6.1 Volumetric Properties and Compressibility Factor Charts

6.1 Volumetric Properties and Compressibility Factor Charts

Thermodynamic calculations on the NCEES PE Chemical Exam frequently involve process streams operating at elevated pressures and non-ambient temperatures where the ideal gas law ($P v = R T$) fails catastrophically. In high-pressure separators, gas pipelines, ammonia synthesis loops, and supercritical extractors, intermolecular attractive and repulsive forces alter fluid density, storage capacity, and compressor horsepower by $20%$ to over $80%$. To predict fluid behavior without experimental PVT data for every unique process condition, chemical engineers apply the compressibility factor ($Z$), the principle of corresponding states, and generalized virial expansions.

---

1. Real Gas Deviations from the Ideal Gas Law

The ideal gas law assumes two fundamental physical postulates:

1. Gas molecules occupy zero physical volume (point masses).
2. Gas molecules exert no intermolecular forces on one another (no attractive dispersion or electrostatic forces, and no hard-sphere steric repulsion).

In real fluids, the intermolecular potential energy $U(r)$ as a function of intermolecular separation distance $r$ is described by the Lennard-Jones 6-12 potential:

$U(r) = 4\epsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^6 \right]$

Where:

• $\epsilon$ is the depth of the attractive potential energy well.
• $\sigma$ is the collision diameter (distance where $U(r) = 0$).
• The $(1/r)^{12}$ term models steep quantum-mechanical Pauli repulsion at close distances (molecular core volume).
• The $(1/r)^6$ term models attractive London dispersion forces at intermediate distances.

Regimes of Deviation

• At low pressures ($P < 2\text{ to }3\text{ bar}$) and high temperatures ($T \gg T_c$): Molecules are separated by large distances ($r \gg \sigma$). Intermolecular forces are negligible, and the molar volume is large. Real gases behave as ideal gases ($Z \approx 1.00$).
• At moderate pressures ($2\text{ bar} < P < 50\text{ bar}$): Molecules are close enough for attractive van der Waals forces to draw them together. The actual pressure exerted on the container walls is less than ideal, and the actual molar volume is smaller than the ideal gas volume. Consequently, $Z < 1.00$.
• At very high pressures ($P > 100\text{ bar}$): Molecules are forced tightly together. The physical volume occupied by the electron clouds of the molecules themselves becomes a substantial fraction of the total container volume. Repulsive forces dominate, resisting further compression. Consequently, $Z > 1.00$.

---

2. The Compressibility Factor ($Z$)

The compressibility factor ($Z$) is a dimensionless thermodynamic property defined as the ratio of the actual molar volume of a real gas to the molar volume of an ideal gas at the identical temperature and pressure:

$Z \equiv \frac{P v}{R T} = \frac{P V}{n R T} = \frac{v}{v^{ig}}$

Where:

• $P$ = absolute pressure ($\text{kPa}$, $\text{bar}$, or $\text{psia}$).
• $v = V / n$ = actual molar volume ($\text{m}^3/\text{kmol}$, $\text{L/mol}$, or $\text{ft}^3/\text{lbmol}$).
• $T$ = absolute temperature ($\text{K}$ or $^\circ\text{R}$).
• $R$ = universal gas constant.
• $v^{ig} = R T / P$ = ideal gas molar volume.

Crucial Handbook Constants for $R$

On the PE exam, picking the correct numerical value and units for $R$ saves vital time:

• $R = 8.31447\text{ J/(mol}\cdot\text{K)} = 8.31447\text{ kPa}\cdot\text{m}^3/(\text{kmol}\cdot\text{K}) = 8.31447\text{ MPa}\cdot\text{cm}^3/(\text{mol}\cdot\text{K})$
• $R = 0.082057\text{ L}\cdot\text{atm}/(\text{mol}\cdot\text{K}) = 82.057\text{ cm}^3\cdot\text{atm}/(\text{mol}\cdot\text{K})$
• $R = 0.083145\text{ L}\cdot\text{bar}/(\text{mol}\cdot\text{K})$
• $R = 10.7316\text{ psia}\cdot\text{ft}^3/(\text{lbmol}\cdot^\circ\text{R})$
• $R = 1.9872\text{ Btu}/(\text{lbmol}\cdot^\circ\text{R}) = 1.9872\text{ cal}/(\text{mol}\cdot\text{K})$

The real gas equation of state using $Z$ is:

$P V = n Z R T \implies v = \frac{Z R T}{P} \implies \rho = \frac{P \cdot MW}{Z R T}$

Where $\rho$ is mass density and $MW$ is molecular weight.

---

3. The Principle of Corresponding States

Directly measuring experimental $PVT$ relationships for thousands of pure chemical compounds and multi-component mixtures across wide ranges of $T$ and $P$ is impractical. The Theorem of Corresponding States, first postulated by Johannes Diderik van der Waals (1881), provides a powerful unifying framework.

Reduced Properties

Thermodynamic variables are rendered dimensionless by scaling them against the fluid's critical point values:

$T_r = \frac{T}{T_c}, \quad P_r = \frac{P}{P_c}, \quad V_r' = \frac{v}{R T_c / P_c}$

Where:

• $T_c$ = critical temperature (absolute temperature, $\text{K}$ or $^\circ\text{R}$).
• $P_c$ = critical pressure (absolute pressure, $\text{MPa}$, $\text{bar}$, or $\text{psia}$).
• $V_r'$ = ideal reduced molar volume (dimensionless).

Two-Parameter Corresponding States

The two-parameter principle states that all pure fluids at the same reduced temperature ($T_r$) and reduced pressure ($P_r$) have the same compressibility factor ($Z$):

$Z = f(T_r, P_r)$

This principle works well for simple fluids—molecules that are spherical and non-polar, such as the noble gases (argon, krypton, xenon) and methane ($\text{CH}_4$). For these substances, the critical compressibility factor is nearly constant: $Z_c = P_c v_c / (R T_c) \approx 0.29$. However, for asymmetric, elongated, or polar fluids (e.g., water, ammonia, alcohols, long-chain hydrocarbons), the two-parameter correlation exhibits substantial errors ($10%$ to $25%$).

---

4. Pitzer Three-Parameter Corresponding States & The Acentric Factor

To correct for molecular non-sphericity and polarity, Kenneth Pitzer (1955) introduced a third corresponding states parameter: the Pitzer acentric factor ($\omega$).

Definition of the Acentric Factor

Pitzer observed that for simple noble gases, the reduced vapor pressure at a reduced temperature of $T_r = 0.70$ is approximately $P_r^{sat} = P^{sat}/P_c \approx 0.10$. Since $\log_{10}(0.10) = -1.00$, Pitzer defined $\omega$ to quantify the deviation of a fluid's vapor pressure from this spherical baseline:

$\omega \equiv -1.000 - \log_{10}\left[ \frac{P^{sat}(T_r = 0.70)}{P_c} \right]$

• Physical Significance: Non-spherical molecules (e.g., n-hexane, decane) have higher surface contact areas and stronger intermolecular forces that suppress vapor pressure at $T_r = 0.70$. Thus, $P^{sat}(T_r=0.70)/P_c < 0.10$, making $\log_{10}[P_r^{sat}] < -1.00$, which yields $\omega > 0$.
• For spherical molecules (argon, krypton, methane): $\omega \approx 0.000$ to $0.011$.
• For elongated alkanes: $\omega$ increases with chain length (ethane = $0.099$, propane = $0.152$, n-butane = $0.200$, n-octane = $0.400$).
• For polar fluids: hydrogen bonding significantly increases $\omega$ (water = $0.344$, ammonia = $0.253$, ethanol = $0.645$).

The Lee-Kesler Generalized Compressibility Expansion

Using Pitzer's acentric factor, the compressibility factor is formulated as a linear Taylor expansion around a simple fluid reference:

$Z = Z^{(0)}(T_r, P_r) + \omega Z^{(1)}(T_r, P_r)$

Where:

• $Z^{(0)}(T_r, P_r)$ is the compressibility factor of a simple fluid (with $\omega = 0$, based on argon/methane).
• $Z^{(1)}(T_r, P_r)$ is the deviation or correction factor representing the change in $Z$ with molecular acentricity.
• Both $Z^{(0)}$ and $Z^{(1)}$ are tabulated in standard NCEES PE Chemical Reference Handbook tables (the Lee-Kesler tables) as functions of $T_r$ and $P_r$.

---

5. Virial Equations of State and Low-Pressure Correlations

The virial equation of state is the only equation of state with a rigorous foundation in statistical mechanics. It expresses the compressibility factor as an infinite power series in molar density ($1/v$) or pressure ($P$):

$Z = 1 + \frac{B(T)}{v} + \frac{C(T)}{v^2} + \frac{D(T)}{v^3} + \dots$

$Z = 1 + B'(T) P + C'(T) P^2 + D'(T) P^3 + \dots$

Where:

• $B(T)$ is the second virial coefficient, accounting for two-body intermolecular collisions and pairwise interactions.
• $C(T)$ is the third virial coefficient, accounting for three-body collisions.
• The coefficients relate via $B' = B / (R T)$ and $C' = (C - B^2) / (R T)^2$.

Truncated Virial Equations

For practical engineering applications at low to moderate pressures:

1. Low-Pressure Truncation (Up to $\sim 15\text{ bar}$ or $P_r < 0.5$): $Z = 1 + \frac{B P}{R T}$
2. Moderate-Pressure Truncation (Up to $\sim 50\text{ bar}$): $Z = 1 + \frac{B}{v} + \frac{C}{v^2}$

Generalized Second Virial Coefficient (Pitzer-Abbott Correlation)

Pitzer and Abbott developed generalized dimensionless correlations for the second virial coefficient as a function of reduced temperature:

$\frac{B P_c}{R T_c} = B^{(0)}(T_r) + \omega B^{(1)}(T_r)$

Where the universal functions $B^{(0)}$ and $B^{(1)}$ are given by:

$B^{(0)}(T_r) = 0.083 - \frac{0.422}{T_r^{1.6}}$

$B^{(1)}(T_r) = 0.139 - \frac{0.172}{T_r^{4.2}}$

Once $B P_c / (R T_c)$ is calculated, the second virial coefficient is obtained from:

$B = \frac{R T_c}{P_c} \left[ B^{(0)}(T_r) + \omega B^{(1)}(T_r) \right]$

And the compressibility factor at pressure $P$ is:

$Z = 1 + \left( \frac{B P_c}{R T_c} \right) \frac{P_r}{T_r} = 1 + \frac{B P}{R T}$

The Boyle Temperature ($T_B$)

The temperature at which the second virial coefficient equals zero ($B(T_B) = 0$) is called the Boyle Temperature. At $T = T_B$, the initial slope of $Z$ versus $P$ is zero ($(\partial Z / \partial P)_{T, P=0} = 0$). At this exact temperature, attractive and repulsive forces balance out, and the real gas exhibits near-ideal behavior ($Z \approx 1.0$) up to substantial pressures ($10-30\text{ bar}$). For simple fluids, $B^{(0)}(T_r) = 0$ at $T_r = (0.422 / 0.083)^{1/1.6} = (5.084)^{0.625} \approx 2.76$.

---

6. Gas Mixtures & Kay's Pseudocritical Method

Process streams in refineries, gas plants, and petrochemical units are almost always multi-component mixtures. To use generalized $Z$-charts without solving complex multi-variable equations of state, W. B. Kay (1936) introduced Kay's Rule.

Kay's Pseudocritical Properties

Kay's rule assumes that a gas mixture behaves as a hypothetical pure fluid whose critical properties are the mole-fraction-weighted averages of the pure-component critical constants:

$T_{pc} = \sum_{i=1}^C y_i T_{c,i}$

$P_{pc} = \sum_{i=1}^C y_i P_{c,i}$

$\omega_m = \sum_{i=1}^C y_i \omega_i$

Where $y_i$ is the mole fraction of component $i$ in the gas phase.

Pseudoreduced Properties

The mixture pseudoreduced temperature and pseudoreduced pressure are evaluated as:

$T_{pr} = \frac{T}{T_{pc}}, \quad P_{pr} = \frac{P}{P_{pc}}$

Using $T_{pr}$, $P_{pr}$, and $\omega_m$, the mixture compressibility factor $Z_m$ is determined directly from the Lee-Kesler tables or Pitzer virial correlation:

$Z_m = Z^{(0)}(T_{pr}, P_{pr}) + \omega_m Z^{(1)}(T_{pr}, P_{pr})$

$\rho_{mix} = \frac{P \cdot MW_{mix}}{Z_m R T}, \quad \text{where } MW_{mix} = \sum_{i=1}^C y_i MW_i$

[!NOTE] Accuracy of Kay's Rule: Kay's rule is reliable when the chemical components in the mixture have similar chemical structures and molecular weights (e.g., mixtures of light hydrocarbons such as methane, ethane, propane, and butane). Errors are typically under $3-5%$. However, for mixtures containing highly dissimilar components (e.g., hydrogen mixed with heavy hydrocarbons, or carbon dioxide and water), Kay's rule can introduce errors exceeding $15%$. In those cases, cubic equations of state with binary interaction parameters ($k_{ij}$) are mandatory.

---

7. Summary Table: Volumetric Property Estimation Methods

---

8. Comprehensive Worked Numerical Example: High-Pressure Natural Gas Storage Drum Sizing

Problem Statement

A natural gas processing facility must design an emergency high-pressure gas storage vessel to hold $10,000\text{ kg}$ of treated pipeline natural gas at an operating pressure of $P = 6.00\text{ MPa}$ ($60.0\text{ bar}$) and a storage temperature of $T = 300.0\text{ K}$ ($26.85^\circ\text{C}$).

Gas Composition and Component Properties:

• Methane ($\text{CH}_4$, Component 1): $y_1 = 0.850$, $MW_1 = 16.043\text{ kg/kmol}$, $T_{c1} = 190.56\text{ K}$, $P_{c1} = 4.599\text{ MPa}$, $\omega_1 = 0.011$
• Ethane ($\text{C}_2\text{H}_6$, Component 2): $y_2 = 0.100$, $MW_2 = 30.069\text{ kg/kmol}$, $T_{c2} = 305.32\text{ K}$, $P_{c2} = 4.872\text{ MPa}$, $\omega_2 = 0.099$
• Propane ($\text{C}_3\text{H}_8$, Component 3): $y_3 = 0.050$, $MW_3 = 44.096\text{ kg/kmol}$, $T_{c3} = 369.83\text{ K}$, $P_{c3} = 4.248\text{ MPa}$, $\omega_3 = 0.152$

Calculate:

1. The apparent molecular weight ($MW_{mix}$) and total molar quantity of gas ($n$) to be stored.
2. The pseudocritical temperature ($T_{pc}$), pseudocritical pressure ($P_{pc}$), and pseudocritical acentric factor ($\omega_m$) using Kay's rule.
3. The pseudoreduced temperature ($T_{pr}$) and pseudoreduced pressure ($P_{pr}$).
4. The generalized second virial coefficient terms $B^{(0)}(T_{pr})$, $B^{(1)}(T_{pr})$, and the compressibility factor $Z_m$.
5. The required physical vessel volume ($V$) in $\text{m}^3$.
6. The volume that would have been predicted if the design engineer had incorrectly assumed ideal gas behavior, and the percentage error in vessel sizing.

---

Step 1: Mixture Molecular Weight & Molar Inventory

Apparent molecular weight: $MW_{mix} = \sum_{i=1}^3 y_i MW_i = (0.850 \times 16.043) + (0.100 \times 30.069) + (0.050 \times 44.096)$ $MW_{mix} = 13.6366 + 3.0069 + 2.2048 = \mathbf{18.848\text{ kg/kmol}}$

Total molar inventory for $10,000\text{ kg}$: $n = \frac{m}{MW_{mix}} = \frac{10,000\text{ kg}}{18.8483\text{ kg/kmol}} = 530.552\text{ kmol} = \mathbf{530,552\text{ mol}}$

---

Step 2: Kay's Pseudocritical Properties

Apply Kay's linear mixing rules: $T_{pc} = \sum_{i=1}^3 y_i T_{c,i} = (0.850 \times 190.56) + (0.100 \times 305.32) + (0.050 \times 369.83)$ $T_{pc} = 161.976 + 30.532 + 18.492 = \mathbf{211.00\text{ K}}$

$P_{pc} = \sum_{i=1}^3 y_i P_{c,i} = (0.850 \times 4.599) + (0.100 \times 4.872) + (0.050 \times 4.248)$ $P_{pc} = 3.9092 + 0.4872 + 0.2124 = \mathbf{4.609\text{ MPa}}$

$\omega_m = \sum_{i=1}^3 y_i \omega_i = (0.850 \times 0.011) + (0.100 \times 0.099) + (0.050 \times 0.152)$ $\omega_m = 0.00935 + 0.00990 + 0.00760 = \mathbf{0.02685}$

---

Step 3: Pseudoreduced Properties

$T_{pr} = \frac{T}{T_{pc}} = \frac{300.0\text{ K}}{211.00\text{ K}} = \mathbf{1.4218}$

$P_{pr} = \frac{P}{P_{pc}} = \frac{6.00\text{ MPa}}{4.609\text{ MPa}} = \mathbf{1.3018}$

---

Step 4: Generalized Compressibility Factor Calculation

Evaluate Pitzer-Abbott generalized second virial functions at $T_{pr} = 1.4218$: $B^{(0)}(T_{pr}) = 0.083 - \frac{0.422}{(1.4218)^{1.6}} = 0.083 - \frac{0.422}{1.7584} = 0.083 - 0.2400 = \mathbf{-0.1570}$

$B^{(1)}(T_{pr}) = 0.139 - \frac{0.172}{(1.4218)^{4.2}} = 0.139 - \frac{0.172}{4.3985} = 0.139 - 0.0391 = \mathbf{+0.0999}$

Dimensionless second virial coefficient: $\frac{B P_{pc}}{R T_{pc}} = B^{(0)}(T_{pr}) + \omega_m B^{(1)}(T_{pr}) = -0.1570 + (0.02685 \times 0.0999) = -0.1570 + 0.00268 = \mathbf{-0.15432}$

Calculate compressibility factor $Z_m$: $Z_m = 1 + \left( \frac{B P_{pc}}{R T_{pc}} \right) \frac{P_{pr}}{T_{pr}} = 1 + (-0.15432) \times \frac{1.3018}{1.4218} = 1 - 0.14130 = \mathbf{0.8587} \approx \mathbf{0.859}$

(Cross-check with Lee-Kesler tables: at $T_r = 1.42, P_r = 1.30$, tabulated $Z^{(0)} = 0.856$ and $Z^{(1)} = 0.110$, giving $Z = 0.856 + 0.0269(0.110) = 0.859$. Both methods show exact agreement).

Because $Z_m = 0.859 < 1.00$, attractive forces between gas molecules pull them closer together, reducing the gas volume to $85.9%$ of its ideal volume.

---

Step 5: Vessel Volume Determination

Using the real gas law ($P V = n Z R T$): $V = \frac{n Z_m R T}{P}$ $V = \frac{(530,552\text{ mol}) \times (0.8587) \times (8.31447\text{ J/(mol}\cdot\text{K)}) \times (300.0\text{ K})}{6.00 \times 10^6\text{ Pa}}$ $V = \frac{1,136,530,000\text{ J}}{6,000,000\text{ N/m}^2} = \mathbf{189.42\text{ m}^3}$

---

Step 6: Ideal Gas Error Comparison

If the engineer had used the ideal gas law ($Z = 1.00$): $V_{ideal} = \frac{n R T}{P} = \frac{(530,552) \times (8.31447) \times (300.0)}{6.00 \times 10^6} = \mathbf{220.59\text{ m}^3}$

$\text{Sizing Overestimation} = \frac{V_{ideal} - V}{V} \times 100\% = \frac{220.59 - 189.42}{189.42} \times 100\% = \mathbf{+16.45\%}$

Designing the pressure vessel for $220.6\text{ m}^3$ instead of $189.4\text{ m}^3$ represents a wasted capacity of $31.2\text{ m}^3$. For a vessel rated at $6.0\text{ MPa}$ ($870\text{ psig}$), this error translates to tens of thousands of pounds of unnecessary high-strength alloy steel fabrication costs.

---

9. Critical PE Exam Traps & Pitfalls

Trap 1: Forgetting to Convert to Absolute Temperature and Pressure
Never evaluate reduced properties using Celsius ($^\circ\text{C}$) or Fahrenheit ($^\circ\text{F}$), or gauge pressure ($\text{psig}$, $\text{barg}$). $T_r = (T[^\circ\text{C}] + 273.15) / T_c[\text{K}]$ and $P_r = (P[\text{psig}] + 14.696) / P_c[\text{psia}]$. A calculation performed with relative temperatures will not match any physical behavior and yields nonsense answers.

Trap 2: Misusing the Truncated Virial Equation at High Reduced Pressures
The truncated virial equation $Z = 1 + B P / (R T)$ is strictly valid only at low to moderate pressures where $P_r < 0.5$. Attempting to use this formula at $P_r = 5.0$ or near the critical point can produce negative compressibility factors or severe underpredictions because it omits higher-order molecular clusters ($C, D$). For $P_r > 0.5$, use Lee-Kesler charts or cubic equations of state.

Trap 3: Confusing Mole Fraction vs. Mass Fraction in Kay's Rule
Kay's pseudocritical rules require mole fractions ($y_i$), not weight/mass fractions ($w_i$). If the exam prompt supplies stream composition in weight percent, you must convert to mole fractions ($y_i = (w_i / MW_i) / \sum (w_j / MW_j)$) before computing $T_{pc}$ and $P_{pc}$.