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I understand what boiling and vaporization is. But what puzzles me is the phase diagram. When I spill a glass of water in my room, it will soon vaporize, though there was normal atmospheric pressure and 20 °C. If you look in phase diagram of water, it should be still liquid at this point.

I understand that molecules of water escape the surface and turn into vapor, but... is the phase diagram of water wrong then? The temperature and pressure didn't change around that spilled water and still it turns into gas, although (looking at the phase diagram), it should be liquid.

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The phase diagram has equilibrium states for pure water, vapor, and both at saturation. You have water exposed to atmospheric gases, so the pressure is not that of pure vapor. The water will evaporate trying to create a partial pressure of vapor equal to the vapor pressure for saturated water at the water temperature. If the surface is open to flow of fresh air, this vapor pressure is not achieved, and the water slowly evaporates away.

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The phase diagram is a diagram that shows water in equilibrium with its vapor throughout a broad temperature range. The water spilled in a puddle in your room is NOT in equilibrium with its surroundings, so the phase diagram does not apply.

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