Theory — Thermodynamic Process Path Builder
This tool builds a multi-step ideal-gas process and computes each step using the
first law of thermodynamics and standard ideal-gas relations. You can reorder steps to “invent” cycles and engines,
then check if the final state matches the initial state (a closed cycle).
1) State variables and ideal gas
A thermodynamic state is described by \((P,V,T)\). For an ideal gas:
If any two of \(P,V,T\) are known (and \(n\) is fixed), the third is determined by the equation above.
2) First law and sign convention
The tool uses the common convention “work done by the gas”:
- \(Q>0\): heat added to the gas
- \(W_{\text{by}}>0\): the gas expands and does work on the surroundings
For an ideal gas, internal energy depends only on temperature:
The calculator models \(C_V\) and \(C_P\) via \(\gamma=\dfrac{C_P}{C_V}\) with \(C_P=C_V+R\), so \(C_V=\dfrac{R}{\gamma-1}\).
3) Step formulas (ideal gas)
Isochoric (\(V=\) const)
Isobaric (\(P=\) const)
Isothermal (\(T=\) const, reversible ideal gas)
Adiabatic (\(Q=0\), reversible)
Reversible adiabatic relations: \(PV^\gamma=\text{const}\) and \(TV^{\gamma-1}=\text{const}\).
4) Path diagrams: \(P\!-\!V\) and \(T\!-\!S\)
The area under a \(P(V)\) curve equals work done by the gas:
If entropy preview is enabled, the tool shows reversible ideal-gas entropy changes:
In \(T\!-\!S\) coordinates, reversible adiabatic steps appear as \(S=\) const, and isothermal steps appear as \(T=\) const.
5) Closed cycles and net changes
A cycle returns to the initial state: \((P_f,V_f,T_f)\approx(P_0,V_0,T_0)\).
For a closed cycle, \(\Delta U_{\text{net}}=0\), so the net heat equals the net work:
University extension: add non-ideal corrections (e.g., van der Waals) or irreversible steps and compare the diagram changes.
