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Kinetic Theory of Gases - Summary

Kinetic Theory of Gases

Molecular Speeds and Ideal Gas Law

  • Ideal gas law: PV = nRT
  • P: Force exerted by molecules on the wall during collisions.
  • V: Available volume for molecules.
  • T: Indicates the speed of molecules.
  • Gas molecules are in random and continuous motion.
  • Collisions redistribute speed among molecules.

Molecular Momentum and Force

  • Change in momentum for one molecule: Δ(mvx) = 2mvx
  • Time between collisions: Δt = \frac{2l}{v_x}
  • Force due to a single molecule: F = \frac{mv_x^2}{l}
  • Average force due to all molecules in x-direction: Fx = \frac{m}{l}N \overline{vx^2}
  • Mean-square of vx: \overline{vx^2} = \frac{(v{x1}^2 + v{x2}^2 + v{x3}^2 + … + v{xN}^2)}{N}

Pressure and Mean Squared Speed

  • \overline{v^2} = \overline{vx^2} + \overline{vy^2} + \overline{vz^2} = 3\overline{vx^2}
  • Pressure: P = \frac{F_x}{A} = \frac{1}{3} \frac{Nm\overline{v^2}}{V}

Mean Translation Energy

  • PV = \frac{1}{3}Nm\overline{v^2} = \frac{2}{3}N(\frac{1}{2}m\overline{v^2})
  • Nm = nM (N = number of molecules, m = mass of each molecule, n = number of moles, M = molar mass)
  • Mean translation energy: E{trans} = \frac{3}{2}kT (per molecule), E{trans} = \frac{3}{2}RT (per mole)

Root-Mean-Square (rms) Speed

  • v_{rms} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}
  • Lighter molecules move at higher rms speed.
  • Different gases have different rms speeds at a given temperature.

Equipartition of Energy

  • E_{trans} = \frac{3}{2}kT = \frac{1}{2}kT + \frac{1}{2}kT + \frac{1}{2}kT (x, y, z directions)
  • Each degree of freedom contributes \frac{1}{2}kT to the energy.
  • Degrees of freedom: Translation, Vibration, Rotation.

Degrees of Freedom

  • Atom: 3 (translation)
  • Molecule (non-linear): 3 (translation), 3 (rotation), 3N-6 (vibration)
  • Molecule (linear): 3 (translation), 2 (rotation), 3N-5 (vibration)

Maxwell Speed Distribution

  • Molecular collisions redistribute speeds.
  • Maxwell distribution describes speeds in x, y, and z directions.

Boltzmann Distribution Law

  • P(h) = P_0 \exp(-\frac{mgh}{kT}) relates air pressure with altitude.
  • Boltzmann factor: \exp(-\frac{E}{kT}) relates the number of molecules in a given state with the energy of that state.
  • \frac{Ni}{Nj} = \exp(-\frac{Ei - Ej}{kT})
  • Fraction of particles in the ith level: p(i) = \frac{Ni}{N} \propto \exp(-\frac{Ei}{kT})

Maxwell Speed Distribution (1D and 3D)

  • p(vx) = K \exp(-\frac{m vx^2}{2kT})
  • p(vx, vy, vz) = (\frac{m}{2\pi kT})^{3/2} \exp(-\frac{m(vx^2 + vy^2 + vz^2)}{2kT})
  • p(v)dv = 4\pi (\frac{m}{2\pi kT})^{3/2} \exp(-\frac{m v^2}{2kT}) v^2 dv

Average Values

  • g(x) = \sum p(xj)g(xj)
  • Most probable speed: v_{mp} = \sqrt{\frac{2RT}{M}} = \sqrt{\frac{2kT}{m}}
  • Average speed: \overline{v} = \sqrt{\frac{8kT}{m\pi}} = \sqrt{\frac{8RT}{\pi M}}
  • Root-mean-square speed: v_{rms} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}
  • v{rms} > v{ave} > v_{mp}

Molecular Collisions

  • Collision cross-section area: \sigma{AB} = \pi d{AB}^2
  • Single-particle collision frequency: z{AB} = \etaB \sigma{AB} v{rel}
  • Total collision frequency: Z{AB} = \etaA z{AB} = \etaA \etaB \sigma{AB} v_{rel}

Collision Frequency

  • v{rel} = \sqrt{(v{A,ave})^2 + (v_{B,ave})^2}
  • If A = B: z{AA} = \sqrt{2} \etaA \sigma{AA} v{A,ave}; Z{AA} = \frac{1}{2} \sqrt{2} \etaA^2 \sigma{AA} v{A,ave}

Mean Free Path

  • \lambda = \frac{v_{ave}}{z} = \frac{1}{\sqrt{2} \eta \sigma}

Molecular Diffusion/Effusion

  • Diffusion: Molecules move from high to low concentration.
  • Effusion: Escape of molecules through a tiny hole.

Graham's Law

  • \frac{r1}{r2} = \sqrt{\frac{M2}{M1}}