GAS LAW
Four roads between two states — and the tolls differ. That difference is why engines exist.
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.
Setup
A cylinder of ideal gas, a piston, and a choice: hold the temperature, hold the pressure, hold the volume, or hold the heat at zero. Same start, four different roads — and the whole point is that the roads cost differently.
State + ledger
Two equations run this module. The state equation pV = nRT pins every configuration the gas can occupy; the first law ΔU = Q − W keeps the energy books, with heat finally granted a column of its own. For a monatomic gas the bank balance is temperature alone.
Solve
Each constraint has a closed form and generally selects its own endpoint from the shared start and the supplied controls. Work is the area under that process-specific road; comparing path-dependent work requires explicitly choosing common endpoints, which this selector does not pretend to do.
Traverse
The piston creeps along its chosen road, quasi-statically — every frame an exact solution of the state equation, no interpolation. The molecule dots keep their number and change only their spacing; the amber area accrues joule by joule as the work is done.
Audit
Audited: pV = nRT holds at every sampled instant on every road; the first law closes to machine precision on all four; and wherever two independent routes to the same number exist — Q via C_p on the isobar, W via pressures and via temperatures on the adiabat — they agree to the last bit.