How does heat travel from one object to another?

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This depends on the conditions of the environment in which the objects are found. For example, the objects can be physically touching each other, they can be separated by a medium (a liquid or gas), or they can be separated by vacuum. Heat will be transferred differently in all three conditions. 

In the first two examples, heat is transferred through massive particles interacting with each other. Heat itself is just another way of talking about how much kinetic energy a molecule has, and that energy can be transferred to another molecule by making them collide. When the objects are in direct contact, they directly collide with each other, such as an object on a hotplate. If they are separated by a medium, then the medium will also participate in the heat transfer, such as when an ice cube dissolves in water. Another comparison would be striking two billiard balls together, compared to hitting pins with a bowling ball. In each case, the energy is distributed to each of the molecules interacting in the system.

In the third example, there aren't really any molecules available in the medium to transfer energy from one object to another. Instead, the process of radiation involves the release of energetic photons, a form of electromagnetic energy, that can be absorbed by the second object, exciting its electrons and therefore increasing its kinetic energy. 

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