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In terms of coulombs law, as the radius increases, the attraction between the negative electrons and the positive protons inside the nucleus decreases. Therefore, the electrons are more easily removed from the atom. In an element like F, the atomic radius is smaller because the zeff is higher for this element. Therefore the atom is more likely to gain electrons than lose them.
Electronegativity is the attractive force between the protons and the electrons. As the atomic radius increases, the protons in the nucleus have a less attractive force to the electrons which is why the electrons are further from the nucleus, expanding the atom, and thus electronegativity decreases. This is why electronegativity and atomic radius have opposite periodic trends.
As you move down a column, more energy levels are added to an atom, causing increased distance between the nucleus and valence electrons. Due to electron shielding (the interference of inner energy shells in the interactions between outer energy shells and the nucleus), electronegativity decreases as the attraction between the nucleus and outer electrons is lowered. Across a row, the atom is more prone to accepting electrons due to the fact that it gets closer and closer to filling its valence shell, causing electronegativity to increase. This relates to radius because, as you move down a column, additional energy shells cause greater radius, while moving across a row causes a tighter pull on the valence shell from the nucleus and thus a smaller nucleus.
An increase in electronegativity means that an atom has a stronger pull on bonding electrons. With larger atomic radii, electrons are further away from the nucleus. It makes sense that they have opposite trends because electrons being further away from the nucleus have less attraction according to the electrostatic potential energy formula.
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