A series expansion for the free energy of the q-state Potts model on a square lattice is used to estimate the specific heat critical exponents. The analysis is based on a series transformation which was suggested by the known solution of the two-state Potts (Ising) model, and which makes optimum use of the duality theorem. The transformed series is quite smooth. Neville tables yield the estimates alpha (2)=0.0001+or-0.0003 for the two-state model, alpha (3)=0.296+or-0.002 for the three-state model, and alpha (4)=0.45+or-0.02 for the four-state model. The value for alpha (3) differs considerably from one reported by Straley and Fisher (1973), and substantially improves compliance with the Rushbrooke inequality. At the time of writing, John Ramshaw was employed at University of Maryland, College Park.