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PHCC 141: Physics for Scientists and Engineers I  Fall 2007
18. Thermal Properties of Matter
Due at 11:59pm on Monday, November 26, 2007
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Equipartition Theorem and Microscopic Motion
Learning Goal:
To understand the Equipartition Theorem and its implications for the mechanical motion of small objects.
In statistical mechanics, heat is the random motion of the microscopic world. The average kinetic or potential energy of each
degree of freedom of the microscopic world therefore depends on the temperature. If heat is added, molecules increase their
translational and rotational speeds, and the atoms constituting the molecules vibrate with larger amplitude about their
equilibrium positions. It is a fact of nature that the energy of each degree of freedom is determined
solely by the temperature
.
The Equipartition Theorem states this quantitatively:
The average energy associated with each degree of freedom in a system at absolute temperature
is
, where
is Boltzmann's constant.
The average energy of the
i
th degree of freedom is
, where the angle brackets represent "average" or "mean" values
of the enclosed variable. A "degree of freedom" corresponds to any dynamical variable that appears quadratically in the energy.
For instance,
is the kinetic energy of a gas particle of mass
with velocity component
along the
x
axis.
The Equipartition Theorem follows from the fundamental postulate of statistical mechanicsthat every energetically accessible
quantum state of a system has equal probability of being populated, which in turn leads to the Boltzmann distribution for a
system in thermal equilibrium. From the standpoint of an introductory physics course, equipartition is best regarded as a
principle that is justified by observation.
In this problem we first investigate the particle model of an ideal gas. An ideal gas has no interactions among its particles, and so
its internal energy is entirely "random" kinetic energy. If we consider the gas as a system, its internal energy is analogous to the
energy stored in a spring. If one end of the gas container is fitted with a sliding piston, the pressure of the gas on the piston can
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