Kinetic Theory
PHYS 4315
R. S. Rubins, Fall 2009
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Kinetic Theory: Introduction
• Features of kinetic theory
• Kinetic theory goes beyond the limitations of classical
thermodynamics by taking into account the structures of
materials.
However, there is a price to be paid in complexity, since
general relationships may be obscured.
• Unlike both classical thermo and statistical mechanics, kinetic
theory may be used to describe non-equilibrium situations.
• Kinetic theory has been particularly useful in describing the
properties of dilute gases.
• It gives a deeper insight into concepts such as pressure,
internal energy and specific heat, and explains transport
processes, such as viscosity, heat conduction and diffusion.
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Classical Dilute Gas Assumptions 1
• Molecules are classical particles with well-defined positions
and momenta.
• A macroscopic volume contains an enormous number of
molecules.
• At STP (0oC and 1 atm), 1 kmole of a gas occupies 22.4 m3,
which is a density of 3 x 1025 molecules/m3.
• Molecular separations are much larger than both their
dimensions and the range of intermolecular forces.
• At STP, the molecular separation is roughly 3 x 10– 9 m.
• The Lennard-Jones or 6-12 potential
V = K [(d/r)12 – (d/r)6],
where K and d are empirical constants is negligible for
molecular separations of 3 x 10– 9 m.
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Classical Dilute Gas Assumptions 2
• The mean free path – the average distance a molecule
moves between collisions is of the order 10– 7 m.
• The only interaction between particles occur during
collisions so brief that their durations may be neglected.
• The container is assumed to have an idealized surface.
• The collisions are elastic, so that momentum and KE are
conserved.
• The molecules are assumed to be distributed uniformly in
both position and velocity direction.
• Molecular chaos exists; that is, the velocity of a molecule is
uncorrelated with its position.
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Kinetic Theory: Pressure
• Macroscopic pressure (classical thermo)
Balancing force F = PA.
• Microscopic pressure (kinetic theory)
Impulse FΔt = Δptot,
where, for a single particle,
Δp = 2mv cosθ.
The piston makes an irregular
(Brownian) motion about its equilibrium
position, because of the random
collisions of molecules with the piston.
Macroscopic
pressure
θ
θ
Microscopic
pressure5
Molecular Flux
A
x
vaveΔt
Simplified calculation
• Assume that the molecules move equally in the ±x, ±y, and ±z
directions; i.e. n/6 move in the +x direction, where n is the
number of molecules per unit volume.
• In time Δt, a total of AnvaveΔt/6 molecules will strike the wall.
• Thus, the molecular flux Φ, the number of molecules striking
unit area of the wall in unit time, is Φ = nvave/6.
• An exact calculations, including integrations over all directions
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and speeds gives Φ = nvave/4.
Molecular Effusion
• Φ = nvave/4, where (from stat. mech.) vave = √(8kT/mπ).
• For an ideal gas, PV = NkT, so that n = N/V = P/kT.
• Thus,
Φ = P/√(2πmkT).
• Imagine two containers containing the same ideal gas at
(P1,T1) and (P2,T2), connected by a microscopic hole of
diameter D << L, where L is the mean free path, which is of
the order 10–7 m at STP.
• The number of molecules passing through the hole is so small
that the pressure and the temperature on each side of the
hole is unchanged over the time of the experiment.
• The condition for equilibrium is the absence of a net flux; i.e.
Φ1 = Φ2 or P1/√T1 = P2/√T2.
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Experimental Problem
Macroscopic hole (D >> L): At equilibrium, P1 = P2, T1 = T2.
Microscopic hole (D << L): At equilibrium, P1/√T1 = P2/√T2.
T2 = 300 K
vacuum
P2
ΔP
Hg
Liquid He
He vapor at
pressure P1
Liquid He at
T1 = 1.3 K
• Under effusion
conditions,
P1/√T1 = P2/√T2 ,
so that
P1 = P2/ √(T1/T2) ,
or
P1 ≈ 0.07 P2.
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Boltzmann’s Transport Equation
• Boltzmann’s H theorem
dH/dt ≤ 0, where
H(t) = ∫d3v f(p,t) log f(p,t) .
• Since H → Hmin, the entropy has the form S = – const. H.
• The equilibrium (Maxwell) distribution is given by
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