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Question:
Why is it that change in internal energy is 0 if heat is 0, since work can still be done without change in heat?
Answer:

When work is done, the internal energy of the system decreases and that of the surroundings increases. The form of this energy need not alwyas be heat. And even if it is heat, it is difficult to sense the change as the amount involved is too small to be sensitive.

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