The fluctuating charge (FQ) force field, a polarizable potential model in which point charges on atomic sites fluctuate in response to the environment, is applied to the aqueous solvation of acetamide and trans- and cis-N-methylacetamide (NMA). Two parameters are assigned to each kind of atom, corresponding to an electronegativity and a hardness. The FQ model accurately reproduces both the gas-phase and aqueous-phase charge distributions of these molecules and is therefore effective in treating the influence of functional group substitutions, conformation changes, and solvent on the charge distribution. The FQ model does this in a way that is computationally efficient by propagating the charges in time using an extended Lagrangian method. Solvation free energy calculations are done using both an explicit (FQ) solvent and a dielectric continuum solvent. Both solvent models predict a negligible free energy difference between trans- and cis-NMA, in agreement with experimental estimates. The explicit solvent calculations find that the free energy difference between NMA and acetamide is 0.5 +/- 0.8 kcal/mol, in good agreement, but of opposite sign, with the measured value of -0.3 +/- 0.4 kcal/mol. The FQ/dielectric continuum calculations find that this free energy difference is sensitive to the details of the solute cavity, varying from 0.4 to 1.5 kcal/mol.