Kinetic-Molecular Theory

Summary

• The kinetic molecular theory (KMT)…
  • is a theory of ideal gases
  • can be used to deduce the properties of gases
  • can be applied to other systems such as free electrons in a metal
  • is sometimes called the molecular kinetic theory (MKT)
• Postulates
  • All matter is composed of particles (molecules in general, but also atoms, ions, and free electrons).
  • Molecules are very small relative to the distances between them.
  • Molecules are in constant random (chaotic) motion.
  • Collisions between molecules are perfectly elastic.
• Equipartition of energy
  • The time-averaged kinetic energy of the molecules in a gas…
    • is divided equally among all the possible degrees of freedom
      • For a monatomic gas, there are 3 degrees of freedom, one for each spatial direction (x, y, z)
      • For a diatomic gas, there are 5 degrees of freedom, one for each spatial direction (x, y, z) plus one for each rotational axis (θ, φ).
    • is equal for every kind of molecule in a mixture of gases
      • On average, heavier molecules move slower and lighter molecules move faster.
• A proper discussion of KMT includes statistics.
  • Time-averaged quantities are written in angle brackets.
• Pressure
  • Absolute pressure is the time-averaged rate per unit of area at which momentum is changed when the molecules of a gas collide with the walls of its container.
• Temperature
  • Absolute temperature is proportional to the time-averaged kinetic energy of the molecules in a gas, ⟨K⟩ ∝ T.

  Mean kinetic energy

  monatomic	diatomic	general
  ⟨K⟩ = 32kT	⟨K⟩ = 52kT	⟨K⟩ = n2kT
  3 spatial, 0 rotational degrees of freedom	3 spatial, 2 rotational degrees of freedom	n degrees of freedom

• Molecular speeds are described by the Maxwell-Boltzmann distribution…

  p(v) =	4v2	⎛ ⎜ ⎝	m	⎞ ⎟ ⎠	32	e	−  mv2 2kT
  	√π		2kT

  where…

  p(v) =	value of the probability distribution [no unit]
  v =	speed of the molecules in an ideal gas [m/s]
  T =	absolute temperature of the gas [K]
  k =	Boltzmann constant,1.380649 × 10−23 J/K
  π =	a familiar transcendental number, 3.141592653…
  e =	a less familiar transcendental number, 2.718281828…

  • The curve of the Maxwell-Boltzmann distribution…
    • resembles a bell curve
    • has a positive skew (most values are greater than the most probable value)
    • has no negative values
    • has a total area under the curve of 1
  • The probability of finding a molecule with a speed in a certain range is equal to the area under the that section of the curve.
  • The most probable speed occurs at the maximum value of the distribution.
  • Higher temperature shifts the peak of the curve…
    • "right" — higher temperature increases the most probable speed
    • "down" — higher temperature increases the statistical dispersion (the curve is flatter and wider)
  • The measures of central tendency are not all the same (v_p < ⟨v⟩ < v_rms).

Molecular Speeds

most probable speed	mean speed	root mean square speed
vp = √ 2kT m	⟨v⟩ = √ 8kT πm	vrms = √ 3kT m
vp = 1 vp	⟨v⟩ =  2  vp √π	vrms = √ 3  vp 2