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Assume that the air pressure and the amount of water in the air stay constant. How can I figure out how much a change in temperature affects the relative humidity?

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Relative humidity equals actual water vapor pressure divided by saturated water vapor pressure. As temperature goes up, saturated vapor pressure goes up as well, and relative humidity will go down, if absolute humidity remains the same. The exact relationship must be measured.

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Relative humidity is just the percentage of what the air at a given temperature can hold. This is given by the Clausius–Clapeyron equation, which rises roughly exponentially with temperature doubling approx every 10degrees C. So if your relative humidity is X, and the saturation vapor pressure at the new temperature is Y times the value at the old temperature, your new (constant volume) humidity is X/Y. You wanted constant pressure, so your absolute humidity is changed by the change in volume, i.e. your humidity also scales inversely with volume, although this second effect is much smaller than the first.

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Actually the "can hold" can more accurately be understood as there is a rate for water vapor condensing, and another rate for droplets evaporating. The dew point is the temperature at which these rates are equal. Dew Point That means it does not actually depend on other gas, such as air, being present. –  Mike Dunlavey Oct 5 '11 at 19:55
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