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What happens to the vapor content if a mixture of gases saturated with water vapor is heated isobarically? Will the amount of liquid water present decrease, yielding up more vapor so as to re-establish equilibrium, or will nothing happen?

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Assuming you have a liquid water reservoir available and the liquid is heated along with the gas, water will evaporate to keep the partial pressure of water equal to the vapor pressure of water at the current temperature. As the total gas mix is maintained at a given pressure, the rest of the gases must expand more than Charles' law would indicate, reducing their partial pressure.

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