Compressibility Factor ($Z$)
Measuring how much a Real Gas deviates from Ideal behavior.
The Ideal Gas Equation ($PV = nRT$) assumes gas molecules have no volume and no intermolecular forces. In reality, gases do have volume and experience attractive/repulsive forces. To measure the extent to which a real gas deviates from ideal behavior, we introduce the Compressibility Factor ($Z$).
1. Mathematical Definition of $Z$
The compressibility factor is defined as the ratio of the actual molar volume of a gas to the molar volume of an ideal gas at the same temperature and pressure.
Because $V_{ideal} = \frac{nRT}{P}$, we can also write this ratio as:
2. The Three Cases of $Z$
Case 1: $Z = 1$ (Ideal Gas)
- The gas behaves exactly like an ideal gas.
- $V_{real} = V_{ideal}$.
- Intermolecular forces are negligible.
Case 2: $Z \gt 1$ (Positive Deviation)
- $V_{real} \gt V_{ideal}$.
- Repulsive forces dominate. The molecules push each other apart, making the gas occupy a larger volume than expected.
- The gas is less compressible (harder to compress) than an ideal gas.
- Occurs mostly at high pressures where molecules are squeezed close together.
Case 3: $Z \lt 1$ (Negative Deviation)
- $V_{real} \lt V_{ideal}$.
- Attractive forces dominate. The molecules pull each other closer, making the gas occupy a smaller volume than expected.
- The gas is more compressible (easier to compress) than an ideal gas.
- Occurs typically at moderate/intermediate pressures.
3. The Graph of $Z$ vs Pressure
Plotting $Z$ against Pressure ($P$) reveals the characteristic behavior of different gases. This graph is heavily tested in exams.
Notice: At very low pressures ($P \to 0$), all curves converge at $Z=1$. Real gases behave ideally at very low pressures!
4. The Boyle Temperature ($T_b$)
Because deviations from ideal behavior depend heavily on temperature, there is a specific 'sweet spot' for every real gas.
- Boyle Temperature ($T_b$): The specific temperature at which a real gas obeys ideal gas laws ($Z \approx 1$) over an appreciable range of pressure.
- Below $T_b$: The gas shows a negative deviation ($Z < 1$) first, then positive ($Z > 1$).
- Above $T_b$: The gas immediately shows a positive deviation ($Z > 1$). This explains why $H_2$ and $He$ always show $Z>1$ at room temp—their Boyle temperatures are extremely low!
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