The science
The case-study files treat either the isothermal limit (PV = const) or the adiabatic limit (TV^(γ−1) = const) of a piston gas. Real gases sit between them, so this lab adds a heat-exchange rate with the walls and lets you slide continuously from one file’s limit to the other.
Equations as delivered in the case-study files
TV^(γ−1)=constant; T(t)=T₀[V₀/(V₀+qt)]^(γ−1) [K].
P(t) V(t) = P₀ V₀ [kPa·m³]; V(t) = V₀ + qt [m³].
P(t)=nRT/[V₀−qt] [Pa]; dP/dt = (q/(nRT))P² [Pa/s] (R=8.314462618 J/(mol·K)).
For slow isothermal compression, PV = P₀V₀ [Pa m³]. ; V(t)=V₀−Au t [m³], with u positive inward. ; dP/dt=PAu/V(t) [Pa/s]; P(t)=P₀V₀/(V₀−Au t) [Pa].
What this lab adds
n = P0V0/(R T0), V(t) = V0 ± q t, P = nRT/V
dT/dt = −(γ−1) T (dV/dt)/V − (T − T_wall)/τ, ratio = t_run/τ
First law: ΔU = n Cv ΔT = W_on + Q, with W_on = −∫P dV and Q = ∫ n Cv (T_wall − T)/τ dt, Cv = R/(γ−1)
Assumptions in the delivered files
- Ideal reversible adiabatic expansion with fixed gas heat capacity ratio and prescribed volume.
- A closed ideal gas expands reversibly at constant temperature under imposed piston motion.
- Ideal dry gas at constant uniform temperature and amount; slowly prescribed volume decrease, no leakage, phase changes, friction or heating. Do not extrapolate beyond V₀−qt > 0.
Honest simplifications of this lab
- Ideal gas with constant γ and a single wall temperature equal to the starting temperature; no friction, leakage or piston mass.
- Several files do not state T0 or P0; where they do not, the lab uses 293 K or 101.325 kPa and says so in the preset note.
- The heat-exchange ratio is a scale-free stand-in for wall conductance; real cylinders need a heat-transfer coefficient and area.
Independent check The page checks the adiabatic end temperature against T0(V0/V)^(γ−1), the isothermal limit against PV = P0V0, step-size convergence and the first law; the test file repeats these with an independent RK4 integrator in Python.
Educational model. Case-study presets reuse the parameter sets, titles and captions of ChatGPT-generated simulations delivered for MySimulator; the physics core here is a richer shared model.