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at extremely high pressures, several thousand atm, will the volume of a gas be greater or less than predicted from the ideal gas law?
At high pressures, the intermolecular forces are more significant and molecular size is more significant between molecules. Therefore, the actual volume will be less than the ideal gas model.
That is correctAt very high pressure volume of real gasses is larger than ideal, due to the size of the gas molecules starting to play a significant role in the total volume.
That's true at moderately high pressure and low temperature, but at extreme pressures the actual volume of the molecules comes into play and the volume ends up being larger than ideal.