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Why is carbon dioxide that comes from dry ice opaque and smoky while the carbon dioxide that we exhale transparent?

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Essentially, because the carbon dioxide sublimates from solid (dry ice) to gas at a very low temperature (roughly −78 °C at 1 atm), it causes water vapour in the air to condense, causing a visible fog. Thus what you are seeing is not carbon dioxide, but rather water.

When we exhale and it is reasonably warm, the carbon dioxide expelled is roughly body temperature (36–37.5 °C) and so we do not see any fog forming.

What happens when we exhale in cold weather is a bit different. Our breath contains a high percentage of water, higher than the percentage of water that can stay in gaseous form at a low temperature. Thus the ‘extra’ water condenses into droplets.

For reference, here is the phase diagram of $\ce{CO2}$:

Phase Diagram of carbon dioxide

And here is the phase diagram of $\ce{H2O}$:

Phase Diagram of H2O

And here is the saturation fraction of water at various temperatures at 1 atm (sea level):

Saturation fraction of H2O by temperature (dew point)

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  • $\begingroup$ How does saturation fraction relate to saturation vapour pressure? $\endgroup$ – gerrit Aug 7 '13 at 10:01
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    $\begingroup$ @gerrit saturation fraction is the concentration while vapour pressure is a pressure. They're related by the formula $p=nRT/V$ where p is pressure, n is amount of the gas in mol, V is volume, T is temperature and R is the gas constant. $\endgroup$ – Tomcat Aug 7 '13 at 10:15

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