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Aims. We investigate the stability of non-isothermal Bonnor-Ebert sphereswith a model that includes a self-consistent calculation of the gastemperature. This way we can discard the assumption of equality between thedust and gas temperatures, and study the stability as the gas temperaturechanges with chemical evolution of the gas.  Methods. We use a gas-grain chemical model including a time-dependenttreatment of depletion onto grain surfaces, which strongly influences the gastemperature as the main coolant, CO, depletes from the gas. Dust and gastemperatures are solved with radiative transfer. For comparison with previouswork, we assume that the cores are deeply embedded in a larger externalstructure, corresponding to visual extinction $A_{\rm V}^{\rm ext}=10$ mag.  Results. We find that the critical non-dimensional radius $\xi_1$ derivedhere is similar to our previous work where we assumed $T_{\rm dust}=T_{\rmgas}$; the $\xi_1$ values lie below the isothermal critical value$\xi_0\sim6.45$, but the difference is less than 10%. Chemical evolution doesnot affect notably the stability condition of low-mass cores (<0.75 $M_\odot$).For higher masses the decrease of cooling owing to CO depletion causessubstantial temporal changes in the temperature and density profiles of thecores. In the mass range 1-2 $M_\odot$ , $\xi_1$ decreases with chemicalevolution, whereas above 3 $M_\odot$ , $\xi_1$ instead increases. We also findthat decreasing $A_{\rm V}^{\rm ext}$ increases the gas temperature especiallywhen the gas is chemically old, causing $\xi_1$ to increase with respect tomodels with higher $A_{\rm V}^{\rm ext}$. The derived $\xi_1$ values are closeto $\xi_0$. The density contrast between the core center and edge variesbetween 8 to 16 depending on core mass and the chemical age of the gas,compared to the constant value $\sim$ 14.1 for the isothermal BES.

On the stability of non-isothermal Bonnor-Ebert spheres. II. The effect of gas temperature on the stability