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We present the evolution of the initial (up to ~ 10 kbar) hydrostatic, pressure dependencies of T_c and of the ambient pressure jump in the heat capacity associated with the superconducting transition as a function of Na - doping in the Ba(1-x)NaxFe2 As2 family of iron-based superconductors. For Na concentrations 0.15 <= x <= 0.9, the jump in specific heat at T_c, Delta C_p, follows the Delta C_p ~ T^3 scaling found for most BaFe2As2 - based superconductors. Pressure dependencies are non-monotonic for x = 0.2 and x = 0.24. For other Na concentrations T_c decreases under pressure in almost linear fashion. The anomalous behavior of the x = 0.2 and x = 0.24 samples under pressure are possibly due to the crossing of the phase boundaries of the narrow antiferromagnetic tetragonal phase, unique for the Ba(1-x)NaxFe2As2 series, with the application of pressure.
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