Read Work as Area and Apply the First Law Path by Path
Thermodynamics questions with P-V diagrams all rest on two ideas: work is the area under the curve, and the first law ties heat, work and internal energy together, with internal energy depending only on the end states. Once those are in place, cycles and process-matching questions are bookkeeping.
First Law and P-V Diagrams is the central idea in this question pattern. Identify the condition in the stem before applying the method below.
First law and sign convention
pick one and keep it throughout. ΔU depends only on the initial and final states, so it is the same for every path between them, while Q and W depend on the path. When a gas goes from state a to state b with Q = 200 J absorbed and W = 80 J done by it, ΔU = 120 J, and any other path between the same states also has ΔU = 120 J.
Work as area on a P-V diagram
Work done by a gas is the area under the P-V curve: positive when the volume grows, negative when it shrinks, and zero on a vertical (constant-volume) line. Over a closed cycle the net work is the enclosed area, positive for a clockwise loop (a heat engine) and negative for an anticlockwise loop (a refrigerator), and ΔU is zero for the whole cycle, so net heat equals net work. For the legs of a cycle, a horizontal segment is isobaric, a vertical segment is isochoric, and a hyperbola is isothermal.
Special processes you can read off
In a free expansion of an ideal gas into a vacuum inside an insulated container the gas does no work, exchanges no heat, and so has ΔU = 0 and unchanged temperature; doubling the volume halves the pressure. Reversing a path changes the sign of ΔU, so if ΔU is +30 cal from i to f it is -30 cal from f to i.
Matching cycles and processes
In matching-table questions, label every leg first (isobaric, isochoric, isothermal, adiabatic), then compute Q, W and ΔU for a monatomic gas with Cv = 3R/2 and Cp = 5R/2. Heat absorbed in the cycle is the sum of the positive Q values of the legs, and the efficiency of the cycle is net work divided by that heat absorbed. Do these leg by leg rather than guessing from the shape of the whole loop.
OlympIQ exam lens
- ✓ΔU = Q - W with W done by the gas (or Q + W with W on the gas); do not mix the two
- ✓ΔU depends on the end states only; Q and W depend on the path
- ✓Work = area under the P-V curve; closed cycle: net work = enclosed area, ΔU = 0
- ✓Isochoric: W = 0 and Q = ΔU; free expansion of an ideal gas in an insulated vessel: W = Q = ΔU = 0
- ✓Label each leg (isobaric, isochoric, isothermal, adiabatic) before computing Q, W and ΔU
Do not assume heat or work is the same along two paths between the same two states just because ΔU is. ΔU is a state function; Q and W are path functions, so each path needs its own calculation.
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