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The compressibility factor Z measures how far a real gas molar volume at a given pressure and temperature deviates from the ideal-gas model.
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Formula
Z dimensionless · p in Pa · Vm in m3/mol · R = 8.314 Pa·m3/(mol·K) · T in K
| Symbol | Meaning | Unit |
|---|---|---|
| Compressibility factor; Z=1 for an ideal gas | dimensionless | |
| Absolute gas pressure | Pa | |
| Molar volume of the gas | m3/mol | |
| Universal gas constant (8.314) | Pa·m3/(mol·K) | |
| Absolute temperature | K |
Z = \frac{pV_m}{RT}Z=pV_m/(RT) compares a real gas state with the ideal-gas equation. Z=1 indicates ideal behavior; Z<1 means attractive interactions reduce measured pV_m relative to RT, while Z>1 can occur when excluded-volume and repulsive effects dominate. Z depends on pressure, temperature and substance. Real gases approach Z=1 at low pressure; near condensation or at high pressure, deviations can become substantial. Z is a state measure, not a complete equation of state. Predicting other conditions requires substance data or a real-gas model.
Z=pV_m/(RT) applies to a measured gas state; Z=1 exactly for an ideal gas and real gases approach it at low pressure.
Compare measured pV_m with the ideal reference RT.
Molar volume with real-gas correction
Z must be known or measured for the substance, pressure and temperature.
p=10 bar, Vm=2.45 L/mol, T=300 K. Find Z.
Z=(1.00×106·2.45×10−3)/(8.314·300)=2450/2494≈0.982.
What is Z for an ideal gas?
From pV_m=RT, Z=1.
Combining pressure in bar with SI gas constant units.
Use Pa and m3/mol or another consistent set of units.
Predicting another temperature from Z alone.
Z depends on state and substance; an equation of state or data table is needed.
Formula cluster
Z quantifies deviation but does not replace a real-gas equation of state.
A measured state has p=10 bar=1.00×106 Pa, Vm=2.45 L/mol=2.45×10−3 m3/mol and T=300 K. Then Z=(1.00×106·2.45×10−3)/(8.314·300)=2450/2494≈0.982. Check: Z is close to 1, so the gas is approximately ideal under these conditions.
Real-gas corrections, gas liquefaction, engineering thermodynamics and checking whether the ideal-gas equation is adequate
Here is how to work through a typical Compressibility Factor for Real Gases (Z = pVₘ/(RT)) task step by step:
Task
p=10 bar, Vm=2.45 L/mol, T=300 K. Find Z.
Solution path
Z=(1.00×106·2.45×10−3)/(8.314·300)=2450/2494≈0.982.
Task
What is Z for an ideal gas?
Solution path
From pV_m=RT, Z=1.
Z=pV_m/(RT) compares a real gas’s molar volume with the ideal-gas value at the same pressure and temperature. For an ideal gas, Z=1. A real value below or above one indicates how molecular interactions and finite volume depart from the ideal approximation. Z is dimensionless and depends on substance, pressure and temperature. It corrects the equation of state but is not a universal constant. State the conditions and source of the Z value used. Report substance, pressure and temperature with Z. The factor corrects only the stated condition; for different conditions it must be determined again or calculated from a suitable real-gas model.
For an ideal gas, set Z=1 and use pV=nRT. This is often reasonable at moderate pressure and far from condensation. If data, an equation of state or tables require a Z value other than one, non-ideality is included. For example, at p=10 bar, Vm=2.45 L/mol and T=300 K, Z≈(10·2.45)/(0.08314·300)=0.982. This small departure is a modest correction; near critical conditions, deviations can be much larger. Report substance, pressure and temperature with Z. The factor corrects only the stated condition; for different conditions it must be determined again or calculated from a suitable real-gas model.
Z<1 means the real molar volume at the stated conditions is smaller than the ideal-gas prediction; attractive interactions often dominate in this region. Z>1 means a larger-than-ideal volume, commonly favoured at high densities by repulsion and molecular size. This interpretation is qualitative and depends on state and substance; the sign alone does not identify every microscopic contribution. Values can vary with pressure and may cross Z=1. Report substance, pressure and temperature with Z. The factor corrects only the stated condition; for different conditions it must be determined again or calculated from a suitable real-gas model.
Use the real-gas form pV=ZnRT instead of pV=nRT. If Z is known, amount can be calculated as n=pV/(ZRT). Substitution is appropriate only when Z matches the same temperature, pressure and composition; sometimes Z must be found iteratively from an equation of state. Also check consistent units for R, p, V and T. If Z is close to one, the ideal approximation may suffice, but required precision should decide. Report substance, pressure and temperature with Z. The factor corrects only the stated condition; for different conditions it must be determined again or calculated from a suitable real-gas model.
No. Z is a thermodynamic state factor defined as pV_m/(RT). A compression ratio often describes a ratio of volumes or pressures in engineering processes. Both can be dimensionless, but their definitions and meanings differ. The letter Z can also denote other quantities in other fields. Read the actual formula and unit convention instead of inferring from the symbol alone. In a gas equation of state, Z describes departure from ideal behaviour. Report substance, pressure and temperature with Z. The factor corrects only the stated condition; for different conditions it must be determined again or calculated from a suitable real-gas model.
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