5.2 Equations of State and Real Gases

5.2 Equations of State and Real Gases

Quick Answer: Use PV = nRT when gases are dilute and far from condensation. Otherwise use Z = PV/(nRT) with critical/reduced properties, or a cubic EOS (van der Waals level) that corrects for molecular volume and attractions. Ideal gas means Z = 1.

Material and energy balances often assume ideal gas for air, nitrogen, and light hydrocarbons at moderate pressure. Qatar gas-processing, LNG, and high-pressure utility services regularly leave that regime. Domain B expects you to know when the ideal gas law fails and how compressibility and simple EOS ideas restore accuracy.

Ideal Gas Law

Forms you must recognize instantly:

R must match units (e.g., 8.314 J/(mol·K) with P in Pa and V in m³). Temperature is absolute (K).

Ideal gas microscopic picture: molecules have negligible volume compared with container volume, and intermolecular forces are negligible except during instantaneous elastic collisions. Internal energy of an ideal gas depends only on temperature (and composition), not on V or P.

When Ideal Gas Works Well

• Low pressure relative to critical pressure (often P_r = P/P_c ≪ 1)
• Temperature not too close to the dew/bubble region for that mixture
• Light gases (H₂, N₂, air, CH₄) at ambient or elevated T and modest P
• Many flue-gas and dilute vent calculations on licensing exams

When Ideal Gas Fails

Rule of thumb (not a hard law): If P is a small fraction of P_c and T is well above T_c (gas far from liquefaction), ideal gas is often acceptable for MCQ engineering estimates. If the problem gives critical data or Z, use them.

Compressibility Factor Z

Define:

Z = PV / (nRT) = Pv / (RT)

Rearranged engineering uses:

• v = ZRT/P (molar volume)
• ρ = PM/(ZRT) (mass density)
• n = PV/(ZRT)

Generalized compressibility charts plot Z versus reduced pressure P_r at constant reduced temperature T_r (and sometimes acentric factor ω as a third parameter). Two fluids at the same T_r and P_r have approximately the same Z (corresponding states).

Critical Properties Intuition

Reduced properties:

T_r = T / T_c , P_r = P / P_c , v_r = v / v_c

Exam skill: given T, P, T_c, P_c, compute T_r and P_r and reason whether Z is near 1. You are not expected to memorize an entire Z chart, but you should know that Z → 1 as P_r → 0 at fixed T_r, and that near T_r ≈ 1 and moderate-to-high P_r, departures are severe.

Cubic Equations of State (Awareness Level)

The van der Waals equation is the prototype cubic EOS:

(P + a/v²)(v − b) = RT

(on a molar basis; a and b are substance-specific parameters).

Expanded in powers of v, the equation is cubic in volume—hence “cubic EOS.” Industrial relatives (Redlich–Kwong, Soave–Redlich–Kwong, Peng–Robinson) keep the same philosophy with improved temperature dependence of attractions and better liquid-density/VLE performance.

What UPDA-level items usually want:

• Name the physical meaning of a and b
• Know that cubic EOS can give up to three real volume roots in the two-phase region (liquid-like, unstable, vapor-like)—equilibrium selection uses equal fugacity, covered with VLE
• Know that ideal gas is recovered as v → ∞ (low density): attractions and b become negligible relative to v
• Do not spend exam time deriving departure functions from scratch unless given a formula

Connecting EOS to Process Calculations

Density-sensitive balances. Volumetric flow meters, pipe Reynolds numbers, and compressor inlet actual m³/h all need real density ρ = PM/(ZRT). Using Z = 1 when Z = 0.8 underpredicts mass flow for a fixed volumetric rate (or mis-sizes equipment).

Isothermal compressibility and work. Ideal-gas isothermal work for closed-system reversible compression is nRT ln(V1/V2) = nRT ln(P2/P1). Real gases need ∫ v dP with v from EOS or Z(P).

Enthalpy and entropy departures. Real-gas h and s differ from ideal-gas values at the same T (and reference P). Departure functions come from EOS; many exam stems simply state “assume ideal gas” or give tabulated h, s.

Worked Example: Ideal Gas vs Z Correction

Problem. Methane-rich gas is stored at 300 K and 80 bar. Approximate critical constants for a pure-methane idealization: T_c ≈ 190.6 K, P_c ≈ 46 bar. Molar mass M = 16 kg/kmol. R = 8.314×10⁻⁵ bar·m³/(mol·K). Compare ideal-gas density to a hypothetical chart reading Z ≈ 0.85 at the corresponding reduced state.

Step 1 — Reduced conditions.

T_r = 300 / 190.6 ≈ 1.57
P_r = 80 / 46 ≈ 1.74

T_r is above 1, so the fluid is supercritical relative to methane’s T_c; still, P_r is not small, so Z may deviate from 1.

Step 2 — Ideal-gas density.

Using ρ = PM/(RT) carefully with consistent R: qualitatively, ideal gas ignores Z.

ρ_ideal = PM / (RT)

Step 3 — Real density.

ρ_real = PM / (ZRT) = ρ_ideal / Z

If Z ≈ 0.85, then ρ_real ≈ ρ_ideal / 0.85 ≈ 1.18 × ρ_ideal.

Exam lesson: At elevated reduced pressure, mass in a vessel or mass flow for a given actual volume can be tens of percent off if you force Z = 1. When a stem supplies Z or critical data, apply them.

Throttling and the Joule–Thomson Effect

Real-gas behaviour becomes visible on the plant floor the moment a gas crosses a valve, orifice, or choke. A throttle has no shaft work and, over a short fitting, negligible heat transfer and negligible kinetic-energy change, so the steady-flow energy balance collapses to the defining condition of a throttle:

ĥ₁ = ĥ₂ — throttling is isenthalpic, not isothermal and not isentropic.

For an ideal gas, enthalpy depends on temperature alone, so isenthalpic expansion means ΔT = 0. Every temperature change you observe across a real throttle is therefore a real-gas effect, quantified by the Joule–Thomson coefficient:

μ_JT = (∂T/∂P)_H

The locus of μ_JT = 0 is the inversion curve, and the temperature below which a gas cools on throttling at a given pressure is its inversion temperature. Most industrial gases — methane, natural gas, propane, nitrogen, carbon dioxide, air — sit below their upper inversion temperature at ordinary conditions and therefore cool when throttled. Hydrogen and helium are the classic exceptions: at ambient temperature they lie above their inversion temperatures and warm on expansion, which is why hydrogen systems must be pre-cooled before a JT stage can liquefy anything.

Why this matters in Qatar

Joule–Thomson cooling is not a textbook curiosity here; it is the working principle of a large part of the country's gas industry.

• LNG and NGL trains use JT valves and expanders as part of liquefaction and dew-point control.
• Wellhead and pipeline chokes drop pressure by tens of bar, and the resulting cooling can drive the stream into the hydrate formation region — the reason for hydrate inhibitor injection and line heaters upstream of chokes.
• Cryogenic embrittlement and frost/ice formation downstream of control valves are direct consequences of the same effect, and they show up in HAZOP low-temperature deviations (Chapter 12).

Throttle versus turbine — an exam favourite

Because the expander removes energy as work while the valve does not, an expander delivers more cooling per bar of pressure drop than a JT valve. Stems that ask which device gives the colder outlet, or which one is "lost work," are testing exactly this distinction: throttling is inherently irreversible and generates entropy even though enthalpy is conserved.

Fast checks

1. "Ideal gas throttled through a valve — outlet temperature?" Unchanged, because h = h(T) only.
2. "Natural gas throttled from 80 bar to 20 bar — outlet temperature?" Lower, because μ_JT > 0 for methane at ordinary conditions.
3. "Hydrogen throttled at 25 °C?" Warmer — the standard trap.
4. "Is throttling isentropic?" No. It is isenthalpic and irreversible; entropy increases.

Mixtures (Exam Awareness)

For gas mixtures at low pressure, Dalton’s law (partial pressure p_i = y_i P) and Amagat’s law are ideal-mixture companions to the ideal gas law. Kay’s rule (pseudocritical T_c,mix ≈ Σ y_i T_c,i, similarly for P_c) is a crude corresponding-states mixture estimate for Z charts. Detailed mixing rules for cubic EOS (kij parameters) are simulator topics—not full UPDA derivations.

UPDA Chemical Exam Workflow

1. Check phase: pure vapor, supercritical, or near dew point?
2. Compare P to P_c and T to T_c (reduced properties if data given).
3. If ideal gas is stated or clearly valid, use PV = nRT and ideal-gas energy relations.
4. If Z or an EOS is given, correct volume/density first; only then do balances.
5. Near VLE, switch mental model to phase equilibrium (Chapter 6), not a single ideal-gas PvT relation for both phases.

Common Traps

• Using °C in PV = nRT instead of K.
• Applying ideal gas to a liquid density estimate.
• Forgetting Z in ρ = PM/(ZRT) when Z is supplied.
• Assuming Z < 1 always (at high P_r, Z often exceeds 1).
• Treating critical temperature as a maximum operating temperature rather than a fluid property fixed for the pure substance.

Section 5.3 builds enthalpy, entropy, and heat capacity on top of these PvT models—ideal-gas Cp/Cv relations first, then awareness that real fluids need departure corrections.