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The question asks if heat is being transferred when "A balloon filled with gas is expanded to a larger volume and lower pressure while maintaining the same temperature." The answer is Yes but I am not too sure how to arrive at this answer. Is it since we need to do work on the system to increase the volume and decrease pressure (the energy put in as work counts as the heat increase)? If this reason was correct is there a mathematical way to show this? If it is not correct then how do you know there is a heat change? Thanks :)
Since it is at the same temperature, you know that Delta U is equal to zero. However, you also know that work is happening because the balloon is expanding, therefore indicating a change in volume. According to Delta U = q + w, for Delta U to be zero, there must also be heat being transferred in order to balance out the work.
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