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Question:

Modern vacuum pumps can evacuate a vessel down to a pressure of 4.0 × 10⁻⁷ atm at room temperature (300 K). Taking R = 8.3 J/mol·K, 1 atm = 10⁵ Pa, and NAvogadro = 6 × 10²³ mol⁻¹, the mean distance between molecules of gas in an evacuated vessel will be of the order of:

0.2 μm

0.2 mm

0.2 nm

0.2 cm

Solution:

P = nkT
n is the number of gas particles per unit volume.
Thus n = P/kT = (4 × 10⁻⁷ × 1.01 × 10⁵)/(8.314 × 300) = 1 × 10⁵/cm³
In 1cm³, the number of molecules are 10⁵
Thus 1 molecule occupies a volume of 10⁻⁵ cm³
This volume will be like a cube with side l = V¹/³ = (10⁻⁵)¹/³ = 0.2 mm