Final answer:
The second ionization energy of sodium is three times greater than that of magnesium due to electron configuration differences and atomic structure. The difference in atomic radii between Na and K is larger than between Rb and Cs because of differences in effective nuclear charges and electron shielding.
Step-by-step explanation:
(a) The second ionization energy of sodium is about three times greater than the second ionization energy of magnesium due to the differences in their electron configurations and atomic structure. Sodium (Na) has an electron configuration of [Ne] 3s^1 and loses an electron to form Na+ with a closed-shell configuration of [Ne]. Magnesium (Mg), on the other hand, has an electron configuration of [Ne] 3s^2 and requires more energy to remove the second electron to form Mg2+. This is because the second electron is being removed from a lower energy level and also requires breaking the electron pairing in the 3s orbital.
(b) The difference between the atomic radii of Na and K is relatively large compared to the difference between the atomic radii of Rb and Cs because of the differences in their effective nuclear charges. While Na and K both have a +1 charge, K has more electrons than Na, resulting in a greater shielding effect and a larger atomic radius. On the other hand, Rb and Cs have larger atomic radii compared to Na and K due to the addition of extra energy levels and increased shielding effect from the inner electrons.