Mention the conditions under which a real gas obeys the ideal gas equation.

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Accepted Answer

Concept understanding — Kinetic Theory of 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:

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.
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.
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.
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.
There are no intermolecular forces.

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

Important

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:

P V = \frac{1}{3} N m \overline{v^2}

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}

This is the bridge between the microscopic world (molecular motion) and the macroscopic world (pressure and volume).

Connecting to Temperature

From experiments, we know that for an ideal gas:

P V = N k_B T

Where $k_B$ is Boltzmann's constant and $T$ is the absolute temperature (in Kelvin). Comparing this with the kinetic equation gives:

\frac{2}{3} N \overline{K} = N k_B T

So:

\overline{K} = \frac{3}{2} k_B T

Note

Temperature is a measure of the average kinetic energy of the molecules. When you heat a gas, you're literally making its molecules move faster. Absolute zero ( $0 \text{ K}$ ) is the temperature at which molecular motion would stop entirely — though quantum mechanics tells us a tiny residual motion always remains.

What This Tells You

Pressure comes from molecular collisions with walls.
Temperature is the average kinetic energy of molecules.
Volume is the space the molecules have to move in.
All gas laws (Boyle's, Charles's, etc.) are consequences of this simple molecular picture.

The kinetic theory is one of the great unifying ideas in physics — it shows that the familiar, measurable world of pressure gauges and thermometers is just the statistical behaviour of countless tiny, invisible particles doing exactly what Newton's laws say they should.

This is exactly the kind of concept that turns up under searches like "Kinetic Theory of Gases class 11 physics syllabus" or "Kinetic Theory of Gases solved examples" — and it belongs squarely in the Class 11 Physics NCERT/CBSE curriculum. Beyond board exams, it's a dependable scoring topic in JEE Main, NEET and state engineering/medical entrance exams once the core logic clicks.

A real gas is composed of molecules that genuinely interact with one another through intermolecular forces, so strictly speaking no real gas is truly ideal. However, a real gas approaches ideal behaviour -- is said to be 'in the ideal state' -- whenever its molecules are spread far enough apart that these intermolecular interactions become negligible compared to the molecules' kinetic energy. This happens under two related conditions: (i) LOW PRESSURE, where the gas occupies a large volume relative to the number of molecules, keeping them far apart on average; and (ii) HIGH TEMPERATURE, where the molecules' kinetic energy is large enough to make any weak intermolecular attraction insignificant by comparison. Additionally, the gas's actual temperature must remain well above its liquefaction temperature -- close to liquefaction, intermolecular attractive forces become dominant (which is exactly what causes the gas to condense into a liquid), and ideal-gas behaviour breaks down sharply. Under low pressure, high temperature, and well away from liquefaction, real gases like nitrogen, oxygen and hydrogen obey PV = nRT to a very good approximation. [!ANSWER] A real gas obeys the ideal gas equation closely at low pressure and high temperature, provided its temperature stays well above its liquefaction temperature.

Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

Showing the 12 most recent of 63 on this concept.

CBSE 2026Set ANNUAL1 markMCQ
$Q. In an ideal gas, molecules possess ______.$
(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.

Since potential energy arises specifically from intermolecular forces (it is the work needed to separate/bring together interacting particles), the absence of such forces in an ideal gas means the molecules have zero potential energy. Their entire mechanical energy is therefore the kinetic energy of translational (and for polyatomic gases, rotational/vibrational) motion:

$\text{Average KE per molecule} = \tfrac{3}{2}k_BT$

✓Final answer
$Option (a) only kinetic energy.$
CBSE 2026Set ANNUAL1 mark
$Q. A monatomic gas molecule has ______ degrees of freedom.$
›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 ordinary temperatures, no vibrational modes (there being no internal bonds). Hence f = 3, and by the law of equipartition of energy its average kinetic energy is (3/2) k_B T.

✓Final answer
$A monatomic gas molecule has 3 degrees of freedom (all translational).$
CBSE 2026Set ANNUAL1 markMCQ
$Q. What is the average kinetic energy of a gas molecule at absolute temperature T?$
(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

Option (d), "(5/2) kT," is the formula for the average total energy of a DIATOMIC gas molecule (3 translational + 2 rotational degrees of freedom), not the average kinetic energy of a generic gas molecule as asked here, so it does not apply to this question regardless of how legibly it is printed.

✓Final answer
$(a) (3/2) kT.$
CBSE 2026Set ANNUAL1 mark
$Q. Fill in the blank: The upper layers of the atmosphere are ________ hot.$
›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 collisions) is very small — a satellite in the upper atmosphere is not scorched despite the high kinetic temperature of the surrounding gas.

✓Final answer
$The upper layers of the atmosphere are very hot (in terms of the average kinetic energy/temperature of their molecules), even though they carry very little actual heat because the gas there is extremely rarefied.$
CBSE 2026Set ANNUAL1 markMCQ
$Q. The average kinetic energy of a molecule of an ideal gas depends on the$
(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,

where k_B is Boltzmann's constant and T is absolute temperature. It does not depend on the nature of the gas, the pressure, or the volume — only on T.

✓Final answer
$(D) temperature.$
CBSE 2026Set ANNUAL1 markMCQ
$Q. The mean square speed of the molecules of a gas at absolute temperature T is proportional to$
(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.

(Note the RMS speed, sqrt(<v^2>), is proportional to sqrt(T), but the MEAN SQUARE speed itself is proportional to T.)

✓Final answer
$(B) T.$
CBSE 2026Set ANN1 markMCQ
$Q. The average kinetic energy of a gas molecule is directly proportional to$
(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,

where kB is Boltzmann's constant and T the absolute temperature. This shows the average KE depends only on temperature (in kelvin), not on the volume, pressure or number of moles taken separately.

✓Final answer
$(c) Temperature.$
CBSE 2025Set ANNUAL1 markMCQ
$Q. According to kinetic theory of gas Kelvin temperature of gas depends upon (A) average translatory kinetic energy of each molecule of gas (B) rotational kinetic energy of each molecule of gas (C) internal potential energy of gas (D) both (A) and (B)$
›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:

$\dfrac{1}{2}m\overline{v^2} = \dfrac{3}{2}k_BT$

where $k_B$ is Boltzmann's constant. This relation defines the kinetic interpretation of temperature — it depends only on the translational motion of molecules, not on rotational kinetic energy (which contributes to internal energy for polyatomic gases but not to this definition of temperature) or internal potential energy (negligible for an ideal gas, where intermolecular forces are ignored).

✓Final answer
$(A) average translatory kinetic energy of each molecule of gas.$
CBSE 2025Set ANNUAL1 markMCQ
$Q. If the temperature of a gas is increased to four times at constant volume then (A) its pressure will become four times (B) root mean square speed of the molecules will become two times (C) mean speed of the molecules will become two times (D) all of these$
›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 $PV = nRT$ : $P \propto T$ , so if $T \to 4T$ , then $P \to 4P$ — statement (a) is true.

Rms speed: $v_{rms} = \sqrt{\dfrac{3RT}{M}} \propto \sqrt{T}$ . If $T \to 4T$ : $v_{rms} \to \sqrt{4}\,v_{rms} = 2v_{rms}$ — statement (b) is true.

Mean speed: $\bar v = \sqrt{\dfrac{8RT}{\pi M}} \propto \sqrt{T}$ — by the same logic, it also becomes 2× — statement (c) is true.

Since all three statements hold, the correct choice is "all of these."

✓Final answer
$(D) all of these.$
CBSE 2025Set ANNUAL1 markMCQ
$Q. The degree of freedom of a rigid molecule is f and atomicity is n. Which of the following is correct? (A) n = 2, f = 5 (B) n = 3, f = 6 (C) n = 1, f = 3 (D) All of these$
›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).

Triatomic non-linear / general polyatomic (rigid) ( $n=3$ ): 3 translational + 3 rotational degrees of freedom, $f = 6$ — matches option (b).

Since each of the three given pairs is a genuinely correct case, the answer covers all of them.

✓Final answer
$(D) All of these.$
CBSE 2025Set ANNUAL1 markMCQ
$Q. At temperature T the rms speed of molecules of any gas is proportional to (A) T (B) \sqrtT (C) T^2 (D) T^3$
›Reveal solutionSolution

RMS speed of gas molecules is proportional to the square root of the absolute temperature.

From kinetic theory: $\dfrac{1}{2}M v_{rms}^2 = \dfrac{3}{2}RT$ (per mole), so:

$v_{rms} = \sqrt{\dfrac{3RT}{M}} \propto \sqrt{T}$

At a fixed molar mass $M$ , rms speed scales as the square root of the absolute temperature $T$ .

✓Final answer
$(B) \sqrtT.$
CBSE 2025Set ANNUAL1 markMCQ
$Q. At absolute zero the speed of molecules of a gas will be (A) very large (B) zero (C) equal to speed at 0°C (D) equal to speed at 100°C$
›Reveal solutionSolution

At absolute zero, the (ideal-gas) kinetic-theory speed of gas molecules becomes zero.

From kinetic theory, $v_{rms} = \sqrt{\dfrac{3RT}{M}}$ . As the absolute temperature $T \to 0$ K:

$v_{rms} \to \sqrt{\dfrac{3R\times0}{M}} = 0$

So, in the idealized picture of kinetic theory, translational molecular motion (and hence pressure) would vanish at absolute zero. (In reality, quantum zero-point motion prevents this exactly, but that is beyond the scope of classical kinetic theory taught at this level.)

✓Final answer
$(B) zero.$

Q.Mention the conditions under which a real gas obeys the ideal gas equation.

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