Why is an exothermic reaction spontaneous?

Exothermic reactions are not necessarily spontaneous.

Take the combustion of magnesium for example:

##2Mg_((s))+ O_(2(g))rarr2MgO_((s))##

##DeltaH## is negative.

Yet a piece of magnesium is quite safe to handle at room temperature.

This is because a very high temperature is needed to make the magnesium burn. The reaction has a very high activation energy.

This is shown in the diagram:

(docbrown.info)

A low activation energy can result in the reaction being spontaneous.

A good example is sodium reacting with water.

The diagram shows two important areas of physical chemistry.

The ##color(red)("red")## arrow relates to thermodynamics and is concerned with initial and final states.

The ##color(purple)("purple")## arrow relates to kinetics and is concerned with how quickly the change from initial to final state is achieved.