Process Path Pv Over Ts Diagram Plotter

Physics Thermodynamics • Advanced Applications and Extensions in Thermodynamics

Written by STEM Calculators Team Published January 8, 2026 Updated February 24, 2026

4. Process Path PV/TS Diagram Plotter

Plot custom multi-step processes/cycles on PV and/or TS diagrams. Includes presets (e.g., Carnot), work/heat shading, an ideal vs. “real-gas EOS” toggle (vdW approximation for PV), interactive pan/zoom, and image export.

Default assumptions for computed TS/energies: ideal gas, reversible processes, constant \(\gamma\). “Real EOS” affects the PV curve shape (via van der Waals), while entropy/energy calculations remain ideal-gas.

Mode + Diagram

Mode: Show: Shading: Progress \(\tau\in[0,1]\):

Tip: In manual mode, drag points directly on the diagram. In builder mode, edit step parameters and re-solve.

Thermo settings (builder mode)

Pressure unit: Moles \(n\): \(\gamma=c_p/c_v\): Equation of state (PV): vdW gas preset: vdW \(a\) \((\mathrm{L^2\cdot P\_unit/mol^2})\): vdW \(b\) (L/mol):

Start pressure \(P_1\): Start temperature \(T_1\) (K):

Start volume is computed from \(P_1,T_1\) (ideal) or solved from vdW if “Real EOS” is selected.

Multi-step builder

Step type	Parameter	Value

0 steps

Supported: isothermal (set \(V_2/V_1\)), adiabatic/isentropic (set \(V_2/V_1\)), isochoric (set \(T_2/T_1\)), isobaric (set \(T_2/T_1\)), polytropic (set exponent \(n\) and \(V_2/V_1\)).

Ready

Steps + results

Build a path and click Solve.

PV diagram

PV path (drag-to-pan, wheel-to-zoom, double-click reset)

Hover: —

TS diagram

TS path (reversible heat area), pan/zoom enabled

Hover: —

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Theory: Process Paths on PV and TS Diagrams

1) What PV and TS diagrams show

PV diagram: plots pressure vs. volume. For quasi-static processes, the boundary work is

\[ W=\int_{1}^{2} P\,dV. \]

Closed cycle: the net work equals the area enclosed by the loop (sign depends on direction).

TS diagram: plots temperature vs. entropy. For a reversible process,

\[ \delta Q_{rev}=T\,dS,\qquad Q_{rev}=\int_{1}^{2} T\,dS. \]

Closed reversible cycle: the net heat equals the area enclosed on a TS diagram, and \(Q_{net}=W_{net}\).

2) Ideal-gas relations used by the builder

The builder assumes an ideal gas with constant \(\gamma=c_p/c_v\):

\[ P V = n R T,\qquad c_v=\frac{R}{\gamma-1},\qquad c_p=\frac{\gamma R}{\gamma-1}. \]

Entropy change between two states can be written (ideal gas):

\[ \Delta s = c_v\ln\!\left(\frac{T_2}{T_1}\right)+R\ln\!\left(\frac{V_2}{V_1}\right) =c_p\ln\!\left(\frac{T_2}{T_1}\right)-R\ln\!\left(\frac{P_2}{P_1}\right). \]

The absolute entropy reference is arbitrary in the plotter; it sets \(S_1=0\) and accumulates \(\Delta S\).

3) Common processes (reversible)

Isothermal (\(T=\) const):

\[ W=nRT\ln\!\left(\frac{V_2}{V_1}\right),\qquad Q=W,\qquad \Delta S=nR\ln\!\left(\frac{V_2}{V_1}\right). \]

Adiabatic / isentropic (\(Q=0,\ \Delta S=0\)):

\[ T V^{\gamma-1}=\text{const},\qquad P V^{\gamma}=\text{const},\qquad W=-\Delta U. \]

Isochoric (\(V=\) const):

\[ W=0,\qquad Q=\Delta U=n c_v (T_2-T_1),\qquad \Delta S=n c_v \ln\!\left(\frac{T_2}{T_1}\right). \]

Isobaric (\(P=\) const):

\[ W=nR(T_2-T_1),\qquad Q=n c_p (T_2-T_1),\qquad \Delta S=n c_p \ln\!\left(\frac{T_2}{T_1}\right). \]

4) Carnot cycle (sample preset)

A Carnot cycle consists of two isotherms (at \(T_h\) and \(T_c\)) and two adiabats. On a TS diagram, isotherms are horizontal lines and adiabats are vertical lines, so the loop is a rectangle.

\[ W_{net}=Q_{in}-Q_{out}=\text{(area enclosed on PV)}=\text{(area enclosed on TS for reversible)}. \]

Real engines are not perfectly reversible, so the “real engine” efficiency is below the ideal cycle prediction.

5) “Real” toggle and advanced notes

The calculator includes a van der Waals EOS option to change the PV curve shape:

\[ P=\frac{nRT}{V-nb}-a\frac{n^2}{V^2}. \]

However, entropy and energy computations still use ideal-gas formulas (constant \(\gamma\)). Fully-consistent real-gas entropy and exergy calculations require more advanced property models.

University extension ideas: exergy destruction shading, irreversibility (entropy generation), and real-gas departure functions.

Steps + results

PV diagram

TS diagram

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