  An ideal Maxwell-Boltzmann gas confined in various rectangular nano domains
is considered under quantum size effects. Thermodynamic quantities are
calculated from their relations with partition function which consists of
triple infinite summations over momentum states in each direction. To get
analytical expressions, summations are converted to integrals for macro systems
by continuum approximation which fails at nanoscales. To avoid both from the
numerical calculation of summations and the failure of their integral
approximations at nanoscale, a method which gives an analytical expression for
single particle partition function (SPPF) is proposed. It's shown that
dimensional transition in momentum space occurs at certain magnitude of
confinement. Therefore, to represent SPPF by lower-dimensional analytical
expressions becomes possible rather than numerical calculation of summations.
Considering rectangular domains with different aspect ratios, comparison of the
results of derived expressions with those of summation forms of SPPF is done.
It's shown that analytical expressions for SPPF give very precise results with
maximum relative errors of around 1%, 2% and 3% at just the transition point
for single, double and triple transitions respectively. Based on dimensional
transitions, expressions for free energy, entropy, internal energy, chemical
potential, heat capacity and pressure are given analytically valid for any
scale.

Statistical Mechanics;Quantum Gases