What can happen to the reaction rate of a system that is heated and then cooled?
Your question isn't quite clear; however, reaction rates increase, generally, with increase in T and decrease with decrease in T. Another way to interpret the problem is that an increase in T may "climb the hill" to a point over the activation energy of the system where the reaction proceeds, then the reaction continues after it is cooled. If this isn't in the ball park, please repost and clarify.
To answer your question, let's assume we are talking about a chemical reaction. When you heat a system, you are increasing its temperature. As a result, the kinetic energy of the molecules in the system also increases.
The reaction rate of a chemical reaction depends on the collision frequency and the energy of the collisions between the reacting molecules. When the temperature is increased, the molecules move faster and collide more frequently. This results in an increased reaction rate because there are more successful collisions.
However, if you cool down the system after heating it, the opposite effect occurs. The decrease in temperature causes the molecules to slow down, reducing the collision frequency and energy. As a result, the reaction rate decreases.
It is important to note that the specific effect of heating and cooling on the reaction rate depends on the particular reaction and its energy requirements. For some reactions, a higher temperature may be necessary to overcome the activation energy, and cooling the system could potentially stop the reaction altogether.
In summary, heating a system generally increases the reaction rate by increasing the kinetic energy and collision frequency of the molecules. On the other hand, cooling the system after heating it reduces the reaction rate by decreasing the kinetic energy and collision frequency.