P–V CYCLE
Four exact isobars and isochores close ΔU and ΔS_gas to zero while Q_net = W_net = the signed rectangular P–V area.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
System + direction
A closed monatomic ideal gas follows one rectangular P–V loop beneath a frictionless piston. Clockwise traversal produces net work; counterclockwise traversal requires net work. The permanent convention is ΔU = Q − W_by. No reservoir temperatures are supplied, so environmental entropy is explicitly not evaluated.
Four exact states
One lower-left temperature, one low volume, and two dimensionless ratios determine every corner through PV = nRT. The graph is a literal rectangle: two constant-volume sides and two constant-pressure sides. Monatomic and diatomic choices change C_v, C_p, and every thermal ledger entry, while leaving the same equation of state.
Four leg ledgers
Each row is signed with the same convention. Isochores have zero work; isobars use PΔV. Internal energy follows temperature, heat closes ΔU = Q − W_by, and gas entropy uses the exact logarithm. The signed hatched rectangle is the geometric cycle work.
One synchronized cycle
The piston, P–V marker, active ledger row, heat and work arrows, cumulative Q, W_by, ΔU, gas entropy, and signed area all read one path coordinate. The six-second traverse is display time only. The gas path is quasistatic and internally reversible; external boundary temperatures are unspecified rather than silently invented.
Cycle + entropy audit
The marker and piston return exactly to A. The four internal-energy changes and four working-gas entropy changes sum to zero because both are state functions. Net heat equals net work, and the work equals both the signed rectangle and an independently oriented shoelace area. None of those closures supplies environmental entropy without a reservoir model.