When an ideal gas is compressed adiabatically, its temperature rises: the molecules on the average have more k

NCERT Exemplar · Q13

Q. When an ideal gas is compressed adiabatically, its temperature rises: the molecules on the average have more kinetic energy than before. The kinetic energy increases, (Note: more than one of the given options may be correct.)

(a) because of collisions with moving parts of the wall only.

(b) because of collisions with the entire wall.

(d) because of redistribution of energy amongst the molecules.

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Concept understanding — Kinetic Theory Explanation

Kinetic Theory Explanation

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 and oxygen — are zipping past you at hundreds of metres per second. You don't feel them because they're too small, and they're moving in every direction at once. But if you put your hand near a hot stove, you suddenly feel heat. Why?

The answer is the kinetic theory of matter. It's a way of explaining what we observe at the human scale (temperature, pressure, heat) by thinking about what's happening at the molecular scale.

The core idea

The kinetic theory says three simple things:

All matter is made of tiny particles (atoms or molecules) that are in constant, random motion.
The particles collide with each other and with the walls of their container — these collisions are perfectly elastic (no energy is lost).
The average kinetic energy of these particles is directly proportional to the temperature of the substance.

That's it. Everything else — pressure, diffusion, the way a gas expands when heated — follows from these three statements.

Building intuition

Think of a single gas molecule bouncing around inside a box. It hits a wall, bounces off, and keeps going. Each time it hits the wall, it exerts a tiny force. Now multiply that by billions of molecules hitting every square centimetre of wall every second. That constant, collective force is what we measure as pressure.

Now heat the box. The molecules move faster — their average kinetic energy increases. They hit the walls harder and more often. Pressure goes up. If the walls can move (like a piston), the gas expands until the pressure inside equals the pressure outside.

This is why a bicycle tyre feels hot after you pump it: you're doing work on the air, compressing it, which increases the average kinetic energy of the molecules — and that's exactly what temperature is.

Note

Temperature is not the total kinetic energy of all molecules — it's the average kinetic energy per molecule. A large cold object can have more total energy than a small hot one, but its molecules move slower on average.

The precise statement

For an ideal gas (a gas where intermolecular forces are negligible and collisions are perfectly elastic), the kinetic theory gives us a direct mathematical link:

\text{Average kinetic energy per molecule} = \frac{3}{2} k_B T

where $k_B$ is Boltzmann's constant ( $1.38 \times 10^{-23} \, \text{J/K}$ ) and $T$ is the absolute temperature in Kelvin.

This means that at the same temperature, all gas molecules — regardless of their mass — have the same average kinetic energy. A light hydrogen molecule moves faster than a heavy oxygen molecule at the same temperature, but their average kinetic energies are equal.

From this, we can derive the ideal gas law:

PV = \frac{1}{3} N m \overline{v^2} = N k_B T

where $P$ is pressure, $V$ is volume, $N$ is the number of molecules, $m$ is the mass of one molecule, and $\overline{v^2}$ is the mean square speed.

PV = nRT

What the theory explains

The kinetic theory isn't just abstract. It explains everyday phenomena:

Evaporation cools you: The fastest molecules escape from a liquid surface, leaving behind slower ones. The average kinetic energy drops — so the temperature drops.
Diffusion: Molecules spread out because they're constantly moving and colliding, gradually mixing with neighbouring molecules.
Brownian motion: Pollen grains jitter under a microscope because they're being bombarded unevenly by invisible water molecules. …

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During adiabatic compression, molecules gain kinetic energy exclusively through collisions with the moving inward wall (the piston), not from the stationary walls or internal redistribution. The correct option is (A).

Kinetic Theory Explanation

When a gas is compressed adiabatically, no heat flows in or out ( $Q = 0$ ). Yet the temperature rises, meaning the average molecular kinetic energy increases. Where does this energy come from?

The answer lies in understanding what happens at the microscopic level during compression. A gas molecule bouncing off a surface behaves like an elastic collision. If the surface is stationary, the molecule rebounds with the same speed it arrived with—no energy change. But if the surface is moving toward the molecule, the rebound speed is higher than the incident speed, and the molecule gains kinetic energy. This is exactly what happens to a tennis ball bouncing off an approaching racket.

In adiabatic compression, the piston (the moving part of the wall) advances into the gas. Molecules colliding with this moving piston pick up extra kinetic energy with each bounce. The stationary walls contribute nothing to the energy increase—they merely reflect molecules elastically. Internal collisions between molecules conserve total kinetic energy and only redistribute it; they cannot create new energy.

Let's examine each option:

Step-by-Step Analysis

Option (A): Collisions with moving parts of the wall only

Consider a molecule of mass $m$ approaching the piston with velocity component $v_x$ perpendicular to the piston. If the piston moves inward with speed $u$ , the molecule's velocity in the piston's reference frame is $(v_x + u)$ . After elastic reflection, it rebounds with velocity $(v_x + u)$ in the opposite direction relative to the piston. Transforming back to the lab frame, the molecule's velocity after collision is $(v_x + 2u)$ .

The kinetic energy change is:

\Delta KE = \frac{1}{2}m(v_x + 2u)^2 - \frac{1}{2}mv_x^2 = 2muv_x + 2mu^2

For typical molecular speeds much larger than piston speed ( $v_x \gg u$ ), the dominant term is $2muv_x > 0$ . Each collision with the moving piston increases molecular kinetic energy. This is the only mechanism by which work done on the gas converts to internal energy.

Option (B): Collisions with the entire wall

The "entire wall" includes both the moving piston and the stationary container walls. Stationary walls reflect molecules elastically with no change in kinetic energy—the velocity component perpendicular to the wall reverses, but the speed remains constant. Only the moving part (the piston) contributes to the energy increase. This option is incorrect because it attributes the energy gain to all walls indiscriminately.
Option (C): Molecules get accelerated in their motion inside the volume …

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Q.When an ideal gas is compressed adiabatically, its temperature rises: the molecules on the average have more kinetic energy than before. The kinetic energy increases, (Note: more than one of the given options may be correct.)

Kinetic Theory Explanation

Step-by-Step Analysis

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