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# 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, 1.3806488 × 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 (vp < ⟨v⟩ < vrms).
• 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
Molecular Speeds