How can I measure when a solid releases the greatest amount of heat per mole?

You can use calorimetry. This is a process in which a
substance is added to water which is in a well insulated container. A thermometer is
placed in the water and when the substance is added the temperature change is recorded
over a period of time. By graphing the change in temperature (on the y-axis) vs. time
(on the x-axis) you can see if the temperature is linear over time ( meaning the change
in temperature is about equal for each time interval you measure) or non-linear. If it
is non-linear you may see a large change initially and then a leveling out of the
temperature over time.