Kinetic Theory of Gases: Distribution Laws

  • Recap: Kinetic Theory of Gases (Part 1)

    • Discussed heat, temperature, and the ideal gas law from a molecular perspective.

    • Examined particles (molecules/atoms) colliding with container walls.

    • Used Newton's laws (momentum, force) to derive energy concepts.

    • Equipartition Theorem: Energy is distributed equally among degrees of freedom.

      • Each degree of freedom possesses an average energy of \frac{1}{2}kT.

    • Average Kinetic Energy (Translational):

      • Monatomic gas (e.g., Helium): \frac{3}{2}kT (3 degrees of freedom for x, y, z motion).

      • Diatomic gas (e.g., Nitrogen, Oxygen, CO): \frac{5}{2}kT (3 translational + 2 rotational degrees of freedom). Rotational modes often pick up two extra degrees of freedom, assuming non-linear rotation around one axis.

      • Polyatomic gas: \frac{6}{2}kT (3 translational + 3 rotational degrees of freedom) or \frac{7}{2}kT if vibrational modes are present (each vibrational mode adds \frac{1}{2}kT for kinetic and \frac{1}{2}kT for potential energy, totaling kT).

    • Root Mean Square (RMS) Velocity (v_{rms}):

      • Derived by equating translational kinetic energy to \frac{1}{2}mv^2: \frac{3}{2}kT = rac{1}{2}m v_{rms}^2.

      • v_{rms} = \sqrt{\frac{3kT}{m}} where m is the mass of a single particle.

      • This is one type of average velocity; other averages (mean velocity) exist but are numerically similar, often differing by a factor close to 3 (e.g., \pi).

  • New Questions for Kinetic Theory (Part 2)

    • Velocity Distribution: How many particles move faster or slower than v_{rms}? More specifically, what is the distribution of velocities among all particles at a given temperature?

      • A simple answer like