Real Gas Equation Tool

Physics Thermodynamics • Advanced Applications and Extensions in Thermodynamics

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

1. Real Gas Equation Tool

Solve real-gas relations using $(P+\dfrac{a}{V^2})(V-b)=RT$ (van der Waals), a truncated virial form, or compressibility $Z=\dfrac{PV}{RT}$ . Includes critical-point values (vdW), iteration for $V$ when needed, and an ideal vs real PV isotherm plot.

Units (default): bar, L/mol, K with

R=0.08314\ \mathrm{L\cdot bar/(mol\cdot K)}

. Inputs support pi, e, sqrt(), sin, cos, exp, log, and *.

Model + Variables

Equation model: Solve for: Gas library (vdW): vdW $a$ (L²·bar/mol²): vdW $b$ (L/mol): Virial $B$ (L/mol): Compressibility $Z$: Pressure $P$ (bar): Molar volume $V$ (L/mol): Temperature $T$ (K): Gas constant $R$: Auto-update plot:

Tip: If you solve for $V$ with van der Waals, the tool uses a robust bracketing search ($V>b$) and picks the gas-like root.

Animation + plot controls

Progress $\tau\in[0,1]$: Play speed: Arrow pulse: Plot $V_{min}$ (L/mol): Plot $V_{max}$ (L/mol): Show ideal curve: Show $Z$ readout:

PV plot: drag-to-pan, wheel-to-zoom, double-click reset.

Ready

Steps

Enter values and click Solve.

Piston schematic (volume change)

Schematic compression/expansion at the chosen model

Phase: —

PV isotherm plot (ideal vs real)

Isotherm at current $T$ with state marker

Hover: (V, P)=…

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Theory: Real Gas Equations (vdW, Virial, and $Z$)

Ideal gases obey $$PV=RT$$ (per mole). Real gases deviate because molecules have finite size and intermolecular forces. This tool includes three common “real-gas” descriptions: van der Waals, a truncated virial model, and a compressibility-factor model.

1) van der Waals equation

van der Waals modifies the ideal gas law with: $$a$$ (attractions) and $$b$$ (excluded volume):

\[ \left(P+\frac{a}{V^2}\right)(V-b)=RT \quad\Longleftrightarrow\quad P=\frac{RT}{V-b}-\frac{a}{V^2}. \]

$b$ reduces available volume: you must have $$V>b$$ .
$a$ lowers pressure (attractions), especially at small $V$ where $a/V^2$ is large.

Critical point for van der Waals

For the vdW model, the critical constants follow closed-form expressions:

\[ V_c=3b,\qquad P_c=\frac{a}{27b^2},\qquad T_c=\frac{8a}{27Rb}. \]

Near and below $T_c$, the vdW isotherm can develop multiple real roots for a given $P$, reflecting the model’s two-phase (liquid–gas) region. The calculator reports the “gas-like” root (largest $V$).

2) Virial equation (truncated)

The virial expansion writes deviations in terms of the compressibility factor $$Z$$ :

\[ Z=\frac{PV}{RT}. \]

A simple truncation in terms of molar volume is:

\[ Z=1+\frac{B}{V}, \qquad P=\frac{RT}{V}\left(1+\frac{B}{V}\right). \]

Here $B$ is a “second virial” correction (often depends on $T$ in more advanced models). This truncation is most accurate at low-to-moderate densities (larger $V$).

3) Compressibility-factor model

If you know or assume a compressibility factor $Z$, you can treat the gas as:

\[ PV = ZRT. \]

This is convenient when $Z$ comes from tables, charts, or correlations (e.g., corresponding states).

Interpreting $Z$

$Z=1$: ideal behavior.
$Z<1$: attractions dominate (measured pressure lower than ideal at the same $T,V$).
$Z>1$: repulsions/excluded volume dominate (pressure higher than ideal).

Why a PV isotherm plot helps

Plotting $P(V)$ at fixed $T$ makes real-gas effects visible: the ideal isotherm scales as $1/V$, while real models can bend above/below it, and vdW can show multiple-$V$ solutions at certain $T,P$.

Units used in this calculator

The default unit system is: bar for pressure, L/mol for molar volume, K for temperature. Then \[ R=0.08314\ \mathrm{L\cdot bar/(mol\cdot K)}. \]

If you change $R$, ensure your $P,V$ units remain consistent with it.

Steps

Piston schematic (volume change)

PV isotherm plot (ideal vs real)

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