Answer:
Barium has a larger atomic radius because it has more energy levels and a larger shielding effect. Silicon has a smaller atomic radius because it has a stronger nuclear charge. Since chlorine's 17 protons are greater than magnesium's 12 protons, chlorine will have a greater effective nuclear charge to draw chlorine's valence electrons closer to the nucleus and, thus, chlorine is expected to have the smaller atomic radius, while magnesium with the lower effective nuclear charge is expected to.