We study the thermodynamics of bromophenyl functionalization of carbon nanotubes with respect to diameter and metallic/insulating character using density-functional theory (DFT). On the one hand, we show that the functionalization of metallic nanotubes is thermodynamically favoured over that of semiconducting ones, in agreement with what binding energy calculations previously suggested. On the other hand, we show that the activation energy for the grafting of a bromophenyl molecule onto a semiconducting zigzag nanotube ranges from 0.72 to 0.75 eV without any clear diameter dependence within numerical accuracy. This implies that this functionalization is not selective with respect to diameter at room temperature, which explains the contradictory experimental selectivities reported in the literature. This contrasts with what is suggested by the clear diameter dependence of the binding energy of a single bromophenyl molecule, which ranges from 1.52 eV for an (8, 0) zigzag nanotube to 0.83 eV for a (20, 0) zigzag nanotube. Also, attaching a single bromophenyl to a nanotube creates states in the gap close to the functionalization site. It therefore becomes energetically favourable for a second bromophenyl to attach close to the first one on semiconducting nanotubes. The para configuration is found to be favoured for resulting bromophenyl pairs and their binding energy is found to decrease with increasing diameter, ranging from 4.35 eV for a (7, 0) nanotube to 2.26 eV for a (29, 0) nanotube. An analytic form for this radius dependence is derived using a tight binding Hamiltonian and first order perturbation theory. The 1/R(2) dependence obtained (where R is the nanotube radius) is verified by our DFT results within numerical accuracy. Finally, bromophenyl pairs are shown to be favoured by only 50 meV with respect to separate moieties on (9, 0) metallic nanotubes, which suggests that pair formation is not significantly favoured on some metallic nanotubes. This result explains the observation of stable isolated moieties at room temperature in nanotube samples containing random nanotube chiralities.