Q.State the law of equipartition of energy. Find the ratio of molar heat capacities of monoatomic and diatomic gases.
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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:
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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.
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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.
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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.
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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.
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There are no intermolecular forces.
The molecules don't attract or repel each other except during collisions. Between collisions, they move freely.
These assumptions define an ideal gas. Real gases deviate from this behaviour at high pressure or low temperature, but the kinetic theory gives an excellent approximation for most everyday conditions.
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:
PV=31Nmv2
Where:
- N = number of molecules
- m = mass of one molecule
- v2 = mean square speed of the molecules (average of the squares of their speeds)
Since the average kinetic energy of a molecule is K=21mv2, we can rewrite this as:
PV=32NK …
The law of equipartition of energy assigns a fixed energy share to each degree of freedom of a gas molecule, and from this the molar heat capacities — and hence their ratio — for monatomic and diatomic gases can be derived. …
By the equipartition law, Cv = (f/2)R; using f = 3 for monatomic and f = 5 for diatomic gases gives γ_mono = 5/3 and γ_dia = 7/5.
Law of equipartition of energy: For a system in thermal equilibrium at absolute temperature T, the total energy is distributed equally among all the degrees of freedom available to the molecules, with each degree of freedom contributing an average energy of (1/2)kT per molecule (equivalently (1/2)RT per mole).
Monatomic gas: A monatomic molecule (e.g. He, Ar) has only translational motion along the 3 spatial directions, so f = 3 degrees of freedom.
Cv = (f/2)R = (3/2)R
Cp = Cv + R (Mayer's relation) = (3/2)R + R = (5/2)R
γ_mono = Cp/Cv = (5/2)R / (3/2)R = 5/3 ≈ 1.67
Diatomic gas: A diatomic molecule (e.g. N2, O2) has 3 translational degrees of freedom plus 2 rotational degrees of freedom (rotation about the two axes perpendicular to the line joining the two atoms; rotation about the bond axis itself is negligible), so f = 5 at ordinary temperatures.
Cv = (f/2)R = (5/2)R
Cp = Cv + R = (5/2)R + R = (7/2)R
γ_dia = Cp/Cv = (7/2)R / (5/2)R = 7/5 = 1.4
…
Showing the 12 most recent of 63 on this concept.
- CBSE 2026Set ANNUAL1 markMCQQ.In an ideal gas, molecules possess ______.(a) only kinetic energy(b) both kinetic energy and potential energy(c) only potential energy(d) neither kinetic energy nor potential energy
›Reveal solutionSolution
An ideal gas is defined by point-mass molecules with no intermolecular forces, so they possess only kinetic energy.
The kinetic theory of gases models an ideal gas on the assumptions that (i) molecules are point masses with negligible volume, and (ii) there are no intermolecular forces of attraction or repulsion between them except during instantaneous elastic collisions.
…
- CBSE 2026Set ANNUAL1 markQ.A monatomic gas molecule has ______ degrees of freedom.
›Reveal solutionSolution
A monatomic gas molecule (a single point-like atom) can only move — translate — along the x, y and z directions, giving it 3 degrees of freedom.
Degrees of freedom = the number of independent ways a molecule can possess energy (independent coordinates needed to specify its motion). A monatomic molecule (e.g. He, Ne, Ar) is treated as a point mass with negligible size, so it can only undergo translational motion along 3 mutually perpendicular axes. It has no rotational degrees of freedom (a point particle has negligible moment of inertia about any axis through its centre) and, at ordi …
- CBSE 2026Set ANNUAL1 markMCQQ.What is the average kinetic energy of a gas molecule at absolute temperature T?(a) (3/2) kT(b) kT(c) (1/2) kT(d) (5/2) kT [option D partly obscured by the page's rotated background watermark digits '301328'; best-effort reading, not fully certain]
›Reveal solutionSolution
The kinetic theory of gases gives the average translational kinetic energy per molecule as (3/2) kT, where k is Boltzmann's constant -- one of the central results of this chapter (also the basis of the law of equipartition of energy: (1/2)kT per translational degree of freedom, and there are 3 such degrees of freedom).
From the kinetic theory of gases, each molecule (treated as a point particle with 3 translational degrees of freedom -- motion along x, y, z) has, on average, (1/2) kT of kinetic energy associated with EACH degree of freedom (the law of equipartition of energy). With 3 translational degrees of freedom:
Average KE per molecule = 3 x (1/2) kT = (3/2) kT
…
- CBSE 2026Set ANNUAL1 markQ.Fill in the blank: The upper layers of the atmosphere are ________ hot.
›Reveal solutionSolution
Even though the upper atmosphere feels cold, the gas molecules there actually move at very high average speeds, corresponding to a very high kinetic (rms) temperature.
Temperature, in the kinetic theory of gases, is a measure of the average translational kinetic energy per molecule: (1/2)m⟨v²⟩ = (3/2)k_BT. In the extremely thin upper layers of the atmosphere, the gas is so rarefied (very low pressure/density) that molecules travel long distances between collisions and can attain very high speeds, corresponding to temperatures of the order of a few thousand kelvin. This is why the upper atmosphere is described as 'hot' in the kinetic-theory sense. However, this does not mean an object placed there would feel hot or would rapidly absorb heat: because the density of molecules is so low, the total heat content (and rate of energy transfer by …
- CBSE 2026Set ANNUAL1 markMCQQ.The average kinetic energy of a molecule of an ideal gas depends on the(a) nature of the gas(b) pressure(c) volume(d) temperature
›Reveal solutionSolution
Average molecular KE = (3/2)kT depends only on temperature. Answer (D).
Kinetic theory gives the average translational kinetic energy of an ideal-gas molecule as:
(average KE) = (3/2) k_B T,
…
- CBSE 2026Set ANNUAL1 markMCQQ.The mean square speed of the molecules of a gas at absolute temperature T is proportional to(a) √T(b) T(c) T^2(d) 1/T
›Reveal solutionSolution
Mean square speed <v^2> is proportional to T. Answer (B).
From kinetic theory, the average translational kinetic energy of a molecule is (1/2) m <v^2> = (3/2) k_B T.
Solving, <v^2> = 3 k_B T/m, which is directly proportional to the absolute temperature T.
…
- CBSE 2026Set ANN1 markMCQQ.The average kinetic energy of a gas molecule is directly proportional to(a) Volume(b) Pressure(c) Temperature(d) Number of moles
›Reveal solutionSolution
The average kinetic energy of a gas molecule is (3/2) kB T, so it is directly proportional to the absolute temperature.
From the kinetic theory of gases, the mean translational kinetic energy of a single molecule is
(1/2) m (v-bar squared) = (3/2) kB T, …
- CBSE 2025Set ANNUAL1 markMCQQ.According to kinetic theory of gas Kelvin temperature of gas depends upon (A) average translatory kinetic energy of each molecule of gas (B) rotational kinetic energy of each molecule of gas (C) internal potential energy of gas (D) both (A) and (B)
›Reveal solutionSolution
The absolute (Kelvin) temperature of a gas is a measure of the average translational kinetic energy of its molecules.
From the kinetic theory of gases, the average translational kinetic energy per molecule is directly proportional to the absolute temperature:
21mv2=23kBT
…
- CBSE 2025Set ANNUAL1 markMCQQ.If the temperature of a gas is increased to four times at constant volume then (A) its pressure will become four times (B) root mean square speed of the molecules will become two times (C) mean speed of the molecules will become two times (D) all of these
›Reveal solutionSolution
Quadrupling the temperature at constant volume makes pressure 4× and both rms speed and mean speed 2× — so all three statements are correct.
At constant volume, from the ideal gas law PV=nRT: P∝T, so if T→4T, then P→4P — statement (a) is true.
Rms speed: vrms=M3RT∝T. If T→4T: vrms→4vrms=2vrms — statement (b) is true.
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- CBSE 2025Set ANNUAL1 markMCQQ.The degree of freedom of a rigid molecule is f and atomicity is n. Which of the following is correct? (A) n = 2, f = 5 (B) n = 3, f = 6 (C) n = 1, f = 3 (D) All of these
›Reveal solutionSolution
All three atomicity–degrees-of-freedom pairings given are correct for rigid molecules, so the answer is "all of these."
For a rigid molecule (ignoring vibration):
- Monatomic (n=1, e.g. He, Ar): only 3 translational degrees of freedom, f=3 — matches option (c).
- Diatomic (n=2, e.g. O₂, N₂): 3 translational + 2 rotational degrees of freedom (rotation about the bond axis is negligible), f=5 — matches option (a). …
- CBSE 2025Set ANNUAL1 markMCQQ.At temperature T the rms speed of molecules of any gas is proportional to (A) T (B) √T (C) T^2 (D) T^3
›Reveal solutionSolution
RMS speed of gas molecules is proportional to the square root of the absolute temperature.
From kinetic theory: 21Mvrms2=23RT (per mole), so:
vrms=M3RT∝T
…
- CBSE 2025Set ANNUAL1 markMCQQ.At absolute zero the speed of molecules of a gas will be (A) very large (B) zero (C) equal to speed at 0°C (D) equal to speed at 100°C
›Reveal solutionSolution
At absolute zero, the (ideal-gas) kinetic-theory speed of gas molecules becomes zero.
From kinetic theory, vrms=M3RT. As the absolute temperature T→0 K:
vrms→M3R×0=0
…
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