Within a discharge tube, the mean free path of electrons in the gas is inversely proportional to the internal pressure. The Crookes dark space occupies half the length of the discharge tube at a pressure of 0.02 mm of mercury. Estimate the pressure at which the dark space will occupy the entire tube.

Step-by-step solution

The length of the Crookes dark space is directly proportional to the mean free path of the electrons. Since the mean free path is inversely proportional to the internal pressure P , the length of the dark space l is also inversely proportional to P . l 1 P l₁ P₁ = l₂ P₂ Let the total length of the discharge tube be L . Given that at P₁ = 0.02 mm of mercury, l₁ = L 2 . For the dark space to occupy the entire tube, l₂ = L . Substituting the values into the equation: L 2 0.02 = L P₂ P₂ = 0.02 2 = 0.01 mm of mercury