Ideal Gas Law for Crowds at High Pressure

  What is the Ideal Gas Law?

The ideal gas law is a simple equation that describes how gases behave:

$$PV = nRT$$

• P = Pressure

• V = Volume

• n = Number of moles (amount of gas)

• R = Universal gas constant

• T = Temperature (in Kelvin)

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It works well for low-pressure, high-temperature gases where the particles are far apart and don’t interact much.

What happens at high pressure?

At high pressure, gas particles are squeezed closer together. Two important effects appear:

• Finite volume of particles: Real particles occupy space, so they can’t be compressed indefinitely.

• Intermolecular forces: Particles attract or repel each other, which the ideal gas law ignores.

These effects cause deviations from the ideal gas law. The gas might compress less than expected.

 How is this corrected?

The Van der Waals equation modifies the ideal gas law to account for high pressure (and low temperature):

$$\Bigg(P + \frac{an^2}{V^2}\Bigg) (V - nb) = nRT$$

• The $a$ term adjusts for intermolecular attractions.

• The $b$ term corrects for the finite volume of gas particles.

 Analogy for crowds
You might be using “crowds” as an analogy:

• At low pressure (like an empty room), people can move freely.

• At high pressure (a packed concert), people are squeezed together — they can’t overlap (finite volume) and they interact (pushing or pulling to find space).

So, the ideal gas law breaks down for crowded gases — just as your assumptions break down for a crowded room.

✅ Key point:
The ideal gas law is a good first approximation, but for high pressures (and low temperatures) real gas behavior must be considered — using corrections like Van der Waals.

The Ideal Gas Law — A Deeper Intro

The ideal gas law, $PV = nRT$, is a cornerstone of classical thermodynamics. It combines three empirical gas laws:

• Boyle’s Law: $P \propto \frac{1}{V}$ (at constant T)

• Charles’s Law: $V \propto T$ (at constant P)

• Avogadro’s Law: $V \propto n$ (at constant P, T)

  
  
  

It assumes:

• Gas molecules have negligible volume.

• There are no forces between gas particles except elastic collisions.

• Collisions with the container walls cause pressure.

This works well for ideal conditions: low pressure and high temperature — where particles are far apart and move freely.

📌 The “Crowd” Analogy

Imagine a gas as a crowd of people in a large hall:

• At low density (low pressure), people (molecules) move freely with plenty of space.

• They hardly bump into each other — no significant interactions.

• The hall (container) feels mostly empty.

This is the ideal gas situation: lots of empty space, no restrictions.

📌 What Changes at High Pressure?

When you compress a gas:

• The “people” get closer — the hall gets crowded.

• They can’t overlap — each person takes up space (finite volume).

• They might push or pull each other to get comfortable (intermolecular forces).

In real gases:

• These volumes and forces matter.

• At high pressure, the volume occupied by gas particles is significant.

• Attractive forces between particles can lower the effective pressure (particles pull each other inward).

📌 Why Does the Ideal Gas Law Fail Here?

Because it ignores reality:

• It assumes point particles — no size.

• It assumes no attractions or repulsions.

When gases are dense (high P) or cold (low T), these assumptions fail.
So, real gases deviate:

• They compress less than predicted.

• They liquefy if cooled enough.

📌 How Do We Correct for Crowds?

We use real gas equations like Van der Waals:

$$\Bigg( P + \frac{an^2}{V^2} \Bigg) (V - nb) = nRT.$$

• $b$ = volume correction (size of particles)

• $a$ = pressure correction (intermolecular attractions)