Residual properties and generalised correlations

Why it matters

Compressor and turbine duties for real gases, heat loads on high-pressure exchangers and the fugacities used in VLE all need H, S and G of non-ideal fluids. Residual properties split each property into an easy ideal-gas part plus a correction computed from PVT data or a correlation, so the same method works for any gas for which you know Tc, Pc and ω.

Key ideas

Residual property. For any extensive molar property M (V, U, H, S, G), the residual is M^R = M − M^ig, where M^ig is the value the same substance would have as an ideal gas at the same T and P. Since H^ig depends only on T, the whole pressure effect on H is in H^R.

Why G^R first. The residual Gibbs energy acts as a generating function: from G^R/RT as a function of T and P, V^R, H^R and S^R all follow by differentiation. G^R/RT is also exactly ln φ, the logarithm of the fugacity coefficient used in phase equilibrium.

Computing residuals from Z. Integrating at constant T from zero pressure (where every gas is ideal and residuals vanish):

• G^R/RT = ∫(Z − 1) dP/P
• H^R/RT = −T∫(∂Z/∂T)_P dP/P
• S^R/R = H^R/RT − G^R/RT So all you need is Z(T, P) from an equation of state or a correlation. Signs: at moderate conditions Z < 1, so G^R, H^R and S^R are usually negative; a real gas has lower enthalpy than the ideal gas at the same T and P because attractive forces lower its energy.

Enthalpy and entropy changes for real gases. Build a three-step path: real state 1 → ideal gas at T₁, P₁ (subtract H₁^R), ideal-gas change from 1 to 2 (Cp^ig integral), ideal gas → real state 2 (add H₂^R). Residuals at very low pressure can be taken as zero.

Generalised correlations (corresponding states). Tc, Pc and the acentric factor ω are enough to estimate Z and residuals for non-polar and slightly polar fluids:

• Pitzer's virial correlation (Abbott equations for B⁰ and B¹) is accurate where Z is nearly linear in P: roughly at low reduced pressure, or for Tr above about 3 at any pressure. Many textbooks give a chart of the region where it applies.
• Lee-Kesler tables give Z⁰, Z¹, (H^R)⁰/RTc, (H^R)¹/RTc and similar for the full Tr, Pr range including liquids. Take values from the tables in your data book; do not invent them.
• They are less reliable for strongly polar or associating fluids (water, alcohols) and quantum gases (H₂, He) unless special corrections are used.

Formulas

M^R = M − M^ig (same T and P) G^R/RT = ∫₀^P (Z − 1) dP/P (constant T) H^R/RT = −T ∫₀^P (∂Z/∂T)_P dP/P (constant T) S^R/R = H^R/RT − G^R/RT Z = 1 + (B⁰ + ω·B¹)·Pr/Tr (Pitzer virial correlation) B⁰ = 0.083 − 0.422/Tr^1.6, B¹ = 0.139 − 0.172/Tr^4.2 dB⁰/dTr = 0.675/Tr^2.6, dB¹/dTr = 0.722/Tr^5.2 G^R/RT = (B⁰ + ω·B¹)·Pr/Tr H^R/(R·Tc) = Pr·[B⁰ − Tr·dB⁰/dTr + ω·(B¹ − Tr·dB¹/dTr)] S^R/R = −Pr·(dB⁰/dTr + ω·dB¹/dTr) Z = Z⁰ + ω·Z¹ (Lee-Kesler; Z⁰, Z¹ from tables) ΔH = H₂^R − H₁^R + ∫Cp^ig dT ΔS = S₂^R − S₁^R + ∫Cp^ig dT/T − R·ln(P₂/P₁)

Symbols: M any molar property; Z compressibility factor; T (K), P (Pa or bar, consistent); Tr = T/Tc, Pr = P/Pc (dimensionless); ω acentric factor; B⁰, B¹ dimensionless reduced second-virial functions; H^R (J/mol), S^R (J/mol·K), G^R (J/mol); Cp^ig ideal-gas heat capacity (J/mol·K); R = 8.314 J/(mol·K).

Worked examples

Example 1 (standard): Z from the Pitzer correlation. Given: propane (Tc = 369.8 K, Pc = 42.48 bar, ω = 0.152) at 400 K and 10 bar. Find Z and V.

1. Tr = 400/369.8 = 1.0817; Pr = 10/42.48 = 0.2354.
2. B⁰ = 0.083 − 0.422/Tr^1.6 = 0.083 − 0.422/1.1338 = −0.2892.
3. B¹ = 0.139 − 0.172/Tr^4.2 = 0.139 − 0.172/1.3906 = 0.0153.
4. Z = 1 + (B⁰ + ω·B¹)·Pr/Tr = 1 + (−0.2892 + 0.152 × 0.0153) × 0.2354/1.0817 = 1 − 0.0624 = 0.9376.
5. V = Z·R·T/P = 0.9376 × 8.314 × 400 / 10⁶ = 3.118 × 10⁻³ m³/mol. Answer: Z = 0.938, V = 3118 cm³/mol

Example 2 (GATE level): residual enthalpy and entropy. Given: the same propane state. Find H^R and S^R, and the enthalpy change when propane is compressed isothermally at 400 K from 0.1 bar (treat as ideal) to 10 bar.

1. dB⁰/dTr = 0.675/Tr^2.6 = 0.675/1.2264 = 0.5504; dB¹/dTr = 0.722/Tr^5.2 = 0.722/1.5041 = 0.4800.
2. H^R/(R·Tc) = 0.2354 × [−0.2892 − 1.0817 × 0.5504 + 0.152 × (0.0153 − 1.0817 × 0.4800)] = 0.2354 × [−0.8846 − 0.0766] = −0.2262.
3. H^R = −0.2262 × 8.314 × 369.8 = −696 J/mol.
4. S^R/R = −Pr·(dB⁰/dTr + ω·dB¹/dTr) = −0.2354 × (0.5504 + 0.0730) = −0.1467, so S^R = −1.22 J/(mol·K).
5. Check: G^R/RT = −0.0624 and H^R/RT = −0.2262/1.0817 = −0.2092; S^R/R = −0.2092 + 0.0624 = −0.1468. Consistent.
6. Isothermal: ∫Cp^ig dT = 0 and H₁^R ≈ 0, so ΔH = H₂^R = −696 J/mol (heat must be removed beyond the work input to hold T constant). Answer: H^R = −696 J/mol, S^R = −1.22 J/(mol·K), ΔH ≈ −0.70 kJ/mol

Common mistakes

• Defining the residual as ideal minus real; the convention is M^R = M − M^ig.
• Comparing real and ideal states at the same T and V instead of the same T and P.
• Forgetting the −R·ln(P₂/P₁) term in ΔS: it belongs to the ideal-gas step, not to the residual.
• Using the Pitzer virial correlation at high reduced pressure or in the liquid region; switch to Lee-Kesler tables or a cubic EOS.
• Using °C in Tr or mixing bar and Pa between P and Pc.
• Reading H^R as zero for an isothermal process; only the ideal-gas part is zero.

For GATE CH

• NAT problems: Z, B or V from the Pitzer correlation with given Tc, Pc and ω; H^R or S^R from a given virial coefficient or Z(T, P) expression.
• Derivation-style questions: residual enthalpy for a gas obeying Z = 1 + BP/RT (giving H^R = P·(B − T·dB/dT)).
• Real-gas ΔH or ΔS across a compressor or throttle using the three-step path.
• Practise keeping track of reduced variables, signs of the residuals and when the virial approach is valid.

Quick check

1. Define the residual enthalpy.
2. Why is G^R considered a generating function?
3. For a gas with Z = 1 + BP/RT and B constant, what is G^R?
4. Is H^R normally positive or negative for a gas at moderate pressure?
5. Which three parameters does the Pitzer correlation need for a pure fluid?

Answers: 1. H^R = H − H^ig at the same T and P. 2. Because V^R, H^R and S^R all follow from G^R/RT by differentiation. 3. G^R = BP. 4. Negative. 5. Tc, Pc and ω.