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••• Derive the ideal gas law without the assumption of no intermolecule collisions. To simplify this calculation, make the radical assumption that the x components of the velocities of all molecules have the same magnitude; that is, assume that all molecules have the same |vx|, with half the molecules moving to the right and the other half moving to the left; make similar assumptions for the y and z directions. Consider a gas of molecules with number density n in a cubical container, and consider only those molecules that are within a short distance Δx of one wall of the container where Δx ≪ λ, the mean free path. At any instant, half of those molecules are heading toward the wall and will collide with the wall in an average time Δt, where Δx/Δt = |vx|. Proceed to compute the average force on the wall due to molecular collisions, and then deduce the ideal gas law.
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