No, the temperature of the system will not stay the same in this scenario. Conservation of energy does not imply that the temperature of a system remains constant during a chemical reaction.
In the given reaction A -> 2B, if there is no enthalpy change, it means that the reaction is thermally neutral. However, this does not mean that there is no energy exchange occurring within the system. The reaction still involves the breaking and formation of chemical bonds, which can result in energy transfer.
When the reaction starts, the conversion of A to B will release energy, typically in the form of heat. This release of energy can cause an increase in the temperature of the system. The actual change in temperature will depend on various factors, including the specific reaction kinetics, reactant concentrations, and heat capacity of the container.
In an isolated or closed container, without any heat exchange with the surroundings, the temperature of the system will generally increase due to the exothermic nature of the reaction (assuming the reaction is exothermic). However, it is important to note that the reaction rate can also be influenced by temperature, so as the temperature rises, the reaction rate may change, potentially leading to dynamic equilibrium.
Overall, conservation of energy does not guarantee a constant temperature during a chemical reaction. The energy released or absorbed during the reaction can cause a change in the system's temperature.