Q.A rigid, thermally isolated container is divided into two chambers by a partition. The left chamber has volume $V_1 = 2.0$ litre and contains $\mu_1 = 4.0$ moles of a gas at pressure $p_1 = 1.00$ atm; the right chamber has volume $V_2 = 3.0$ litre and contains $\mu_2 = 5.0$ moles of the same gas at pressure $p_2 = 2.00$ atm. The partition is removed and the gas is allowed to mix and reach equilibrium. Because the container is isolated, the total internal energy of the gas is conserved. Calculate the pressure of the mixture after equilibrium is reached.

The Ideal Gas Law: From Intuition to Equation

Imagine you're blowing up a balloon. You feel the resistance as you push more air in. The balloon gets tighter, harder to squeeze. Now imagine leaving that balloon in a hot car — it might even pop. Or take it to the top of a mountain, and it suddenly looks half-deflated.

These everyday experiences are telling you something deep about gases: their pressure, volume, temperature, and the amount of gas inside are all connected. The Ideal Gas Law is the single equation that captures that connection.

The Four Players

Every gas has four measurable properties:

• Pressure ($P$) — how hard the gas pushes on its container (like the tightness of the balloon)
• Volume ($V$) — how much space the gas occupies (the size of the balloon)
• Temperature ($T$) — how hot the gas is (measured in Kelvin, not Celsius)
• Amount ($n$) — how many gas particles are present (measured in moles)

The Ideal Gas Law says: if you know any three of these, you can calculate the fourth. It's the master relationship.

The Precise Statement

Where $R$ is the universal gas constant. Its value depends on the units you use, but the most common one for exams is:

This means: if pressure is in atmospheres (atm), volume in litres (L), amount in moles (mol), and temperature in Kelvin (K), then $R = 0.0821$.

Why It Makes Physical Sense

The equation $PV = nRT$ isn't just a random formula — it's a compact summary of three simpler laws that were discovered earlier:

1. Boyle's Law (pressure-volume relationship): At constant $n$ and $T$, $P \propto 1/V$. Squeeze a gas into half the volume, pressure doubles.
2. Charles's Law (volume-temperature relationship): At constant $n$ and $P$, $V \propto T$. Heat a gas, it expands.
3. Avogadro's Law (amount-volume relationship): At constant $P$ and $T$, $V \propto n$. More gas particles need more space.

The Ideal Gas Law combines all three into one clean statement.

What "Ideal" Means

Real gases don't always follow this law perfectly. At very high pressures or very low temperatures, gas particles start interacting with each other and taking up significant space themselves. The "ideal" gas is a simplified model where:

• Particles have negligible volume
• No forces act between particles (except during collisions)
• Collisions are perfectly elastic …