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By compressing $\ce{CO2 (g)}$ the gas gains a lot of heat. When the hot compressed carbon dioxide is left to cool, it attains the temperature of the surrounding material. When it is released into the pressure outside (about 100kPa), it will get so cold (actual temperature depending on how much the carbon dioxide was compressed), that it will freeze (below 78.5 degrees celcius).

How much pressure is needed to make dry ice at 293K (20 degrees celcius)?

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"" If that is left to cool, it will become so cold"" if it is left to cool, it will attain the temperature of the surrounding material. "so cold that" is not physical. – Georg Mar 3 '14 at 20:34
@Georg When I re-read this I see that it is a bit unclear written. I updated the question. Now it should be easier to understand what I meant. Thanks for your clarification. – Friend of Kim Mar 3 '14 at 20:43

Isn't that a bit different?

At 293 K, you need about 56 bar to liquefy carbon dioxide. If you then release the pressure, a part of the liquid evaporates again while cooling down the rest until it solidifies at around -80 °C.

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Does the carbon dioxide have to be liquid to turn into dry ice? You say that it is the evaporation that cools some of the gas down to -80 degrees. Is it the evaporation or the drop in pressure that makes it cold? – Friend of Kim Mar 3 '14 at 20:50

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