The Schrödinger wave equation with the generalized Cornell potential model parameterized for a thermal medium at finite temperature was studied using Nikiforov Uvarov method. The bound state energy and mass spectra were obtained. The results were applied for the charmonium and bottomonium mesons to offer phenomenological predictions for their mass spectra, in order to understand their disassociation property of their mass spectra within the temperature range 0.025 GeV ≤ T ≤ 0.41 GeV, with the critical the temperature \({T}_{c}\) as 0.169 GeV. Also, the Hellmann–Feynman Theorem was used to obtain the expectation values of the square of inverse of position \(\langle {r}^{-2}\rangle \) , the kinetic energy \(\langle T\rangle \) , and the square of momentum \(\langle {p}^{2}\rangle \) for the quantum state as the charmonium and bottomonium. The phenomenological predicted results were compared with experimental data, yielding a percentage error of approximately 2.11% for charmonium and 0.84% for bottomonium. The mass spectra for the charmonium meson for the quantum states showed no complete indication of disassociation, for temperature values up to \(2.43{T}_{c}\) , while for the bottomonium meson there was a complete disassociation of its mass spectrum for all their corresponding quantum states, for temperature values at \(2.43{T}_{c}\) . The expectation values showed similar characteristics between the charmonium and bottomonium mesons for same quantum states, with the charmonium meson leading only in the expectation values for \(\langle T\rangle \) , but lagging behind the bottomonium meson in the expectation values for \(\langle {r}^{-2}\rangle \) and \(\langle T\rangle \) .
