A horizontal tube with closed ends is rotated with a constant angular velocity `omega` about a vertical axis passing through one of its ends. The tube contains carbon dioxide at a temperature `T = 300 K`. The length of the tube is `l = 100 cm`. Find the value `omega` at which the ratio of molecular concentrations at the opposite ends of the tube is equal to `eta = 2.0`.

Explanation

The potential energy associated with each molecule is : `-(1)/(2) m omega^2 r^2`
and there is a concentration variation
`n(r) = n_0 exp((m omega^2 r^2)/(2 kT)) = n_0 exp ((m omega^2 r^2)/(2 RT))`
Thus `eta = exp ((M omega^2 l^2)/(2 R T)`or `omega = sqrt((2 R T)/(Ml^2) 1n eta)`
Using `M = 12 + 32 = 44 gm, l = 100 cm, R = 8.31 xx 10^7 (erg)/(.^@K), T = 300`,
we get `omega = 280` radians per second.