  In this work, we demonstrate the inappropriateness of the Boltzmann-Gibbs
log-formulation of the physical Clausius entropy $S$ in connecting
thermodynamics and phase space statistics. To achieve our goal, we study
thermodynamically the simple case of ideal gases embedded in a finite heat bath
and compare the derived equations with the phase space statistical ones
obtained within the canonical ensemble. We then show that the results of the
aforementioned approaches contradict to each other even in the dilute gas limit
(infinite heat bath) if we request the identification of the Boltmann-Gibbs
formula $\ln(Density\,\, of\,\, States)$ with the thermodynamic entropy $S$.
