Langmuir monolayers of long-chain fatty acids display a wide range of interesting flow and phase behaviors, but the correlation between these behaviors varies. In this paper, we report two interesting behaviors in a Langmuir monolayer of 2-hydroxy-tetracosanoic acid: phase coexistence between the L-2h and L-2' phases over a large pressure range and a peak in the viscosity and elastic modulus of the material as a function of pressure. The two behaviors are confirmed over a range of temperatures for which the location of the undistorted lattice appears to vary relative to the range of coexistence for the two phases. Evidence for a correlation between the phase behavior and the peak in the mechanical properties is presented.