In order to obtain a better fit for the observational data, various cosmological models have been developed in the literature. For this purpose, we propose an adiabatic expansion of the Universe with a polytropic fluid matter-radiation content and a varying cosmological constant. Basic motivation of this study basically differs from the previous models because our model does not neglect the pressure of the matter content of the Universe. We present the details of our new derivations by obtaining the polytropic matter density and luminosity distances. After constructing the model, two large collections of observational data of supernovae type Ia (SNe Ia) are conducted to diagnose the model. Examining the fitness of our new models of a polytropic universe with the observational data, we find a significant influence of polytropic matter on the cosmological parameters of the expanding universe. We find that our flat constant ? model and varying ? model describes the SNe Ia data significantly better than the other models including the standard model for Gold Data and Union2.1 Data, respectively.