  Lie-algebraic and quantum-algebraic techniques are used in the analysis of
thermodynamic properties of molecules and solids. The local anharmonic effects
are described by a Morse-like potential associated with the $su(2)$ algebra. A
vibrational high-temperature partition function and the related thermodynamic
potentials are derived in terms of the parameters of the model. Quantum
analogues of anharmonic bosons, $q$-bosons, are introduced and used to describe
anharmonic properties of molecules and solids. It is shown that the quantum
deformation parameter is related to the fixed number of anharmonic bosons and
the shape of the anharmonic potential. A new algebraic realization of the
$q$-bosons, for the case of $q$ being a root of unity is given. This
realization represents the symmetry of a linear lattice with periodic boundary
conditions.
