The Compressibility Factor Z

Use Z to measure and correct real-gas deviation from PV = nRT using reduced properties and generalized charts.

Material & Energy BalancesChemical Engineering Year 1Free preview
⏱️ About 16 min

PV = nRT is almost right — so how do you apply a clean correction without solving a complicated EOS every time?

💡
The big idea: The compressibility factor Z corrects ideal-gas calculations: V_real = Z·(nRT/P). Z depends primarily on reduced pressure and temperature for many gases.
🎯 By the end, you'll be able to
  • Define Z and interpret Z relative to 1
  • Compute reduced pressure P_r and reduced temperature T_r
  • Describe the generalized compressibility chart conceptually
  • Compute real-gas volume using a provided Z value
📎 Helpful to know first

Definition: Compressibility Factor

The compressibility factor Z measures how much a real gas deviates from the ideal gas law.

Rearranging gives a very useful correction form: V = Z(nRT/P).

\[ Z \;=\; \frac{PV}{nRT} \;=\; \frac{P\bar{v}}{RT} \]
Z = 1 for an ideal gas. Real gases have Z ≠ 1 depending on conditions and molecular interactions.

Interpreting Z (What It Means Physically)

  • Z = 1: ideal-gas behavior
  • Z < 1: attractions dominate (gas is more compressible than ideal; smaller volume than ideal prediction)
  • Z > 1: repulsions/excluded volume dominate (gas is less compressible than ideal; larger volume than ideal prediction)

Both Z < 1 and Z > 1 occur in real systems.

\[ P_r = \frac{P}{P_c}, \qquad T_r = \frac{T}{T_c} \]
Reduced pressure and reduced temperature used in corresponding-states correlations and generalized compressibility charts.

Principle of Corresponding States (Why Charts Work)

The principle of corresponding states says that many gases show similar behavior at the same reduced conditions (same Pr and Tr). Engineers exploit this by using:

  • a generalized compressibility chart that plots Z versus Pr for several Tr curves, or
  • equations/correlations that approximate those charts.

In practice, you compute Pr, Tr, then read/estimate Z, then compute V.

⚠️ Pitfall: Z is not always less than 1

A common misconception is “real gases always have Z < 1.” Not true: at high Pr (or sufficiently high density), repulsions can dominate and Z often becomes > 1.

📝 Worked example: A real gas is in a vessel at P = 50.0 bar and T = 450 K. Suppose a generalized compressibility chart gives Z = 0.85 at these conditions. Compute the molar volume v (L/mol). Use R = 0.08314 L·bar/(mol·K).
  1. For a real gas: v = Z (RT/P).
  2. Compute RT: (0.08314 L·bar/(mol·K))(450 K) = 37.413 L·bar/mol
  3. Compute RT/P: 37.413 / 50.0 = 0.74826 L/mol
  4. Apply Z: v = 0.85(0.74826) = 0.63602 L/mol
✓ v_real = 0.636 L/mol (with Z = 0.85 at 50.0 bar, 450 K)
✏️ Practice: Methane at P = 100 bar and T = 400 K has Z = 1.10 (assume this Z is provided). Compute the molar volume v (L/mol). Use R = 0.08314 L·bar/(mol·K).
L/mol
Solution
  1. Use v = ZRT/P.
  2. RT = (0.08314)(400) = 33.256 L·bar/mol
  3. RT/P = 33.256 / 100 = 0.33256 L/mol
  4. v = 1.10(0.33256) = 0.365816 L/mol

Check your understanding

1. The compressibility factor Z is defined as:
By definition, Z = PV/(nRT).
2. If Z = 0.80 at fixed P and T, then compared to the ideal-gas prediction, the real-gas molar volume is:
v_real = Z v_ideal, so Z < 1 means v_real is smaller than v_ideal at the same P and T.
🔎 Related lessons
The van der Waals Equation · soon
✅ Key takeaways
  • Z = PV/(nRT) measures deviation from ideality (Z = 1 ideal gas)
  • Real-gas volume correction: V = Z(nRT/P)
  • Reduced properties: P_r = P/Pc and T_r = T/Tc support corresponding-states charts/correlations
  • Z can be less than 1 or greater than 1 depending on which interactions dominate
➡️ Z is a compact correction, but you often want a direct equation of state you can compute from molecular constants or critical properties. Next: the classic cubic EOS — van der Waals.
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