Real Gases vs. Ideal Gases: Limitations of the Ideal Gas Model

1. The Ideal Gas: A Model Built on Assumptions

The ideal gas law is a powerful simplification that stems from the Kinetic Molecular Theory (KMT). KMT describes a gas as a large number of submicroscopic particles (atoms or molecules) in constant, rapid, random motion. The properties we observe, like pressure and temperature, are just the macroscopic consequences of these microscopic actions.

To see how these ideas build up to the ideal gas law and, more importantly, what simplifications are made along the way, let's watch a short video.

From the video and the principles of KMT, we can distill the ideal gas model down to two critical, and ultimately incorrect, assumptions about gas molecules:

1. The volume of gas molecules is negligible compared to the total volume of the container.
2. There are no intermolecular forces of attraction or repulsion between gas molecules.

While other assumptions exist (like perfectly elastic collisions), these two are the primary culprits behind the deviation of real gases from ideal behavior.

2. When the Model Breaks Down: Causes of Deviation

Real gases, unlike their ideal counterparts, consist of molecules that do have a finite size and do attract each other. The ideal gas law works well only when conditions are such that we can safely ignore these two facts. So, when can't we ignore them?

This is a central concept for your JEE preparation. The following reading from a study material for IIT JEE explains this very clearly.

Let's break down the two failure conditions based on what you just read.

Failure Condition 1: High Pressure

At high pressures, molecules are forced very close together.

• Impact on the 'Negligible Volume' Assumption: The total volume of the container becomes small, so the volume occupied by the molecules themselves is no longer a negligible fraction of the total. The "free space" for molecules to move in is significantly less than the container's volume, $V$. This effect tends to make the gas less compressible than an ideal gas.

Failure Condition 2: Low Temperature

At low temperatures, molecules move much more slowly.

• Impact on the 'No Intermolecular Forces' Assumption: Slower-moving molecules spend more time near each other during fly-bys. This gives the weak intermolecular attractive forces (like van der Waals forces) a chance to have an effect. These attractions pull molecules together, slightly reducing the force of their collisions with the container walls. This leads to a measured pressure that is lower than what the ideal gas law would predict. This effect makes the gas more compressible than an ideal gas.

In summary:

• High Pressure: The finite volume of molecules becomes significant.
• Low Temperature & High Pressure: Intermolecular forces become significant.

Therefore, real gases deviate most from ideal behavior at high pressures and low temperatures—precisely the conditions under which gases are close to liquefaction. Conversely, a real gas behaves most like an ideal gas at low pressures and high temperatures.

3. Quantifying Deviation: The Compressibility Factor (Z)

To measure the extent of deviation, we use a correction factor called the compressibility factor, Z. It's defined as:

$Z=\frac{PV}{nRT}$

For an ideal gas, $PV=nRT$, so $Z$ is always equal to 1. For a real gas, $Z$ can be greater or less than 1.

• When Z < 1: $PV<nRT$. The gas is more compressible than an ideal gas. This happens at moderate pressures where intermolecular attractive forces dominate. They pull the molecules together, reducing the volume (or pressure) compared to the ideal case.
• When Z > 1: $PV>nRT$. The gas is less compressible than an ideal gas. This happens at very high pressures where the finite molecular volume effect dominates. The "repulsive" effect of particles taking up space outweighs the attractive forces.

The following graph illustrates this behavior for several real gases.

Conclusion and Preview

Today, we've gone beyond simply using the ideal gas law and have critically examined its foundations. You can now identify the specific assumptions from the Kinetic Theory of Gases that limit the ideal gas model and explain when and why these limitations become important for real gases.

Key Takeaways:

• The ideal gas model is based on two main simplifying assumptions: gas molecules have negligible volume and no intermolecular forces.
• These assumptions fail under conditions of high pressure (when molecular volume becomes significant) and low temperature (when intermolecular forces have a greater effect).
• The deviation of a real gas from ideal behavior is quantified by the compressibility factor, Z.
  • $Z<1$ implies attractive forces are dominant.
  • $Z>1$ implies repulsive forces/finite molecular volume is dominant.

This entire discussion naturally leads to a question: Can we create a better equation of state that accounts for these real-world effects? The answer is yes.

Preview of the Next Lesson:
In our next lesson, we will be introduced to the van der Waals equation. This is a modification of the ideal gas law that introduces specific correction terms to account for both finite molecular volume and intermolecular forces. We will learn how these correction terms, a and b, relate directly to the limitations we've discussed today. To get a head start, you can watch the beginning of this Khan Academy video, which elegantly derives the logic for these corrections.