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Polarizability refers to the ability of an atom or molecule to undergo temporary distortion or deformation in its electron cloud when subjected to an external electric field. In other words, it is a measure of how easily the electron cloud can be perturbed or polarized.

To compare polarizabilities, we need to consider the size of the atoms, given that larger atoms have more diffuse electron clouds, making them more deformable. As we go down a group in the periodic table, atomic size increases. In the given set, the order of atomic size is \(\mathrm{N} < \mathrm{P} < \mathrm{As} < \mathrm{Sb}\). Thus, the most polarizable atom is antimony (\(\mathrm{Sb}\)), as it is the largest atom.

Polarizability generally increases with the size and the number of electrons in the molecule. Comparing the molecules given, 1. \(\mathrm{GeCl}_{4}\) has larger central atom Ge compared to \(\mathrm{CH}_{4}\) and \(\mathrm{SiCl}_{4}\), which have C and Si as central atoms respectively, with 4 electronegative Cl atoms, making it more polarizable. 2. \(\mathrm{CH}_{4}\) has the smallest central atom (C), with 4 small H atoms, and hence has the least polarizable electron cloud. 3. \(\mathrm{SiH}_{4}\) would be more polarizable than \(\mathrm{CH}_{4}\) due to Si being larger than C. 4. \(\mathrm{SiCl}_{4}\) would be more polarizable than \(\mathrm{SiH}_{4}\) due to the larger Cl atoms compared to H atoms. 5. \(\mathrm{GeBr}_{4}\) would be the most polarizable molecule among the group, due to the larger central atom Ge and the even larger Br atoms. In short, the order of increasing polarizability is \(\mathrm{CH}_{4} < \mathrm{SiH}_{4} < \mathrm{SiCl}_{4} < \mathrm{GeCl}_{4} < \mathrm{GeBr}_{4}\).

Boiling points are directly related to the strength of intermolecular forces between molecules. Polarizability influences these intermolecular forces, with more polarizable molecules having stronger dispersion forces. In general, as polarizability increases, boiling point increases. Hence, following the increasing order of polarizability obtained in step 3, the order of boiling points should be: \(\text{BP}_{\text{CH}_4} < \text{BP}_{\text{SiH}_4} < \text{BP}_{\text{SiCl}_4} < \text{BP}_{\text{GeCl}_4} < \text{BP}_{\text{GeBr}_4}\).