P–V WORK
W_by = ∫P dV, signed by direction; ΔU = Q − W_by closes while one computed piston state draws the entire ledger.
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 + sign
A closed monatomic ideal gas sits below a frictionless piston. Choose one reversible constraint and one endpoint ratio. Heat into the gas is positive; work done by the gas is positive, so the permanent convention is ΔU = Q − W_by. The dependent endpoint is solved from the constraint instead of accepting an incompatible P₂,V₂,T₂ triplet.
The selected path
The ideal-gas surface supplies P = nRT/V; the selected reversible constraint cuts one exact path through it. Monatomic and diatomic gases differ only through their constant molar heat capacities and γ here. The diatomic model keeps rotations active and vibrations frozen, an approximation stated rather than hidden.
Signed work area
Work done by the gas is the oriented integral ∫P dV. Expansion traces toward larger volume and gives positive W_by; compression retraces toward smaller volume and gives negative W_by. The polygon closes on the true P = 0 baseline, and arrow plus hatch direction carry the sign redundantly.
One reversible traverse
The piston position, P–V cursor, growing curve, signed work area, and partial Q, W_by, and ΔU all come from one process coordinate. Six display seconds do not assert a physical duration: reversibility idealizes an arbitrarily slow path, and reversible heat transfer requires a continuum of reservoirs infinitesimally close to the gas temperature.
First-law audit
At the endpoint, ΔU = Q − W_by closes with the declared sign convention. Numerical quadrature independently checks the exact work, a separate heat identity checks Q, and the endpoint validator checks both PV = nRT and the selected path invariant. The model is ideal-gas, constant-heat-capacity, quasistatic, frictionless, and excludes the singular V → 0 limit.