1. Criteria for No Unpaired Electrons: This occurs in elements belonging to Group 2 (Alkaline Earth Metals) and Group 18 (Noble Gases) because their valence shells ($s^2$ or $s^2p^6$) are full.
2. Identification of Elements:
• Helium ($Z=2$): $1s^2$ (Noble Gas)
• Beryllium ($Z=4$): $1s^2 2s^2$ (Group 2)
• Neon ($Z=10$): $1s^2 2s^2 2p^6$ (Noble Gas)
• Magnesium ($Z=12$): $[Ne] 3s^2$ (Group 2)
• Argon ($Z=18$): $[Ne] 3s^2 3p^6$ (Noble Gas)
• Calcium ($Z=20$): $[Ar] 4s^2$ (Group 2)
3. Counting: There are exactly
6 such elements in the range $Z=1$ to $Z=20$. Elements in other groups (like Group 14 Carbon or Group 16 Oxygen) have partially filled $p$ subshells, leading to unpaired electrons.