A gas undergoes two different processes from state A to state B: one isothermal expansion, one adiabatic expansion. Which statement is true?
ABoth ΔU and Q are identical for the two processes since the endpoints are the same
BΔU is identical for both processes; Q and W differ between them
CΔU differs because the irreversibility of each process affects the internal energy
DQ is the same for both processes; only W differs depending on path
ΔU = U_B − U_A depends only on the thermodynamic states A and B (not on the path), so it is identical for both processes. Q and W separately are path-dependent (inexact): in the adiabatic process Q = 0 by definition, while in the isothermal process Q ≠ 0; correspondingly W differs too. The first law dU = đQ − đW expresses this: two inexact, path-dependent quantities whose difference always yields the same path-independent result. This is the central application of the exact/inexact distinction.
Question 2 Multiple Choice
Internal energy U is a state function. Which statement correctly follows from this?
AdU is inexact because it depends on the heat and work exchanged, which are path-dependent
BdU is exact only for reversible processes; it is inexact for irreversible ones
CdU is exact, so integrating it between two equilibrium states always gives the same result regardless of path
DdU is exact, but only when the process is both quasi-static and adiabatic
State functions have exact differentials by definition: integrating an exact differential between two points gives the same result regardless of path. Since U depends only on the current thermodynamic state (not on history or process), dU is exact and ΔU is path-independent. The reversibility of the process is irrelevant to whether dU is exact — U is a state function under all conditions. What changes between reversible and irreversible paths is how Q and W are individually distributed, not ΔU.
Question 3 True / False
An inexact differential like đQ cannot be integrated — it is mathematically undefined as an integral.
TTrue
FFalse
Answer: False
Inexact differentials can absolutely be integrated along a specific path. You can calculate the heat Q absorbed during a particular isothermal expansion by integrating đQ along that path and obtain a well-defined numerical result. What you cannot do is evaluate the integral using only the initial and final states — unlike exact differentials, the result depends on which path you take. Inexact means path-dependent, not unintegrable.
Question 4 True / False
For a differential expression M dx + N dy to be exact, the cross-partial derivatives must satisfy ∂M/∂y = ∂N/∂x.
TTrue
FFalse
Answer: True
This is the exactness criterion (integrability condition). When ∂M/∂y = ∂N/∂x, there exists a potential function F(x,y) such that M = ∂F/∂x and N = ∂F/∂y, and the integral along any path between the same endpoints gives the same result. When the condition fails, no such potential exists. In thermodynamics, applying this criterion to dU = T dS − P dV yields the Maxwell relations — the entire machinery of which is simply the exactness condition applied to thermodynamic state functions.
Question 5 Short Answer
Why do thermodynamicists write đQ and đW with a bar through the d, rather than dQ and dW? What would be wrong with writing dQ?
Think about your answer, then reveal below.
Model answer: Writing dQ would imply Q is a state function — that there exists a function Q(state) whose exact differential is dQ, making ΔQ path-independent. But heat is not a state function: how much heat a system exchanges depends entirely on the process (path), not just on the initial and final states. The notation đQ signals an inexact differential: there is no underlying function Q, and the integral ∫đQ is path-dependent. The same logic applies to đW.
The notational distinction enforces a conceptual one: dU is legitimate because U exists as a function of state variables (T, V, etc.), while Q and W do not. The first law dU = đQ − đW is an equation with one exact differential on the left and two inexact ones on the right whose difference cancels the path-dependence. Using dQ would be a mathematical claim — that a potential function Q exists — which is false in general.