The acoustic relaxation is one important nature of gas, which is caused by the sound propagation in the polyatomic molecule gas. It is the basic relaxation process, which arises from the translational-vibrational degree of freedom (V-T) and the molecular energy transfer between different vibrational degrees of freedom (V-V) separately. By studying the molecule energy transition model of the basic acoustic relaxation processes of gas, we propose an analytic model reflecting the correspondence between effective specific heat capacity and relaxation time in this paper. Compared with the existing relaxation model, the analytic model provides the corresponding relationship between the vibrational specific heat capacity and the relaxation time in V-T and V-V. The solution procedure of the analytic model illustrates that the higher vibrational energy level is the determinant of the basic relaxation process. The effective heat capacity is the foundation of acoustic relaxation attenuation spectrum of gas. The relaxation attenuation spectra result from the analytic model in this paper, which is modified by fine-tuning the collision diameter of the gas molecule, are more consistent with the experiment data than with the existing theoretical value. It proves the correctness and validity of the analytic model.