  The quantum corrections related to the ideal gas model that are often
considered are those which are related to the particles nature: bosons or
fermions. These corrections lead respectively to the Bose-Einstein and
Fermi-Dirac statistics. However, in this work, other kinds of corrections which
are related to the quantum nature of phase space are considered. These
corrections are introduced as improvement in the expression of the partition
function of an ideal gas. Then corrected thermodynamics properties of the gas
are deduced. Both the non-relativistic quantum and relativistic quantum cases
are considered. It is shown that the corrections in the non-relativistic
quantum case may be particularly useful to describe the deviation from
classical behavior of a Maxwell-Boltzmann gas at low temperature and in
confined space. These corrections can be considered as including the
description of quantum size and shape effects. For the relativistic quantum
case, the corrections could be relevant for confined space and when the thermal
energy of each particle is comparable to their rest energy. The corrections
appear mainly as modifications in the thermodynamic equation of state and in
the expressions of the partition function and thermodynamic functions like
entropy, internal energy, and free energy. Classical expressions are obtained
as asymptotic limits.
