Real Gases

Real gases deviate from ideal-gas behaviour because of finite molecular size and intermolecular forces. Deviations are quantified by the compressibility factor Z = pV_m/(RT) (Z = 1 for an ideal gas). Two common approaches to model real gases are (1) equations of state (EOS) such as van der Waals, Redlich–Kwong, Peng–Robinson that include parameters accounting for attraction and excluded volume, and (2) virial expansions (in powers of density or pressure) whose coefficients (B, C, ...) encode intermolecular interactions. Thermodynamic "departure" or "residual" functions (e.g., H^R = H_real − H_ideal) measure how much a real gas property differs from the ideal-gas value at the same T and p. Fugacity f and the fugacity coefficient φ = f/p quantify the effective chemical potential of a real gas (φ → 1 in the ideal limit). Important phenomena for real gases include critical behaviour, phase equilibria, and Joule–Thomson throttling (temperature change on isenthalpic expansion), which depend on the balance of repulsive and attractive forces encoded in EOS parameters or virial coefficients.

Governing FormulaKey relations (SI units): - Compressibility factor: Z = p V_m / (R T) - van der Waals EOS (molar form): p = R T / (V_m - b) - a / V_m^2 - van der Waals critical constants: V_c = 3 b, p_c = a / (27 b^2), T_c = 8 a / (27 R b), Z_c = 3/8 - Second virial (pressure form) low-pressure expansion: Z ≈ 1 + B(T) p / (R T) - Fugacity coefficient (residual Helmholtz/Gibbs): ln φ = G^R / (R T) = (μ_real − μ_ideal) / (R T) - van der Waals maximum inversion temperature (approximate): T_max = 2 a / (R b) (approximate criterion for Joule–Thomson inversion) - Ideal-gas limit: φ → 1, residual properties → 0 Assumptions: single-component pure substance unless otherwise stated; R = 8.314462618 J·mol^−1·K^−1 when numeric evaluation is required.

Knowledge Check

10 Questions

1.Which compressibility-factor condition indicates that attractive intermolecular forces predominate at the specified T and p?

2.For a van der Waals gas with parameters a = 1.50 Pa·m^6·mol^−2 and b = 4.00×10^−5 m^3·mol^−1, use the approximate van der Waals criterion for the maximum inversion temperature T_max = 2 a / (R b). Evaluate T_max (use R = 8.314462618 J·mol^−1·K^−1).

3.A real gas at T = 300 K has a molar residual Gibbs energy G^R = G_real − G_ideal = −500 J·mol^−1. What is the fugacity coefficient φ (φ = f/p)? Use ln φ = G^R/(R T) and R = 8.314462618 J·mol^−1·K^−1.

4.At T = 300 K the second virial coefficient for a gas is B(T) = −1.20×10^−4 m^3·mol^−1. Estimate the compressibility factor Z at p = 5.0 MPa using the low-pressure virial pressure-series truncation Z ≈ 1 + [B(T) p]/(R T). Use R = 8.314462618 J·mol^−1·K^−1.

5.For a van der Waals fluid with a = 3.50 Pa·m^6·mol^−2 and b = 4.00×10^−5 m^3·mol^−1, what is the critical pressure p_c (use p_c = a / (27 b^2))? Express the answer in MPa.

6.Which statement about the fugacity coefficient φ of a real gas is correct?

7.What is the value of the compressibility factor at the critical point predicted by the van der Waals EOS (Z_c = p_c V_c /(R T_c))?

8.A particular van der Waals gas has maximum inversion temperature T_max = 9000 K. If a gas sample at T = 300 K undergoes an isenthalpic (Joule–Thomson) expansion at moderate pressures, what qualitative temperature change should be expected?

9.Which of the following statements is true for an ideal gas (relative to a real gas at the same T and p)?

10.At T = 350 K and p = 2.0 MPa a real gas has compressibility factor Z = 0.85. Compute the molar volume V_m (m^3·mol^−1) using V_m = Z R T / p. Use R = 8.314462618 J·mol^−1·K^−1.