Classical ideal-gas entropy limits ($S_m=S_0-R\ln (p/p_0)+C_{mp}\ln (T/T_0)$)

For constant positive molar heat capacity, the ideal gas entropy grows without bound on isothermal dilution to unbounded volume, corresponding to $p\to 0$. This is a meaningful limiting configuration count, not an infinite entropy in a finite container. Extrapolation to $T\to 0$ at fixed positive pressure instead predicts negative infinite entropy. That extrapolation fails: phase changes, quantum statistics and temperature-dependent heat capacities prevent a real equilibrium substance from retaining the same classical formula down to zero temperature.