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If Z is the compressibility factor, which expression correctly represents the van der Waals equation for a real gas at low pressure?

1. Z = 1 + R*T / (P*b)
2. Z = 1 - a / (Vₘ * R * T)
3. Z = 1 - P*b / (R*T)
4. Z = 1 + P*b / (R*T)

Correct answer: Z = 1 - a / (Vₘ * R * T)

Solution

Van der Waals equation: (P + a/Vₘ²)(Vₘ - b) = RT. Expanding: P*Vₘ - Pb + a/Vₘ - ab/Vₘ² = RT, so P*Vₘ = RT + Pb - a/Vₘ + ab/Vₘ². Dividing by RT: Z = 1 + Pb/(RT) - a/(Vₘ*RT) + ab/(Vₘ²*RT). At low pressure Vₘ is large (Vm ~ RT/P), so Pb/(RT) is first order in P and a/(Vₘ*RT) ~ aP/(RT)² is second order. The dominant deviation at low pressure is -a/(Vₘ*RT), giving Z = 1 - a/(Vₘ*RT).

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