At the Boyle temperature, the second virial coefficient is zero, and the gas behaves ideally at low pressures, so Z = 1. For a van der Waals gas, T_B = a/(Rb), where attractive and repulsive effects balance.

 Z = PV/(nRT). For one mole, Z = PV/RT. Given P = 5 × 10⁵ Pa, V = 0.05 m³, T = 500 K, R = 8.314 J/mol·K:

Z = (5 × 10⁵ × 0.05) / (8.314 × 500) = 25000 / 4157 ≈ 0.801.

 In the van der Waals equation, (P + a/V²)(V – b) = RT, the term a/V² corrects for attractive forces and becomes significant when volume V decreases (i.e., at high pressures), as Z deviates further from 1.

When Z < 1, the gas has a lower PV product than predicted by the ideal gas law, indicating that attractive forces dominate, making the gas more compressible.

The compressibility factor at the critical point for any van der Waals gas is Z_c = 3/8 = 0.375, independent of the specific values of a and b.

 At high temperatures, molecular kinetic energy dominates, reducing intermolecular forces. At low pressures, molecular volume is negligible compared to total volume, making the gas behave ideally (Z ≈ 1).

For a van der Waals gas, Zc = Pc. Vc / (RT_c). Using critical constants Pc = a/(27b²), Vc = 3b, Tc = 8a/(27Rb):

Zc = (a/(27b²) × 3b) / (R × 8a/(27Rb)) = 3/8 = 0.375.

 When Z > 1, the gas is less compressible than an ideal gas, indicating that repulsive forces (due to molecular volume) dominate, causing the gas to occupy more volume than predicted by the ideal gas law.

For an ideal gas, PV = nRT, so Z = PV/(nRT) = 1 under all conditions of pressure and temperature.

The compressibility factor is Z = PV/(nRT), where P is pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is the absolute temperature. For one mole, Z = PV/RT. It measures deviation from ideal gas behavior (Z = 1 for an ideal gas).