Within a discharge tube, the mean free path of electrons in the gas...

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 $P$.

$l \propto \dfrac{1}{P} \Rightarrow l_1 P_1 = l_2 P_2$

Let the total length of the discharge tube be $L$.

Given that at $P_1 = 0.02 \text{ mm}$ of mercury, $l_1 = \dfrac{L}{2}$.

For the dark space to occupy the entire tube, $l_2 = L$.

Substituting the values into the equation:

$\dfrac{L}{2} \times 0.02 = L \times P_2$

$P_2 = \dfrac{0.02}{2} = 0.01 \text{ mm}$ of mercury