Distribution of Molecular SpeedsThe distribution of molecular speeds in an ideal gas can be found from Eq. (9.11) by
making the substitutions^12 m^2 dmdThe result for the number of molecules with speeds between and disn() d 4 N
3 2
^2 em(^2) 2 kT
d (9.14)
This formula, which was first obtained by Maxwell in 1859, is plotted in Fig. 9.3.
The speed of a molecule with the average energy of ^32 kTis
rms^2 (9.15)
since 21 m^2 ^32 kT. This speed is denoted rmsbecause it is the square root of the average
of the squared molecular speeds—the root-mean-square speed—and is not the same
as the simple arithmetical average speed . The relationship between and rmsde-
pends on the distribution law that governs the molecular speeds in a particular sys-
tem. For a Maxwell-Boltzmann distribution the rms speed is about 9 percent greater
than the arithmetical average speed.
3 kT
m
RMS speed
m
2 kT
Molecular-speed
distribution
Statistical Mechanics 303
Equipartition of Energy
A
gas molecule has three degrees of freedomthat correspond to motions in three independ-
ent (that is, perpendicular) directions. Since the average kinetic energy of the molecule is ^32 kT
we can associate ^12 kTwith the average energy of each degree of freedom: ^12 mx^2 ^12 my^2 ^12 mz^2
^12 kT. This association turns out to be quite general and is called the equipartition theorem:
The average energy per degree of freedom of any classical object that is a member of a
system of such objects in thermal equilibrium at the temperature T is^12 kT.
Degrees of freedom are not limited to linear velocity components—each variable that appears
squared in the formula for the energy of a particular object represents a degree of freedom. Thus
each component iof angular velocity (provided it involves a moment of inertia Ii), is a degree
of freedom, so that ^12 Ii
i
2
^12 kT. A rigid diatomic molecule of the kind described in Sec. 8.6
therefore has five degrees of freedom, one each for motions in the x,y, and zdirections and two
for rotations about axes perpendicular to its symmetry axis.
A degree of freedom is similarly associated with each component siof the displacement
of an object that gives rise to a potential energy proportional to (si)^2. For example, a one-
dimensional harmonic oscillator has two degrees of freedom, one that corresponds to its kinetic
energy ^12 mx^2 and the other to its potential energy ^12 K(x)^2 , where Kis the force constant. Each
oscillator in a system of them in thermal equilibrium accordingly has a total average energy of
2(^12 kT) kTprovided that quantization can be disregarded. To a first approximation, the con-
stituent particles (atoms, ions, or molecules) of a solid behave thermally like a system of classi-
cal harmonic oscillators, as we shall see shortly.
The equipartition theorem also applies to nonmechanical systems, for instance to thermal
fluctuations (“noise”) in electrical circuits.
bei48482_Ch09.qxd 2/4/02 11:51 AM Page 303