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How would a system which exchanges energy but does not exchange matter between the system and the surroundings be classified?

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Step-by-step explanation:

When a system exchanges only energy and there is no exchange of matter between the system and surrounding then this means the system is closed.

For example, when we placed hot water in a closed jar then after a certain hours the water inside the container becomes cold. This is because energy in the form of heat has been exchanges with the atmosphere or surroundings.

Hence, there is no exchange of matter between the system and surrounding.

User Nicklas Nygren
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If a system allows the exchange of energy but does not permit exchange of matter between the system and surroundings, this system is classified as a CLOSED system. The energy that flows in and out of the system boundaries is in the form of heat. 
User TheRoadrunner
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