  This work investigates the magnetic properties and the magnetocaloric effect
in the spin-1 Blume-Capel model. The study was carried out using the mean-field
theory from the Bogoliubov inequality to obtain the expressions of free energy,
magnetization and entropy. The magnetocaloric effect was calculated from the
variation of the entropy obtained by the mean-field theory. Due to the
dependence on the external magnetic field and the anisotropy included in the
model, the results for the magnetocaloric effect provided the system with
first-order and continuous phase transitions. To ensure the results, the
Maxwell relations were used in the intervals where the model presents
continuous variations in magnetization and the Clausius-Clapeyron equation in
the intervals where the model presents discontinuity in the magnetization. The
methods and models for the analysis of a magnetic entropy change and
first-order and continuous magnetic phase transitions, such as mean-field
theory and the Blume-Capel model, are useful tools in understanding the nature
of the magnetocaloric effect and its physical relevance.
