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The object of this article is to present known information about the relation
between temperature and energy in a manner accessible to physical chemistry
students. Temperature, T,
can be defined as
 or  .
Combining this with the Boltzmann expression
gives an operational definition of temperature:
The temperature of a
collection of N harmonic oscillators, representing
vibrations of diatomic molecules, containing
n quanta of energy is calculated using
The validity
of this definition of temperature can
be demonstrated by calculating both the
temperature and the vibrational contribution
to heat capacity for N2 and showing that
they agree well with experiment. The calculated
heat capacity is not constant with temperature,
demonstrating that, contrary to the
ideal gas case, temperature is not generally
proportional to energy. Temperature can
be calculated exactly even for systems
containing many moles of oscillators. The
lowest possible nonzero vibrational temperature
of a mole of hydrogen molecules is calculated
to be 109 K. A two-level system is used
to illustrate negative Kelvin temperatures.
That system also provides an analogy to
account for the observation that systems
with a limited numbers of energy levels,
such as F2, (i) have heat capacities that
reach a maximum, then decrease with temperature;
and (ii) have a maximum heat capacity that
is not a multiple of R/2.
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