The mean free path of electrons in a gas in a discharge tube is inversely proportional to the pressure inside it. The Crookes dark space occupies half the length of the discharge tube when the pressure is 0.02 mm of mercury. Estimate the pressure at which the dark space will fill the whole tube.

#### Solution

Let the mean free path of the electrons be L and pressure inside the tube be P.

It is given that the mean free path of electrons and the pressure inside the tube are related as:

\[L \propto \frac{1}{P}\]

Here, L = half of tube's length

Pressure inside the tube, P = 0.02 mm of Hg

As pressure 'P' becomes half, mean free path of electrons, L, doubles. So, the whole tube is filled with Crookes dark space.

Thus, the pressure required for filling the whole tube with Crookes dark space,

\[P' = \frac{0 . 02}{2} = 0 . 01\] mm of Hg