Diatomic molecules like hydrogen have energies due to both translational as well as rotational motion. From th

NCERT Exemplar · Q10

pV = \dfrac{2}{3}E

(a) the total energy per unit volume.

(b) only the translational part of energy because rotational energy is very small compared to the translational energy.

(c) only the translational part of the energy because during collisions with the wall pressure relates to change in linear momentum.

(d) the translational part of the energy because rotational energies of molecules can be of either sign and its average over all the molecules is zero.

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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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In the kinetic-theory relation $pV = \dfrac{2}{3}E$ , the quantity $E$ is the total translational kinetic energy of the gas molecules -- it comes purely from the change in linear momentum of molecules striking the container wall. Only option (C) gives the correct physical reason; option (D) is wrong (rotational KE is never negative) and option (B), though it names the right quantity, gives a false reason (rotational energy is not negligible compared to translational). The correct answer is (C) only.

Where $pV = \dfrac{2}{3}E$ comes from

Picture a cubical box of gas molecules, each with velocity components $v_x, v_y, v_z$ . When a molecule collides elastically with a wall perpendicular to the $x$ -axis, only its $x$ -component of momentum reverses: from $mv_x$ to $-mv_x$ , a change of $2mv_x$ . The pressure on that wall is built entirely from this linear-momentum transfer -- nothing about the molecule's rotation enters, because rotation about the molecule's own centre of mass does not change the centre-of-mass (linear) momentum that actually strikes the wall.

Summing over all molecules and directions, the standard kinetic-theory result is

p = \frac{1}{3}\frac{N}{V}m\langle v^2\rangle

Multiplying both sides by $V$ :

pV = \frac{1}{3}Nm\langle v^2\rangle = \frac{2}{3}\left(\frac{1}{2}Nm\langle v^2\rangle\right)

The bracketed term is exactly the total translational kinetic energy of all $N$ molecules. So by definition, $E \equiv \frac{1}{2}Nm\langle v^2\rangle$ , giving $pV = \frac{2}{3}E$ .

Checking each option

(A) "the total energy per unit volume" -- Wrong. $E$ is the total translational kinetic energy of all molecules in the box, not an energy density, and it excludes rotational/vibrational energy.

(B) "only the translational part... because rotational energy is very small compared to translational energy" -- The conclusion happens to be correct, but the reason is false. For a diatomic gas like hydrogen, equipartition gives average translational KE $=\tfrac32 k_BT$ per molecule and average rotational KE $=k_BT$ per molecule -- so rotational energy is $\tfrac23$ of translational, not negligible at all. Since the statement must be judged on its full reasoning, (B) is not an acceptable justification. …

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Q.Diatomic molecules like hydrogen have energies due to both translational as well as rotational motion. From the equation in kinetic theory pV=23EpV = \dfrac{2}{3}EpV=32​E, EEE is (Note: more than one of the given options may be correct.)

Where pV=23EpV = \dfrac{2}{3}EpV=32​E comes from

Checking each option

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Q.Diatomic molecules like hydrogen have energies due to both translational as well as rotational motion. From the equation in kinetic theory $pV = \dfrac{2}{3}E$ , $E$ is (Note: more than one of the given options may be correct.)

Where $pV = \dfrac{2}{3}E$ comes from

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