The increasing demand for sustainable and low-cost catalysts for biofuel upgrading has stimulated research into the utilization of agricultural wastes such as rice husk and eggshell for catalyst synthesis. This paper discusses the preparation, characterization, and catalytic potential of NiCo/SiO₂and CaO/Al₂O₃. Catalysts synthesized from rice husk and eggshell, respectively. Rice husk serves as an abundant renewable source of silica, while eggshell is a rich source of calcium carbonate that can be converted into calcium oxide through calcination. The synthesized catalysts were characterized using X-ray diffraction (XRD), X-ray fluorescence (XRF), Brunauer–Emmett–Teller (BET) surface area analysis, and thermogravimetric analysis (TGA) to evaluate their structural, thermal, and surface properties. Results revealed that the catalysts possess high surface area, mesoporous structures, and good thermal stability suitable for heterogeneous catalytic applications. NiCo/SiO₂ catalyst exhibited enhanced surface area and porosity due to silica derived from rice husk, while The CaO/Al₂O₃ catalyst showed excellent basic characteristics. The study highlights the importance of converting agricultural waste into valuable catalytic materials for biofuel production and environmental sustainability. The utilization of waste-derived catalysts offers economic, environmental, and industrial benefits by reducing waste disposal problems and promoting sustainable energy development.

The relationship between reaction rate constant and half-life is fundamental to understanding chemical kinetics and predicting reaction behaviour. Although expressions describing half-life for zero-, first-, and second-order reactions are well established, comparative visualization across different reaction orders is seldom presented. This study provides a mathematical comparison of half-life as a function of reaction rate constant and the corresponding inverse relationship for zero-, first-, and second-order reactions. Calculations were performed using standard integrated rate equations assuming identical initial reactant concentration of 1 M. The resulting plots demonstrate that half-life decreases nonlinearly with increasing rate constant for all reaction orders. At any given rate constant, second-order reactions exhibit the longest half-life, followed by first-order and zero-order reactions. Conversely, the rate constant decreases with increasing half-life, producing inverse relationships characteristic of each kinetic order. The results emphasize the distinct kinetic behaviour associated with reaction order and provide a useful graphical framework for interpreting experimental kinetic data. These findings may serve as references for professionals involved in reaction kinetics.