How to derive the order of average intermolecular spacing of air molecules in the room temperature and standard pressure?
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The question is as above. I remember the answer to be of order 10−3cm10−3cm. However I cannot see how to derive this order of magnitude from the contents of thermodynamics. Could anyone please help me
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If you assume air is an ideal gas (as is often done at STP), then one mole occupies
V=nRTP=1∗R∗298 K1 atm=0.0245 m3V=nRTP=1∗R∗298 K1 atm=0.0245 m3.
Thus, each molecule occupies
V/Na=4∗10−26 m3V/Na=4∗10−26 m3.
The spacing between molecules is on the order of the cube root of that, or 3 nm.
This is a simplification; in reality, there are attractive interactions and size exclusion effects. The former are probably more important, so you might interpret the above result as an upper bound.
Edit: Mark suggests the direct approach of looking up the density of air. According to his sources, the volume of a mole of air is 0.0227 m^3 -- so as expected, the ideal gas law overpredicts the spacing slightly (by about 3%).
V=nRTP=1∗R∗298 K1 atm=0.0245 m3V=nRTP=1∗R∗298 K1 atm=0.0245 m3.
Thus, each molecule occupies
V/Na=4∗10−26 m3V/Na=4∗10−26 m3.
The spacing between molecules is on the order of the cube root of that, or 3 nm.
This is a simplification; in reality, there are attractive interactions and size exclusion effects. The former are probably more important, so you might interpret the above result as an upper bound.
Edit: Mark suggests the direct approach of looking up the density of air. According to his sources, the volume of a mole of air is 0.0227 m^3 -- so as expected, the ideal gas law overpredicts the spacing slightly (by about 3%).
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