Chapter 12: Kinetic Theory Short Questions | physics Class 11 CBSE

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Chapter 12: Kinetic Theory Short Questions | physics Class 11 CBSE | ByteNotes.in

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Question 1

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State Avogadro's hypothesis and explain how kinetic theory justifies it.

Avogadro's hypothesis states that equal volumes of all gases at the same temperature and pressure contain the same number of molecules.
Kinetic theory shows that PV = NkBT, so if P, V, and T are the same for two gases, the number of molecules N must also be the same.
This makes Avogadro's hypothesis a direct consequence of the ideal gas equation derived from kinetic theory, rather than just an empirical observation.

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Question 2

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What is meant by the mean free path of a gas molecule? On what factors does it depend?

The mean free path (l) is the average distance a molecule travels between two successive collisions with other molecules.
It depends inversely on the number density n of the gas: higher the density, shorter the mean free path.
It also depends inversely on the square of the molecular diameter d, given by l = 1/(√2 π n d²), so larger molecules have shorter mean free paths.

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Question 3

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Define root mean square speed. How does it depend on the temperature and molecular mass of the gas?

Root mean square speed (vrms) is the square root of the mean of the squared speeds of all molecules in a gas: vrms = √(v̄²).
It is given by vrms = √(3kBT/m), showing that vrms increases with the square root of the absolute temperature T.
vrms decreases with the square root of the molecular mass m, meaning lighter molecules have higher rms speeds at the same temperature.

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Question 4

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State Dalton's law of partial pressures. How is it derived from the kinetic theory of gases?

Dalton's law states that the total pressure of a mixture of non-reactive ideal gases equals the sum of the partial pressures of each constituent gas.
From kinetic theory, P = (1/3)[n₁m₁v̄₁² + n₂m₂v̄₂² + ...] for a mixture; in equilibrium each gas contributes its own partial pressure independently.
In thermal equilibrium, the average kinetic energies of molecules of all gases are equal, so each gas obeys PiV = niVkBT, and their pressures simply add.

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Question 5

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What assumptions are made in the kinetic theory of an ideal gas?

Gas consists of a large number of identical molecules in incessant random motion, and the volume of the molecules is negligible compared to the volume of the gas.
All collisions between molecules and between molecules and walls are perfectly elastic, so total kinetic energy is conserved.
Intermolecular forces are negligible except during brief collisions, and the time of collision is negligible compared to the time between collisions.

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Question 6

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What is the kinetic interpretation of temperature? What does it imply about the temperature of absolute zero?

The kinetic interpretation states that the absolute temperature T of a gas is directly proportional to the average translational kinetic energy per molecule: ½mv̄² = (3/2)kBT.
Temperature is therefore a measure of the random molecular motion of a gas, connecting the macroscopic thermodynamic variable T to a molecular quantity.
At absolute zero (T = 0 K), the average kinetic energy of molecules would be zero, implying all molecular motion would cease—a state that cannot be fully reached in practice.

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Question 7

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State the law of equipartition of energy. What is the energy associated with each degree of freedom?

The law of equipartition of energy states that in thermal equilibrium at temperature T, the total energy is equally distributed among all energy modes of a molecule.
Each translational or rotational degree of freedom contributes an average energy of ½ kBT to the total molecular energy.
Each vibrational mode contributes kBT (= 2 × ½ kBT) because it has both kinetic and potential energy modes, each contributing ½ kBT.

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Question 8

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How many degrees of freedom does a diatomic molecule have at moderate temperatures? Justify your answer.

A diatomic molecule has 5 degrees of freedom at moderate temperatures: 3 translational and 2 rotational.
The 3 translational degrees correspond to motion of the centre of mass along the x, y, and z axes.
The 2 rotational degrees correspond to rotation about two axes perpendicular to the molecular axis; rotation about the bond axis itself has negligible moment of inertia and does not contribute at moderate temperatures for quantum mechanical reasons.

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Question 9

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Write the expression for pressure derived from kinetic theory. Explain what each symbol represents.

The pressure of an ideal gas is P = (1/3) n m v̄², where n is the number of molecules per unit volume (number density) and m is the mass of one molecule.
v̄² is the mean of the squared speeds of all molecules, also written as ⟨v²⟩, averaged over the entire distribution of molecular velocities.
This expression is derived by calculating the momentum transferred per unit time per unit area when molecules collide elastically with the walls of their container.

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Question 10

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Why does a gas leaking from a cylinder take a long time to fill a room, even though gas molecules travel at speeds comparable to the speed of sound?

Although gas molecules have very high speeds (~485 m/s for air), their mean free path is very small (about 2.9 × 10⁻⁷ m at STP).
Molecules undergo billions of collisions per second, constantly changing direction, so their net displacement is much slower than their individual speeds.
This incessant deflection due to collisions means the effective rate of diffusion across the room is far slower than the molecular speed would suggest.

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Question 11

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Show that the relation Cp − Cv = R holds for any ideal gas.

For an ideal gas, the internal energy depends only on temperature: U = f × (½ kBT) × NA per mole, where f is the total number of degrees of freedom.
At constant volume, all heat supplied raises internal energy: Cv = dU/dT. At constant pressure, additional work P dV = R dT per mole is done, so Cp = Cv + R.
This gives Cp − Cv = R regardless of the atomicity or the number of degrees of freedom, confirming it is a universal result for all ideal gases.

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Question 12

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What is the value of Cv, Cp, and γ for a monatomic ideal gas? Derive using the equipartition law.

A monatomic gas has only 3 translational degrees of freedom, so internal energy per mole U = 3 × (½ kBT) × NA = (3/2)RT.
Cv = dU/dT = (3/2)R ≈ 12.5 J mol⁻¹ K⁻¹ and Cp = Cv + R = (5/2)R ≈ 20.8 J mol⁻¹ K⁻¹.
The ratio of specific heats γ = Cp/Cv = (5/2)R / (3/2)R = 5/3 ≈ 1.67, which agrees well with experimental values for noble gases like He and Ar.

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Question 13

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Why do heavier molecules in a gas mixture have lower average speeds than lighter ones at the same temperature?

In a mixture of gases in thermal equilibrium, the average kinetic energy per molecule is the same for all species: ½ m₁v̄₁² = ½ m₂v̄₂² = (3/2)kBT.
Therefore v̄ ∝ 1/√m, meaning speed is inversely proportional to the square root of the molecular mass.
A heavier molecule has greater inertia and needs more energy to achieve the same speed, so at equal temperatures, heavier molecules move more slowly on average.

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Chapter 12: Kinetic Theory | Short Questions

State Avogadro's hypothesis and explain how kinetic theory justifies it.

What is meant by the mean free path of a gas molecule? On what factors does it depend?

Define root mean square speed. How does it depend on the temperature and molecular mass of the gas?

State Dalton's law of partial pressures. How is it derived from the kinetic theory of gases?

What assumptions are made in the kinetic theory of an ideal gas?

What is the kinetic interpretation of temperature? What does it imply about the temperature of absolute zero?

State the law of equipartition of energy. What is the energy associated with each degree of freedom?

How many degrees of freedom does a diatomic molecule have at moderate temperatures? Justify your answer.

Write the expression for pressure derived from kinetic theory. Explain what each symbol represents.

Why does a gas leaking from a cylinder take a long time to fill a room, even though gas molecules travel at speeds comparable to the speed of sound?

Show that the relation Cp − Cv = R holds for any ideal gas.

What is the value of Cv, Cp, and γ for a monatomic ideal gas? Derive using the equipartition law.

Why do heavier molecules in a gas mixture have lower average speeds than lighter ones at the same temperature?

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