Sommerfeld has given an apparent case of a perpetual motion machine of the second kind. This consists of a Carnot engine employing liquid water and operating between the normaland anomalous regions of thermal expansion. His explanation of the paradox is shown to be incomplete when the temperature of maximum density is pressure-dependent. To analyze thiscase a simple thermodynamic model for a substance with a density extremum is given; this model yields a reasonable approximation to the data for water. Standard thermodynamicproperties of the system are computed and useful approximate forms given. Various Carnot cycles and a non-trivial “two-process” cycle are then shown in thep−T,T−s, andp−vplanes. Sommerfeld's paradox is resolved by showing that a Carnot cycle qualitatively similar to thatin his problem involvesexpansionsforbothisothermal processes. Theoretical implications of the analysis and applications to sound waves and shock waves are briefly discussed.