= Classical ideal-gas entropy limits
{title2=$S_m=S_0-R\ln(p/p_0)+C_{mp}\ln(T/T_0)$}
For constant positive <molar heat capacity>, the <ideal gas> <entropy> grows without bound on isothermal dilution to unbounded <volume>, corresponding to $p\to0$. This is a meaningful limiting configuration count, not an infinite <entropy> in a finite container. Extrapolation to $T\to0$ 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>.
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