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

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

Long Answer

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Hard

Question 1

Long Form

Derive an expression for the pressure exerted by an ideal gas on the walls of its container using the kinetic theory. State all assumptions clearly.

Assumptions: molecules are in incessant random motion; all collisions are elastic; intermolecular forces are negligible; the time of collision is negligible compared to the time between collisions.
Consider a molecule of mass m with x-component of velocity vx hitting the wall of area A. It rebounds elastically, so momentum imparted to wall = 2mvx. In time Δt, molecules within distance vxΔt reach the wall; on average, half move toward it, giving N hits = ½ n A vx Δt.
Total momentum transferred to wall in Δt: Q = 2mvx × ½ n A vx Δt = n m vx² A Δt. Force = Q/Δt = n m vx² A, so pressure P = n m vx².
Summing over all molecules, P = n m v̄x². By isotropy of a gas, v̄x² = v̄y² = v̄z² = (1/3) v̄², giving P = (1/3) n m v̄².
This result is independent of the shape of the container (by Pascal's law) and of the specific velocity distribution; the expression depends only on the mean squared speed.
The derivation shows that pressure is a statistical result arising from the collective impact of an enormous number of rapidly moving molecules, not from any single collision.

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Question 2

Long Form

Explain the kinetic interpretation of temperature and derive the relation between average kinetic energy and absolute temperature. Also derive the expression for vrms.

Starting from the kinetic theory pressure P = (1/3) n m v̄², multiplying both sides by volume V: PV = (1/3) N m v̄² = (2/3) × N × (½ m v̄²), where N is the total number of molecules.
Comparing with the ideal gas equation PV = NkBT: (2/3) × N × (½ m v̄²) = NkBT, which gives ½ m v̄² = (3/2) kBT.
This is the kinetic interpretation of temperature: the absolute temperature T is directly proportional to the average translational kinetic energy per molecule, independent of the nature or mass of the gas.
The result also shows that the internal energy of an ideal gas E = N × (3/2) kBT depends only on T, not on pressure or volume.
For the rms speed: since ½ m vrms² = (3/2) kBT, we get vrms = √(3kBT/m) = √(3RT/M₀), where M₀ is the molar mass in kg/mol.
At T = 300 K for nitrogen (M₀ = 28 × 10⁻³ kg/mol), vrms = √(3 × 8.314 × 300 / 0.028) ≈ 516 m/s, which is of the order of the speed of sound in air.

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Question 3

Long Form

Using the law of equipartition of energy, derive the values of Cv, Cp, and γ for (i) monatomic gases, (ii) diatomic gases treated as rigid rotators, and (iii) diatomic gases with vibrational modes. Tabulate your results.

Law of equipartition: each degree of freedom contributes ½ kBT; each vibrational mode contributes kBT. For a mole, multiply by NA (so R = NkB per term).
Monatomic gas (f = 3 translational): U = (3/2)RT; Cv = (3/2)R = 12.5 J mol⁻¹ K⁻¹; Cp = (5/2)R = 20.8 J mol⁻¹ K⁻¹; γ = 5/3 ≈ 1.67.
Rigid diatomic gas (f = 5: 3 translational + 2 rotational): U = (5/2)RT; Cv = (5/2)R = 20.8 J mol⁻¹ K⁻¹; Cp = (7/2)R = 29.1 J mol⁻¹ K⁻¹; γ = 7/5 = 1.40.
Non-rigid diatomic (adds 1 vibrational mode contributing kBT = 2 × ½ kBT): U = (7/2)RT; Cv = (7/2)R; Cp = (9/2)R; γ = 9/7 ≈ 1.29.
In all cases Cp − Cv = R, the Mayer relation, because work R dT per mole must be done against constant pressure in addition to raising internal energy.
The predicted values agree well with experimental data for He, Ne, Ar, H₂, O₂, and N₂ at ordinary temperatures, validating the equipartition law, though polyatomic gases and gases at high temperatures show larger Cv due to more vibrational modes being active.

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Question 4

Long Form

Derive the expression for the mean free path of a gas molecule. Estimate its value for air at STP and comment on its significance relative to molecular size.

Consider a molecule of diameter d moving with average speed ⟨v⟩. In time Δt, it sweeps a cylinder of volume π d² ⟨v⟩ Δt, within which any other molecule's centre will cause a collision.
If n is the number density, the number of collisions in time Δt is n π d² ⟨v⟩ Δt. The time between collisions τ = 1/(n π d² ⟨v⟩). A more exact treatment accounting for relative motion of all molecules replaces ⟨v⟩ by √2 ⟨v⟩.
Thus τ = 1/(√2 π n d² ⟨v⟩) and the mean free path l = ⟨v⟩ τ = 1/(√2 π n d²).
At STP for air: n = (0.02 × 10²³)/(22.4 × 10⁻³) = 2.7 × 10²⁵ m⁻³. With d = 2 × 10⁻¹⁰ m: l = 1/(√2 × π × 2.7 × 10²⁵ × 4 × 10⁻²⁰) ≈ 2.9 × 10⁻⁷ m.
This is approximately 1500 times the molecular diameter d, confirming that molecules travel long distances relative to their own size between collisions.
The large mean free path explains gaseous behaviour: molecules are effectively free between collisions. It also explains why diffusion is slow despite high molecular speeds, why gases cannot be confined without a container, and it forms the basis for relating macroscopic transport properties (viscosity, thermal conductivity) to molecular parameters.

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Question 5

Long Form

Compare the behaviour of real gases with that of an ideal gas. Discuss the conditions under which deviations from ideal behaviour become significant, using the P-V diagram.

An ideal gas obeys PV = µRT exactly at all pressures and temperatures; for an ideal gas, PV/µT = R is a constant independent of P.
Real gases deviate from ideal behaviour, especially at high pressures and low temperatures. The ratio PV/µT approaches R only in the limit of low pressure and high temperature, as shown in Fig. 12.1 of the chapter.
At high pressures, molecular volumes are no longer negligible compared to container volume, and the molecules are close enough for intermolecular attractions to matter, reducing pressure below the ideal value.
At low temperatures near the liquefaction point, intermolecular attractions are comparable to kinetic energies, causing significant departure from PV = µRT; the gas may even liquefy.
From the P-V graph (Fig. 12.2), experimental P-V curves for steam at various temperatures deviate from Boyle's law (dotted lines) at high pressures, but converge toward the ideal prediction at lower pressures.
The kinetic theory is strictly valid only for ideal gases where intermolecular forces can be neglected. Corrections such as the van der Waals equation are needed for real gases at non-ideal conditions.

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Question 6

Long Form

Analyse how the kinetic theory of gases is consistent with Avogadro's hypothesis, Boyle's law, and Charles' law, showing each as a special case of the ideal gas equation.

The ideal gas equation PV = NkBT is derived from kinetic theory, where N is the number of molecules and kB is the Boltzmann constant.
Avogadro's hypothesis: if P, V, and T are the same for two gas samples, then N must be the same (from PV = NkBT). This confirms that equal volumes of all gases at the same T and P contain the same number of molecules.
Boyle's law: fixing N and T in PV = NkBT gives PV = constant. Pressure and volume are inversely related at constant temperature—a direct derivation of Boyle's law.
Charles' law: fixing N and P in PV = NkBT gives V = (NkB/P) T, so V ∝ T at constant pressure—Charles' law follows immediately.
Dalton's law of partial pressures: for a mixture, P = (n₁ + n₂ + ...) kBT = P₁ + P₂ + ..., where each Pi = nikBT is the partial pressure, showing the total is the sum of partial pressures.
This demonstrates the extraordinary power of kinetic theory: a single microscopic model simultaneously explains and unifies all the classical gas laws, giving each a molecular interpretation rather than leaving them as separate empirical observations.

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Question 7

Long Form

Describe the degrees of freedom of monatomic, diatomic, and polyatomic molecules. Explain why the rotation of a diatomic molecule about the bond axis is not counted at moderate temperatures.

Degrees of freedom are the number of independent coordinates needed to completely specify the configuration of a molecule. Each independent quadratic energy term counts as one degree.
A monatomic molecule (e.g., He, Ar) has 3 translational degrees of freedom—motion of its centre of mass along x, y, and z axes. Rotation is negligible because atoms are point-like with negligible moment of inertia.
A diatomic molecule (e.g., O₂, N₂) has 3 translational degrees, plus 2 rotational degrees (rotation about the two axes perpendicular to the bond). At moderate temperatures, total f = 5.
Rotation about the bond axis (along the molecular axis) has a very small moment of inertia. By quantum mechanical arguments, the energy quantum needed to excite this rotation is very large compared to kBT at moderate temperatures, so this mode is effectively frozen out.
A non-linear polyatomic molecule has 3 translational + 3 rotational degrees of freedom = 6, plus f vibrational modes. A linear polyatomic molecule has 3 translational + 2 rotational + f vibrational modes.
This classification forms the basis of the equipartition-based prediction of specific heats, and understanding which modes are active at a given temperature is essential for correctly predicting Cv and γ.

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Question 8

Long Form

Explain the specific heat capacity of solids using the law of equipartition of energy. How does the Dulong-Petit law arise, and where does it fail?

In a solid, each atom is bound to its equilibrium position and can vibrate in all three spatial dimensions. Each dimensional vibration constitutes one vibrational mode.
Each vibrational mode has two energy terms: kinetic energy (½ m(dy/dt)²) and potential energy (½ ky²). By equipartition, each contributes ½ kBT, so each vibrational mode contributes kBT.
For a mole of solid with NA atoms vibrating in 3 dimensions: U = 3 × kBT × NA = 3RT. The molar specific heat C = dU/dT = 3R ≈ 24.9 J mol⁻¹ K⁻¹.
This is the Dulong-Petit law, which states that the molar specific heat of all solids at ordinary temperatures is approximately 3R. Experimental values for Al, Cu, Pb, Ag, and W all agree closely with this prediction.
The law fails for carbon (diamond) and some other light-atom solids at room temperature, where the measured specific heat is much lower than 3R. This is because the vibrational energy quanta are large relative to kBT, so quantum effects suppress the contribution of certain modes—classical equipartition overestimates Cv.
At sufficiently high temperatures, even carbon approaches 3R, confirming that the classical law is asymptotically correct and that quantum mechanics (specifically the Einstein and Debye models) is needed for a complete description at low temperatures.

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Question 9

Long Form

Discuss the historical development of atomic theory from Kanada and Democritus to Dalton, and explain how Gay Lussac's law and Avogadro's hypothesis are connected to the molecular theory of matter.

Ancient speculation about atomic structure appeared independently in India (Kanada's Vaiseshika school, ~6th century BC, postulating eternal indivisible paramanu) and Greece (Democritus, ~4th century BC, proposing atoms differing in shape and size). However, these were philosophical conjectures without quantitative experimental support.
John Dalton (~1800) proposed the modern atomic theory to explain the laws of definite proportions (any compound has fixed mass ratios of constituents) and multiple proportions (for a fixed mass of one element, masses of the other are in small integer ratios).
Gay Lussac's law (early 19th century) stated that when gases combine chemically, their volumes are in ratios of small integers. Avogadro's hypothesis (equal volumes of all gases at same T and P contain the same number of molecules) explains Gay Lussac's law when combined with Dalton's theory.
Since elements often exist as molecules, Dalton's atomic theory is equivalently called the molecular theory of matter. Kinetic theory provides the physical justification for Avogadro's hypothesis through PV = NkBT.
The atomic picture remained controversial even at the end of the 19th century; it was fully vindicated by Einstein's 1905 analysis of Brownian motion, which gave a direct measure of Avogadro's number and confirmed the reality of atoms.
Modern electron microscopes and scanning tunnelling microscopes now allow direct imaging of atoms, confirming all the historical predictions and establishing the size of atoms at approximately 10⁻¹⁰ m.

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Question 10

Long Form

Analyse why the temperature of a gas in a cylinder rises when the gas is compressed by a piston. Use the kinetic theory analogy of a ball hitting a moving bat to explain this phenomenon.

When a ball hits a stationary wall, it rebounds with the same speed (elastic collision). When it hits a massive bat moving toward it with speed V, the ball's rebound speed increases: if the ball's speed relative to the bat is V + u (before collision), after collision it moves away at V + u relative to the bat, giving a speed of V + (V + u) = 2V + u relative to the ground.
The analogy to gas compression: when a piston moves inward, it acts like the moving bat. Gas molecules hitting the approaching piston rebound with greater speed, gaining kinetic energy.
Since temperature is proportional to average molecular kinetic energy (½ mv̄² = (3/2)kBT), an increase in molecular speeds directly raises the gas temperature.
The faster the piston moves (higher compression rate), the more kinetic energy is imparted per collision, and the greater the temperature rise—consistent with the adiabatic compression relation in thermodynamics.
Conversely, when gas expands and pushes a piston outward, molecules hitting the receding piston lose energy, slowing down. The temperature of the expanding gas falls.
This microscopic explanation via kinetic theory is fully consistent with the macroscopic first law of thermodynamics: work done on the gas during compression appears as an increase in internal energy, raising the temperature.

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Chapter 12: Kinetic Theory | Long Questions

Derive an expression for the pressure exerted by an ideal gas on the walls of its container using the kinetic theory. State all assumptions clearly.

Explain the kinetic interpretation of temperature and derive the relation between average kinetic energy and absolute temperature. Also derive the expression for vrms.

Using the law of equipartition of energy, derive the values of Cv, Cp, and γ for (i) monatomic gases, (ii) diatomic gases treated as rigid rotators, and (iii) diatomic gases with vibrational modes. Tabulate your results.

Derive the expression for the mean free path of a gas molecule. Estimate its value for air at STP and comment on its significance relative to molecular size.

Compare the behaviour of real gases with that of an ideal gas. Discuss the conditions under which deviations from ideal behaviour become significant, using the P-V diagram.

Analyse how the kinetic theory of gases is consistent with Avogadro's hypothesis, Boyle's law, and Charles' law, showing each as a special case of the ideal gas equation.

Describe the degrees of freedom of monatomic, diatomic, and polyatomic molecules. Explain why the rotation of a diatomic molecule about the bond axis is not counted at moderate temperatures.

Explain the specific heat capacity of solids using the law of equipartition of energy. How does the Dulong-Petit law arise, and where does it fail?

Discuss the historical development of atomic theory from Kanada and Democritus to Dalton, and explain how Gay Lussac's law and Avogadro's hypothesis are connected to the molecular theory of matter.

Analyse why the temperature of a gas in a cylinder rises when the gas is compressed by a piston. Use the kinetic theory analogy of a ball hitting a moving bat to explain this phenomenon.

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