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imagine the ice cube as the system, and its immediate surroundings as the surroundings. in order for the ice cube to melt, it needs to absorb heat from its surroundings in order to reach its melting point. thus, the phase change is endothermic because the system takes heat from its surroundings (delta H is positive)
The ice cube needs energy in the form of heat to melt it further and further. By adding energy to the ice cube via the heat of your hand if you hold it, the reaction is endothermic. The ice cube absorbs the heat energy in order to melt it. An endothermic reaction is defined as a process that absorbs heat (p.279 in the textbook).
If you focus in on the ice cube, you can see that it MUST be absorbing heat in order to melt. Thus, the system of the ice cube must be absorbing heat from its surroundings (the hand), and will have a positive delta H (change in enthalpy). It is helpful to know that an endothermic reaction (one in which a system GAINS heat) has a positive delta H value. So, the ice cube is experiencing an endothermic reaction while the hand, which is losing heat, is experiencing an exothermic reaction.
If you think of placing an ice cube on your hand, the ice cube melts because the heat of your hand is transferred to the ice cube. In other words, your hand provides the input of heat needed to melt the ice cube, and since heat must be put in in order for this melting reaction to proceed, this reaction is endothermic.
The ice would be the system, and the hand would be its surroundings. When the ice absorbs energy(heat) and reaches to its melting point, it melts, therefore the heat absorbed comes from surroundings, and the phase change is endothermic, and the hand that loses heat would cool down.
An ice cube melting is an endothermic reaction because the ice cube must take in heat in order to begin melting. It is endothermic because heat must go into the ice cube ("en"dothermic - "in"to the ice cube) instead of the ice cube emanate heat which would be an exothermic reaction.
The characterization of exothermic and endothermic reactions is with respect to energy going in or out of a system. In this case, the ice cube would be considered the system while the hand holding the ice cube is the surroundings. As such, the heat energy from your hand is transferred to the ice cube, causing it to melt. In other words, heat is being transferred into your system (ice cube) from the surroundings (hand), making the melting of the ice cube endothermic (endothermic roughly meaning heat energy going in (to the system)).
One way to think about this is in terms of the average kinetic energy of the water molecules. When water is a solid in the form of ice, the average kinetic energy of the water molecules is low. When water is a liquid, the average kinetic energy of the water molecules is larger. Thus, in order for phase change from solid to liquid to occur, energy is required. Thus, the reaction is endothermic.
As described above, ice melting is an endothermic reaction because it requires heat for the reaction to take place. As a result, surroundings, in Dr. Lavelle's example, one's hand, will cool down because the heat required for the reaction is absorbed from the surroundings (conservation of energy).
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