combination of state functions, and any such combination is itself a state function.
Temperature is a path function because heating a gas raises its temperature.
False — the change in depends on the process, but itself is fully determined by the current state; the value at a state is unique, which makes it a state function.
Because entropy change is calculated using , entropy is a path function.
False — the clever point is that dividing reversible heat by turns an inexact differential into an exact one, so is a state function (see 2.5.07-Entropy-as-State-Function).
Spot the error
"A student writes and calls it the First Law for cycles."
The First Law over a cycle is only because ; the equality is derived from being a state function, so stating it without that reasoning hides the actual content.
"Since , we conclude and ."
Error — one equation cannot fix two unknowns; infinitely many pairs give (e.g. and , or and ). You must know the process type first.
"For an ideal gas, internal energy depends on , so is a path function."
Two errors — for an ideal gas depends only on (not ), and even a -dependence would not make path-dependent, since is itself a state variable.
"In free expansion , so no work is possible in expansion."
Error — only because in this specific path; the same expansion done reversibly yields nonzero work, which is exactly why work is a path function.
"For an adiabatic process , therefore work is now a state function equal to ."
Error — is path-independent, but only fixes the value on that adiabatic path; a different (non-adiabatic) route between the same states gives a different , so stays a path function.
"Heating water at constant gave the same by two routes, proving is a state function."
Error — the routes agreed only because and both ended at the same state at constant ; switch one leg to a different pressure or add stirring work and changes.
"The symbol đ and mean the same thing, just different fonts."
Error — marks an exact differential (integrates to between endpoints); đ (or ) marks an inexact differential (needs a path to integrate). The distinction is the whole subject.
Why questions
Why do we write but in the First Law?
Because is a state function (its differential is exact, integrable between endpoints alone), while is a path function (inexact, needing the full route), and the notation encodes that difference.
Why does Euler's reciprocity tell us a differential is exact?
Because it is the condition guaranteeing is the total differential of some function ; when it holds, a state function exists and the integral is path-independent. Details in 4.1.02-Exact-and-Inexact-Differentials.
Why can two paths with the same endpoints give different heat but never different ?
is fixed by the endpoints since is a state function; and individually adjust with the route, but the First Law forces their difference to always land on the same .
Why does the mountain-elevation analogy fail if pushed to describe heat?
Elevation is like a state function (endpoint-only), so it models , , ; heat is like distance walked, which depends on the trail — you cannot read "distance" off a map point, just as you cannot read off a state.
Why is so useful in chemistry even though is a path function?
Because most reactions run in open flasks at constant atmospheric pressure, fixing that constraint pins to the state function , letting us tabulate reaction heats independent of apparatus. See 2.5.03-Internal-Energy-and-Enthalpy.
Why does a reversible path generally exchange more heat than an irreversible one between the same isothermal endpoints?
A reversible expansion does maximum work against a matching external pressure, and since isothermally, that maximum work is supplied by maximum heat absorbed; the irreversible path does less work and needs less heat. See 3.2.04-Reversible-vs-Irreversible-Processes.
Edge cases
Constant-volume process: is work zero because became a state function?
No — because removes the mechanism for work on this path; it is a special value of a path function, not a promotion to state function.
Adiabatic irreversible compression vs adiabatic reversible compression to the same final volume: same ?
Not generally to the same final state — with , , and the two paths do different work, so they reach different final temperatures; you cannot force identical endpoints and identical on both.
Free expansion of an ideal gas into vacuum: what are , , ?
(no opposing pressure), (ideal gas, isothermal here), so ; yet the reversible route between the same two volumes gives nonzero and — the endpoints are shared but the path quantities differ.
A cycle in which : what is and ?
always for a cycle, so the First Law forces ; the net heat in becomes net work out, the operating principle of engines.
Degenerate "path" where initial and final states are identical but the process took a long journey: what is ?
for the system because is a state function and the endpoints coincide, even though was traversed along a long route; the surroundings' entropy may still have changed.
Can heat be negative and still be "the same kind of quantity" as positive heat?
Yes — sign only encodes direction (out of the system when negative); it does not change the fact that is a path function, since sign and magnitude both depend on the process.
If someone hands you only , , , of a gas, can you state its ? Its ?
You can state (up to a reference), because it is a state function fixed by those variables; you cannot state , because "the heat of a resting state" is meaningless — heat exists only during a process.