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    Why is temperature inversely proportional to the number of moles

    I assume that you're talking about a gas at constant pressure, otherwise temperature doesn't affect number of moles at all (like in a solid, as no matter how much you heat it, it remains constant unless you evaporate it).

    In a system of constant pressure, there is a given amount of moles of gas in a particular space (i.e. 1M of gas in 1m^3 of space). Increasing temperature increases the speed at which the gas particles move, increasing the pressure. As the system has to be at constant pressure, the area has to increase (1M in 4m^3), so the number of moles per m^3 decreases (0.25M in 1m^3). Hope this helps.
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