Q.If $c$ is r.m.s. speed of molecules in a gas and $v$ is the speed of sound waves in the gas, show that $c/v$ is constant and independent of temperature for all diatomic gases.

Kinetic Theory of Gases

Imagine you're sitting in a quiet room. The air around you feels still — but it isn't. Every second, billions of tiny particles (molecules of nitrogen, oxygen, and others) are zipping past you at hundreds of metres per second. They're constantly crashing into each other and into the walls, your skin, the furniture. You don't feel each individual hit because the molecules are so small and the collisions happen so fast. But collectively, those countless tiny impacts produce something you do feel: pressure.

That's the core intuition behind the kinetic theory of gases. It says: all the macroscopic properties of a gas — pressure, temperature, volume — can be explained by the motion of its molecules.

The Big Idea

Instead of treating a gas as a continuous, smooth substance (like a fluid), the kinetic theory treats it as a swarm of tiny, hard, perfectly elastic balls in constant, random motion. "Perfectly elastic" means that when two molecules collide, no kinetic energy is lost — they bounce off each other like ideal billiard balls, not like sticky clay.

From this simple picture, we can derive the gas laws (Boyle's, Charles's, Avogadro's) and even calculate things like the speed of sound in a gas.

The Five Assumptions (The Precise Statement)

For a gas to behave according to the kinetic theory in its simplest form, we make these assumptions:

1. A gas consists of a very large number of molecules.

   The number is so huge that we can use statistics — individual molecules don't matter, only averages do.

2. The molecules are in constant, random motion.

   They move in straight lines until they hit something (another molecule or a wall). There's no preferred direction.

3. The molecules are point masses.

   Their actual size is negligible compared to the distance between them. In other words, the volume of the molecules themselves is tiny compared to the volume of the container.

4. Collisions are perfectly elastic.

   No kinetic energy is lost when molecules collide with each other or with the walls. Total energy of the system stays constant.

5. There are no intermolecular forces.

   The molecules don't attract or repel each other except during collisions. Between collisions, they move freely.

How It Explains Pressure

Pressure is the force per unit area exerted by the gas on the walls of its container. In the kinetic picture:

• A molecule moving toward a wall hits it and bounces back.
• During the collision, the wall exerts a force on the molecule to reverse its momentum.
• By Newton's third law, the molecule exerts an equal and opposite force on the wall.
• Multiply that by the billions of collisions happening every second, and you get a steady, measurable pressure.

If you heat the gas, the molecules move faster. They hit the walls harder and more often — pressure increases. If you compress the gas into a smaller volume, molecules hit the walls more frequently — pressure increases again.

The Key Result: The Kinetic Equation

From these assumptions, we can derive a relationship between pressure $P$, volume $V$, and the average kinetic energy of the molecules. The result is:

Where:

• $N$ = number of molecules
• $m$ = mass of one molecule
• $\overline{v^2}$ = mean square speed of the molecules (average of the squares of their speeds)

Since the average kinetic energy of a molecule is $\overline{K} = \frac{1}{2} m \overline{v^2}$, we can rewrite this as:

$P V = \frac{2}{3} N \overline{K}$ …