  Due to one of the most representative contributions to the energy in diatomic
molecules being the vibrational, we consider the generalized Morse potential
(GMP) as one of the typical potential of interaction for one-dimensional
microscopic systems, which describes local anharmonic effects. From Eckart
potential (EP) model, it is possible to find a connection with the GMP model,
as well as obtain the analytical expression for the energy spectrum because it
is based on $S\,O\left(2,1\right)$ algebras. In this work we find the
macroscopic properties such as vibrational mean energy $U$, specific heat $C$,
Helmholtz free energy $F$ and entropy $S$ for a heteronuclear diatomic system,
along with the exact partition function and its approximation for the high
temperature region. Finally, we make a comparison between the graphs of some
thermodynamic functions obtained with the GMP and the Morse potential (MP) for
$H\,Cl$ molecules.
