• The behaviour of Kinetic theory explain by considering that the gas consist rapid movement of tiny atomic particles.
  • The kinetic theory was developed by mainly Maxwell, Boltzmann and some others.

MOLECULAR NATURE OF MATTER-

  • The discovery of “Matter is made from atoms” by Richard Feynman, a physicist of 20th century.
  • Atomic Hypothesis- Each and every things made form atoms which move around in perpetual motion.
  • Perpetual motion- A system which move without stop and always move, but it is imaginary.
  • The size of atom is about 1 angstrom(10-10m).
  • Liquid can flow because the distance between atoms of liquid is almost same and not rigidly fixed like solid.
  • But in gas the interatomic distance more, and mean free path of gas is thousands of angstrom.
  • Mean Free Path- The average distance where the atom or molecule can move or travel without colliding.

GASES BEHAVIOUR-

  • The molecules of gases are very far to each other, so their interaction almost negligible.
  • Only two molecules can collide, it is exceptional case.
  • The relation of gas between their pressure, temperature and volume is PV = KT, where t is temperature( in Kelvin), K is constant for given sample which varies with volume of gas.
  • K is proportional for to number of molecule, N is sample, then K = N K. That means K is same for all gases. It is called Boltzmann constant, denoted by KB.
  • (P1V2)/(N1T1) = (P1V2)/(N2T2) = constant = KB, where N is same for every gas and P, V and T are same.
  • Avogadro ‘s Hypothesis- It states that equal volumes of gases, at the same temperature and pressure, contain an equal number of molecules.
    • Thus the number is 22.4 liter.
    • No. of atoms in 1 mole is 6.022 * 1023. This number is called Avogadro ‘s constant, denoted by NA. The amount of substance is called mole. [You Learnt in chapter 1].
    • The mass of 22.4 liter any ideal gas is equal to its molecular gas in gram at Standard Temperature 273K and Pressure 1atm.
  • Finally, we get the equation where μ is number of mole, T is absolute temperature(in K) and R = NA KB is universal constant.
  • R= 8.314 J mol-1 K-1.
  • μ = N / NA = M / M0, where N is molecule which contain by gas, M0 is molar mass, NA is Avogadro number, M is mass of gas.
  • PV = KB NT or P = KB nT, where n is no. of molecule of per unit volume(or number density), KB is Boltzmann constant(1.38 * 10-23 JK-1). We also write as P = ρRT / M0, where ρ is density of gass.
  • Ideal gas- It is a theoretical gas where particles move randomly, have no volume, and no intermolecular forces, obeying the ideal gas law: PV = n RT
  • Boyle ‘s Law- It state that for a fixed mass of gas at constant temperature, the pressure (PPP) is inversely proportional to its volume (V):
    • P ∝ 1 / V
    • or PV=constant.
  • Charles ‘s Law- It states that for a fixed mass of gas at constant pressure, the volume (V) is directly proportional to its absolute temperature (T):
  • V ∝ T
  • or V / T = constant.
  • Partial pressure of a gas is the pressure it would exert if it alone occupied the entire volume of the mixture.
  • Dalton’s Law of Partial Pressures states that the total pressure of a gas mixture is the sum of the partial pressures of all individual gases:
  • Ptotal = P1 + P2 + P3 +…P

Key Concepts of Kinetic Theory of Gases

  1. Molecular Picture of Matter:
    • Gases consist of a huge number of molecules in random motion.
    • Molecular separation is much larger than molecular size (approx. 10 times).
  2. Interactions and Collisions:
    • Interactions between molecules are negligible except during collisions.
    • Collisions are elastic, meaning no loss of kinetic energy or momentum.
    • Molecules collide with each other and container walls, changing their velocity.

Pressure of an Ideal Gas

  1. Pressure Derivation:
    • Gas molecules in a cube of side lll, move with velocity components vx​,vy​,vz​).
    • When a molecule collides elastically with a wall, the x-component of its velocity reverses.
    • Momentum change per collision: −2mvx​.
    • Momentum imparted to the wall per unit time depends on molecular velocity and number density (n).
  2. Expression for Pressure:
    • Force per unit area (Pressure P) is proportional to the square of the molecular velocity:
      • P= 1/3 ​nm⟨v2⟩
    • The factor 1/3 arises due to isotropic velocity distribution across x,y,z directions.
  3. Isotropic Nature:
    • Velocities in all directions are symmetric:
      • ⟨v2⟩= ⟨v2​x⟩+⟨v2y​⟩+⟨v2​z⟩

Kinetic Interpretation of Temperature

  1. Energy and Temperature:
    • Average translational kinetic energy: ⟨Kinetic Energy⟩==3/2 ​KB​T
      • KB​: Boltzmann constant.
      • T: Absolute temperature.
    • Internal energy of an ideal gas is purely kinetic and depends only on T.
  2. Relation Between Macroscopic and Microscopic Variables:
    • PV=2/3 ​E, linking pressure and volume to molecular energy.
  3. Root Mean Square (RMS) Speed:
    • vrms​= ⟨v2⟩1/2 =(3KB​T/m)​1/2
    • Lighter molecules move faster at the same temperature.

Gas Mixtures and Partial Pressures

  1. Dalton’s Law of Partial Pressures:
    • Total pressure in a mixture: P=(n1​+n2​+…)KB​T
    • Molecules of all gases have the same average kinetic energy if they are at the same T.

Practical Insights

  1. Diffusion:
    • Rate of diffusion (r) is inversely proportional to molecular mass(molecular mass)1/2​.
    • r1​/r2 ​= M2​/M1​ (Graham’s Law).
  2. Applications:
    • Gas separation: Faster molecules (lighter mass) diffuse more efficiently through porous materials.
    • Example: Enrichment of 235U in nuclear processes.
  3. Temperature Changes During Compression/Expansion:
    • Compressing gas: Molecules gain kinetic energy (temperature rises).
    • Expanding gas: Molecules lose kinetic energy (temperature falls).
  4. Sports Analogy:
    • Using a heavy cricket bat allows higher rebound speeds, akin to molecular collisions with a massive wall.

Important Formulae for Competitive Exams

  1. Pressure:P= 1/3 ​nm⟨v2⟩
  2. Kinetic Energy: K.E. (per molecule)= 3/2 ​KB​T
  3. Root Mean Square Speed: vrms​= [(3KB​T)/M]​1/2
  4. Graham’s Law: r1/r2​​= ​ (M2/M1​)1/2​​​​​

Key Takeaways for Problem-Solving

  1. Identify Gas Laws:
    • Relate P,V,T using ideal gas equations.
    • Use kinetic theory to interpret macroscopic behaviors.
  2. Consider Molecular Properties:
    • Use molecular mass for speed comparisons.
    • Apply energy and momentum conservation during collisions.
  3. Work With Units:
    • Keep units consistent (e.g., mass in kg, velocities in m/s).
  4. Understand Assumptions:
    • Elastic collisions, negligible intermolecular forces, isotropic velocity distribution.

Law of Equipartition of Energy (Simplified)

Core Concept:

The law of equipartition of energy states that in thermal equilibrium at temperature T, the total energy of a system is equally distributed across all its possible energy modes (degrees of freedom). Each degree of freedom contributes an average energy of  ½ ​KB​T, where  ​KB​​ is the Boltzmann constant.

Key Points:

  1. Kinetic Energy of a Gas Molecule:
    • For a gas molecule moving in 3D space, its kinetic energy is: ϵ= ½ mv2x + ½ mv2y + ½ mv2z
    • Here vx​,vy​,vz​​ are the velocity components along x,y,z-axes.
    • At thermal equilibrium, the average energy per degree of freedom is: ⟨1/2 ​mv2x​⟩= ½ ​KB​T.
  2. Degrees of Freedom:
    • Degrees of freedom refer to the independent ways a molecule can move or store energy.
    • Translational motion (movement in space):
      • 1 degree of freedom per direction (e.g., x,y,z).
      • A molecule in 3D has 3 translational degrees of freedom.
    • Rotational motion (spinning about an axis):
      • Monatomic molecules: No rotational degrees of freedom.
      • Diatomic molecules: 2 rotational degrees of freedom.
      • Polyatomic molecules: 3 rotational degrees of freedom.
    • Vibrational motion:
      • Each vibrational mode contributes 2 degrees of freedom (kinetic + potential energy).
  3. Energy Distribution:
    • Translational or rotational motion contributes ½ ​KB​T per degree of freedom.
    • Vibrational motion contributes 2 * ½ KBT= kBT2 because of the kinetic and potential energy components.
  4. Examples:
    • Monatomic gas (e.g., Argon, Helium):
      • 3 translational degrees of freedom.
      • Total energy per molecule:E=3/2 KBT
    • Diatomic gas (e.g., Oxygen, Nitrogen):
      • 3 translational + 2 rotational = 5 degrees of freedom.
      • Total energy per molecule: E=5/2 KBT
    • Polyatomic gas:
      • 3 translational + 3 rotational + f vibrational degrees of freedom.
      • Total energy: E= (3/2 ​+3/2​+f)kB​T
  5. Specific Heat and Energy:
    • Molar internal energy (U): U=degrees of freedom× ½ ​RT, where R=NAk​B​, the gas constant.
    • Specific Heat Capacity (Cv​​) at constant volume: Cv​= 1/n ∂U/​∂T
  6. Special Cases:
    • At higher temperatures, vibrational modes get excited, increasing energy storage capacity.
    • Real gases often deviate from theoretical predictions due to complex molecular interactions.

Additional Knowledge for Competitive Exams:

  1. Key Formulae:
    • PV=μRT (Ideal gas law).
    • Cp​−Cv​=R (Relation for specific heats).
    • γ= ​Cp​​/Cv​​ (Ratio of specific heats).
  2. Degrees of Freedom Table-
Types of gasTranslationalRotationalVibrationalTotal DoF
Monatomic3003
Diatomic321 (temp. dependent)5-7
Polyatomic33Varies>6

3. Important Constants:

  • R= 8.314J mol−1K−1
  • kB​=1.38×10−23J K−1

4.Applications:

  • Predicting specific heats for gases and solids.
  • Understanding molecular speeds: vrms​= (3kB​T/m​)1/2
  • Calculating mean free paths:l= 1/(2​πnd2)1/2

These simplified points provide a solid understanding of the law of equipartition of energy and its implications, especially useful for solving competitive questions.

These all are the notes of chapter 12 in physics. And after some time you get important questiions and NCERT solutions HERE. *#THANKS FOR VISITING, VISIT AGAIN#* 😊