The following numbers are the ratios of second ionization energy to first ionization energy: Na: \(\quad 9.2\) P: \(1.8\) Mg: \(\quad 2.0\) \(\mathrm{S}: \quad 2.3\) Al: \(\quad 3.1\) \(\mathrm{Cl}: \quad 1.8\) Si: \(\quad 2.0\) Ar: \(1.8\) Explain these relative numbers.

Ionization energy is the amount of energy required to remove an electron from a gaseous atom. There are mainly three key factors that affect ionization energies: 1. Distance of electrons from the nucleus 2. Effective nuclear charge 3. Electron shielding By considering these factors, we can analyze each element's electron configuration to explain these numbers.

Each element's electron configuration affects its ionization energies: 1. Na (Sodium): [Ne]3s^1 2. P (Phosphorus): [Ne]3s^23p^3 3. Mg (Magnesium): [Ne]3s^2 4. S (Sulfur): [Ne]3s^23p^4 5. Al (Aluminum): [Ne]3s^23p^1 6. Cl (Chlorine): [Ne]3s^23p^5 7. Si (Silicon): [Ne]3s^23p^2 8. Ar (Argon): [Ne]3s^23p^6

Now, let's attempt to explain the given ratios: 1. Na: The first ionization energy of Na corresponds to removing its only 3s electron, while the second ionization energy involves breaking a much more stable [Ne] noble gas configuration. Hence, the second ionization energy is considerably higher than the first, resulting in a large ratio of 9.2. 2-7. P, Mg, S, Al, Cl, and Si: In all these cases, the second ionization energy involves removing an electron from either the same subshell (3s or 3p) or another subshell with similar energy (3s to 3p or vice versa). Thus, while there is a difference in the energies, the difference is not as large as in the case of Na. The ratios range from 1.8 to 3.1. 8. Ar: The first ionization energy corresponds to removing an electron from a full 3p subshell, while the second ionization corresponds to breaking a stable [Ne] noble gas configuration. Moreover, the loss of another electron would make the atom more positively charged, thus making it even more difficult to remove an electron in the second ionization. However, the first ionization energy is significantly high, as the stable noble gas configuration of Ar is hard to break. As a result, the second ionization energy is not too much greater than the first one, resulting in a ratio of 1.8.

In conclusion, the relative numbers of the second ionization energy to first ionization energy for each element are explained by their electron configurations and the factors affecting ionization energies: distance from the nucleus, effective nuclear charge, and electron shielding.