Class 11 Physics Notes Chapter 13 (Chapter 13) – Examplar Problems (English) Book

Alright class, let's delve into Chapter 13: Kinetic Theory. This is a crucial chapter, connecting the macroscopic properties of gases (like pressure, temperature, volume) to their microscopic behavior (motion of molecules). Understanding this well is vital for many competitive exams.

Chapter 13: Kinetic Theory - Detailed Notes for Exam Preparation

1. Introduction & Molecular Nature of Matter:

• Matter is composed of atoms/molecules.
• In gases, molecules are far apart and in continuous, random motion.
• This random motion is responsible for the properties we observe (pressure, temperature).

2. Behaviour of Gases & Gas Laws:

• Boyle's Law (Constant T): For a fixed mass of gas at constant temperature, pressure is inversely proportional to volume.
  • P ∝ 1/V or PV = constant
  • P₁V₁ = P₂V₂
• Charles's Law (Constant P): For a fixed mass of gas at constant pressure, volume is directly proportional to its absolute temperature (in Kelvin).
  • V ∝ T or V/T = constant
  • V₁/T₁ = V₂/T₂
• Gay-Lussac's Law (Constant V): For a fixed mass of gas at constant volume, pressure is directly proportional to its absolute temperature.
  • P ∝ T or P/T = constant
  • P₁/T₁ = P₂/T₂
• Ideal Gas Equation: Combines the gas laws for a fixed amount (n moles) of gas.
  • PV = nRT
  • Where:
    • P = Pressure (Pa or N/m²)
    • V = Volume (m³)
    • n = Number of moles
    • R = Universal Gas Constant (≈ 8.314 J mol⁻¹ K⁻¹)
    • T = Absolute Temperature (K)
  • Also expressed as: PV = N k<0xE2><0x82><0x99>T
  • Where:
    • N = Total number of molecules
    • k<0xE2><0x82><0x99> = Boltzmann Constant (R/N<0xE2><0x82><0x90>) ≈ 1.38 × 10⁻²³ J K⁻¹
    • N<0xE2><0x82><0x90> = Avogadro's Number ≈ 6.022 × 10²³ mol⁻¹
• Dalton's Law of Partial Pressures: The total pressure exerted by a mixture of non-reacting gases is the sum of the partial pressures that each gas would exert if it occupied the same volume alone.
  • P<0xE1><0xB5><0x97><0xE1><0xB5><0x92><0xE1><0xB5><0x97><0xE1><0xB5><0x82><0xE1><0xB5><0x85> = P₁ + P₂ + P₃ + ...

3. Kinetic Theory of an Ideal Gas - Assumptions:

• Gas consists of a large number of identical molecules, which are point masses (size is negligible compared to intermolecular distances).
• Molecules are in continuous, random motion, obeying Newton's laws.
• Collisions between molecules and with the walls of the container are perfectly elastic (kinetic energy and momentum are conserved).
• The duration of collision is negligible compared to the time between collisions.
• There are no intermolecular forces of attraction or repulsion except during collisions.
• The volume occupied by the molecules is negligible compared to the total volume of the gas.

4. Pressure Exerted by an Ideal Gas:

• Pressure is due to the continuous collisions of gas molecules with the walls of the container.
• The derived expression for pressure is:
  • P = (1/3) * (N/V) * m * v<0xE1><0xB5><0xA3><0xE1><0xB5><0x98><0xE1><0xB5><0x9B>²
  • Or P = (1/3) * ρ * v<0xE1><0xB5><0xA3><0xE1><0xB5><0x98><0xE1><0xB5><0x9B>²
  • Where:
    • N = Total number of molecules
    • V = Volume
    • m = Mass of one molecule
    • ρ = Density of the gas (Nm/V)
    • v<0xE1><0xB5><0xA3><0xE1><0xB5><0x98><0xE1><0xB5><0x9B>² = Mean square speed of molecules
    • v<0xE1><0xB5><0xA3><0xE1><0xB5><0x98><0xE1><0xB5><0x9B> = √( (v₁² + v₂² + ... + v<0xE2><0x82><0x99>²) / N ) is the Root Mean Square (RMS) speed.

5. Kinetic Interpretation of Temperature:

• From PV = N k<0xE2><0x82><0x99>T and P = (1/3) * (N/V) * m * v<0xE1><0xB5><0xA3><0xE1><0xB5><0x98><0xE1><0xB5><0x9B>², we get:
  • (1/3) * m * v<0xE1><0xB5><0xA3><0xE1><0xB5><0x98><0xE1><0xB5><0x9B>² = k<0xE2><0x82><0x99>T
• Average translational kinetic energy of one molecule:
  • KE<0xE1><0xB5><0x82><0xE1><0xB5><0x97> = (1/2) * m * v<0xE1><0xB5><0xA3><0xE1><0xB5><0x98><0xE1><0xB5><0x9B>² = (3/2) * k<0xE2><0x82><0x99>T
• Key Conclusion: The average translational kinetic energy of a gas molecule is directly proportional to the absolute temperature of the gas. This provides a microscopic definition of temperature.
• At absolute zero (T=0 K), theoretically, molecular motion ceases.

6. Molecular Speeds:

• RMS Speed (v<0xE1><0xB5><0xA3><0xE1><0xB5><0x98><0xE1><0xB5><0x9B>): Root Mean Square speed.
  • v<0xE1><0xB5><0xA3><0xE1><0xB5><0x98><0xE1><0xB5><0x9B> = √(3k<0xE2><0x82><0x99>T / m) = √(3RT / M) (where M is molar mass)
• Average Speed (v<0xE1><0xB5><0x82><0xE1><0xB5><0x97>): Arithmetic mean of the speeds of molecules.
  • v<0xE1><0xB5><0x82><0xE1><0xB5><0x97> = √(8k<0xE2><0x82><0x99>T / πm) = √(8RT / πM)
• Most Probable Speed (v<0xE1><0xB5><0x98><0xE1><0xB5><0x96>): Speed possessed by the maximum number of molecules.
  • v<0xE1><0xB5><0x98><0xE1><0xB5><0x96> = √(2k<0xE2><0x82><0x99>T / m) = √(2RT / M)
• Relationship: v<0xE1><0xB5><0xA3><0xE1><0xB5><0x98><0xE1><0xB5><0x9B> > v<0xE1><0xB5><0x82><0xE1><0xB5><0x97> > v<0xE1><0xB5><0x98><0xE1><0xB5><0x96> (Ratio ≈ 1.732 : 1.596 : 1.414)

7. Law of Equipartition of Energy:

• Statement: For any dynamical system in thermal equilibrium, the total energy is distributed equally amongst all its degrees of freedom, and the average energy associated with each degree of freedom per molecule is (1/2) k<0xE2><0x82><0x99>T.
• Degrees of Freedom (f): The total number of independent coordinates (or ways) required to specify the position and configuration of a system completely.
  • Monatomic Gas (e.g., He, Ne, Ar): f = 3 (only translational motion along x, y, z axes).
    • Average energy per molecule = (3/2) k<0xE2><0x82><0x99>T
  • Diatomic Gas (e.g., O₂, N₂, H₂):
    • At low/moderate temperatures: f = 5 (3 translational + 2 rotational). Rotational KE is negligible about the axis joining the two atoms.
      • Average energy per molecule = (5/2) k<0xE2><0x82><0x99>T
    • At high temperatures: f = 7 (3 translational + 2 rotational + 2 vibrational [1 KE + 1 PE]). Vibrational modes become active.
      • Average energy per molecule = (7/2) k<0xE2><0x82><0x99>T
  • Polyatomic Gas (Non-linear, e.g., H₂O, NH₃): f = 6 (3 translational + 3 rotational).
    • Average energy per molecule = (6/2) k<0xE2><0x82><0x99>T = 3 k<0xE2><0x82><0x99>T (Vibrational modes add at higher temperatures).
  • Polyatomic Gas (Linear, e.g., CO₂): f = 5 (3 translational + 2 rotational) - similar to diatomic at moderate temperatures.

8. Specific Heat Capacity:

• Molar Specific Heat at Constant Volume (C<0xE1><0xB5><0x97>): Heat required to raise the temperature of 1 mole of gas by 1 K at constant volume.
  • Internal Energy (U) for n moles = n * f * (1/2) RT
  • C<0xE1><0xB5><0x97> = (1/n) * (dU/dT) = (f/2) R
• Molar Specific Heat at Constant Pressure (C<0xE2><0x82><0x99>): Heat required to raise the temperature of 1 mole of gas by 1 K at constant pressure.
  • From First Law of Thermodynamics (dQ = dU + dW), at constant pressure dQ = dU + P dV. For 1 mole, dQ = C<0xE2><0x82><0x99> dT, dU = C<0xE1><0xB5><0x97> dT, and P dV = R dT (from PV=RT).
  • C<0xE2><0x82><0x99> dT = C<0xE1><0xB5><0x97> dT + R dT
  • Mayer's Relation: C<0xE2><0x82><0x99> - C<0xE1><0xB5><0x97> = R
  • C<0xE2><0x82><0x99> = (f/2) R + R = ( (f+2)/2 ) R
• Ratio of Specific Heats (Adiabatic Index, γ):
  • γ = C<0xE2><0x82><0x99> / C<0xE1><0xB5><0x97> = ( (f+2)/2 ) R / ( (f/2) R ) = (f+2) / f = 1 + (2/f)
• Values for different gases:
  • Monatomic (f=3): C<0xE1><0xB5><0x97> = (3/2)R, C<0xE2><0x82><0x99> = (5/2)R, γ = 5/3 ≈ 1.67
  • Diatomic (f=5): C<0xE1><0xB5><0x97> = (5/2)R, C<0xE2><0x82><0x99> = (7/2)R, γ = 7/5 = 1.40
  • Diatomic (f=7, high temp): C<0xE1><0xB5><0x97> = (7/2)R, C<0xE2><0x82><0x99> = (9/2)R, γ = 9/7 ≈ 1.29
  • Polyatomic (Non-linear, f=6): C<0xE1><0xB5><0x97> = 3R, C<0xE2><0x82><0x99> = 4R, γ = 4/3 ≈ 1.33

9. Mean Free Path (λ):

• The average distance travelled by a molecule between two successive collisions.
• λ = 1 / (√2 * π * n * d²)
  • Where:
    • n = Number density of molecules (N/V)
    • d = Diameter of the molecule
• Mean free path is inversely proportional to number density (and hence pressure, at constant T) and the square of the molecular diameter.
• It is directly proportional to temperature (at constant P).

10. Brownian Motion:

• The random, zig-zag motion of particles suspended in a fluid (liquid or gas) resulting from their collision with the fast-moving molecules of the fluid.
• Provides visible evidence for the existence and motion of molecules as postulated by kinetic theory.

Multiple Choice Questions (MCQs)

1. According to the kinetic theory of gases, the average translational kinetic energy of gas molecules is directly proportional to:
   (a) Pressure of the gas
   (b) Volume of the gas
   (c) Absolute temperature of the gas
   (d) Mass of the gas molecule

2. The RMS speed of molecules of a gas at absolute temperature T is proportional to:
   (a) T
   (b) √T
   (c) 1/√T
   (d) T²

3. The ideal gas equation PV = nRT relates pressure, volume, temperature, and the number of moles (n). The constant R is known as:
   (a) Boltzmann Constant
   (b) Avogadro's Number
   (c) Universal Gas Constant
   (d) Planck's Constant

4. For a diatomic gas like Oxygen (O₂) at moderate temperatures, the number of degrees of freedom is typically considered to be:
   (a) 3
   (b) 5
   (c) 6
   (d) 7

5. The ratio of specific heats (γ = C<0xE2><0x82><0x99>/C<0xE1><0xB5><0x97>) for a monatomic gas is:
   (a) 1.40
   (b) 1.33
   (c) 1.67
   (d) 1.50

6. Which of the following statements is NOT an assumption of the kinetic theory of ideal gases?
   (a) Collisions between molecules are perfectly elastic.
   (b) The volume occupied by molecules is negligible compared to the container volume.
   (c) Molecules exert significant attractive forces on each other except during collisions.
   (d) Molecules are in continuous, random motion.

7. If the absolute temperature of an ideal gas is doubled and the pressure is halved, the volume of the gas will:
   (a) Remain unchanged
   (b) Be doubled
   (c) Be halved
   (d) Become four times

8. The mean free path (λ) of a gas molecule is inversely proportional to:
   (a) Temperature (T)
   (b) Square of the molecular diameter (d²)
   (c) Volume (V)
   (d) Mass of the molecule (m)

9. According to the law of equipartition of energy, the average energy associated with each degree of freedom per molecule is:
   (a) k<0xE2><0x82><0x99>T
   (b) (1/2) k<0xE2><0x82><0x99>T
   (c) (3/2) k<0xE2><0x82><0x99>T
   (d) RT

10. Which of the following represents the correct relationship between RMS speed (v<0xE1><0xB5><0xA3><0xE1><0xB5><0x98><0xE1><0xB5><0x9B>), average speed (v<0xE1><0xB5><0x82><0xE1><0xB5><0x97>), and most probable speed (v<0xE1><0xB5><0x98><0xE1><0xB5><0x96>)?
    (a) v<0xE1><0xB5><0xA3><0xE1><0xB5><0x98><0xE1><0xB5><0x9B> > v<0xE1><0xB5><0x98><0xE1><0xB5><0x96> > v<0xE1><0xB5><0x82><0xE1><0xB5><0x97>
    (b) v<0xE1><0xB5><0x98><0xE1><0xB5><0x96> > v<0xE1><0xB5><0x82><0xE1><0xB5><0x97> > v<0xE1><0xB5><0xA3><0xE1><0xB5><0x98><0xE1><0xB5><0x9B>
    (c) v<0xE1><0xB5><0xA3><0xE1><0xB5><0x98><0xE1><0xB5><0x9B> > v<0xE1><0xB5><0x82><0xE1><0xB5><0x97> > v<0xE1><0xB5><0x98><0xE1><0xB5><0x96>
    (d) v<0xE1><0xB5><0x82><0xE1><0xB5><0x97> > v<0xE1><0xB5><0xA3><0xE1><0xB5><0x98><0xE1><0xB5><0x9B> > v<0xE1><0xB5><0x98><0xE1><0xB5><0x96>

Answers to MCQs:

1. (c) Absolute temperature of the gas
2. (b) √T
3. (c) Universal Gas Constant
4. (b) 5
5. (c) 1.67
6. (c) Molecules exert significant attractive forces on each other except during collisions. (Ideal gas assumption is NO intermolecular forces except during collision).
7. (d) Become four times (Use P₁V₁/T₁ = P₂V₂/T₂ => V₂ = (P₁/P₂) * (T₂/T₁) * V₁ = (P₁/(P₁/2)) * (2T₁/T₁) * V₁ = 2 * 2 * V₁ = 4V₁)
8. (b) Square of the molecular diameter (d²) (Also inversely proportional to number density/pressure)
9. (b) (1/2) k<0xE2><0x82><0x99>T
10. (c) v<0xE1><0xB5><0xA3><0xE1><0xB5><0x98><0xE1><0xB5><0x9B> > v<0xE1><0xB5><0x82><0xE1><0xB5><0x97> > v<0xE1><0xB5><0x98><0xE1><0xB5><0x96>

Make sure you understand the derivation of pressure, the connection between temperature and kinetic energy, and how degrees of freedom influence specific heats. These are frequently tested areas. Good luck with your preparation!