Ideal Gas and Van der Waals

Van der Waals Equation of State vs. Ideal Gas — P-V Isotherms

Gas Model

Gas

a = 3.61 L²·atm/mol² · b = 0.0429 L/mol · T_c = 304.1 K

Solve for

Parameters

Volume

Container volume

0.0110000

Temperature

Absolute gas temperature

-273.15000

Amount of substance

Moles of gas (or mass if molar mass is known)

0.00110000

Results

Pressure

—

atm

Compressibility Factor (Z)

—

P-V Diagram (Isotherm)

Fundamentals & Explanation

Ideal Gas vs. Real Gas

The ideal gas law describes the behavior of a hypothetical gas where molecules do not interact and occupy no volume. The equation of state is:

PV = nRT

where P is the pressure, V the volume, n the amount of substance in moles, R the universal gas constant (0.0820574 L·atm·mol⁻¹·K⁻¹, NIST value rounded to 7 significant figures), and T the absolute temperature in Kelvin.

Van der Waals Equation

In 1873, Johannes Diderik van der Waals proposed a modification that incorporates two corrections to the ideal model:

Intermolecular attraction (constant a): real molecules attract each other, reducing the effective pressure.
Exclusion volume (constant b): molecules have finite volume, reducing the available space.

\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT

The constants a and b are specific to each gas and are experimentally determined. Gases with strong intermolecular forces (like H₂O) have high a values, while noble gases like He have values close to zero.

Compressibility Factor (Z)

The compressibility factor measures the deviation from ideal behavior:

Z = \frac{PV}{nRT}

Z = 1: ideal behavior.
Z < 1: attractive forces dominate (the gas occupies less volume than expected).
Z > 1: repulsive forces dominate (the gas occupies more volume than expected).

Critical Temperature

The critical temperature (T_c) is the temperature above which a gas cannot be liquefied by compression alone, regardless of the applied pressure. When T < T_c, the Van der Waals equation may predict regions where liquid and gas phases coexist (Van der Waals loop), and the equation loses quantitative validity in that region.

Solve Formulas

Pressure:

P = \frac{nRT}{V - nb} - \frac{an^2}{V^2}

Volume (cubic):

PV^3 - (nbP + nRT)V^2 + an^2V - abn^3 = 0

Temperature:

T = \frac{(P + \frac{an^2}{V^2})(V - nb)}{nR}

Van der Waals Constants

Gas	a	b	T_c (K)
He	0.0346	0.0238	5.2
N₂	1.370	0.0387	126.2
O₂	1.364	0.0318	154.6
CO₂	3.610	0.0429	304.1
H₂O	5.460	0.0305	647.1

Units: a in L²·atm/mol², b in L/mol.

Reference

Atkins, P. W., & de Paula, J. (2014). Atkins' Physical Chemistry (10th ed.). Oxford University Press. Chapter 1: The properties of gases.