Ionization energy is the energy required to remove an electron from an isolated gaseous atom. The first ionization energy (IE₁) removes the outermost electron. The second ionization energy (IE₂) removes the next electron, which often comes from a more stable, inner shell, making IE₂ significantly higher than IE₁.

The greatest difference between IE₁ and IE₂ will occur for an element that achieves a very stable electronic configuration after losing its first electron. The most stable configurations are those of noble gases.

Let's analyze the given elements and their electron configurations:

Potassium (K), Atomic Number: 19

Electron Configuration: $1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}4s^1$

After losing 1 electron (K⁺): $1s^{2}2s^{2}2p^{6}3s^{2}3p^6$ (Stable Argon configuration)

Removing a second electron would break this stable noble gas core. Therefore, IE₂ for Potassium is extremely high compared to its relatively low IE₁. The difference (IE₂ - IE₁) is very large.

Calcium (Ca), Atomic Number: 20

Electron Configuration: $1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}4s^2$

After losing 1 electron (Ca⁺): $1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}4s^1$ (Not a noble gas configuration)

Losing the second electron gives Ca²⁺: $1s^{2}2s^{2}2p^{6}3s^{2}3p^6$ (Stable Argon configuration). The jump from IE₁ to IE₂ is significant, but not as extreme as for Potassium because the first electron removed is not from a stable core.

Barium (Ba), Atomic Number: 56

Similar to Calcium but with higher principal quantum number. Its configuration ends in $6s^2$. Losing one electron gives a half-filled subshell, which is somewhat stable. Losing the second electron gives a stable Xe core. The difference is large, but the absolute ionization energies are lower than for lighter elements, and the stability jump is not as pronounced as from s¹ to noble gas.

Scandium (Sc), Atomic Number: 21

Electron Configuration: $\left[\mathrm{Ar}\right]3d^{1}4s^2$

The first electron removed is from the 4s orbital. The second electron is removed from the 3d orbital. While there is an increase, it does not involve achieving a noble gas configuration. The difference is not as drastic.

Conclusion: Potassium (K) shows the greatest difference. Its first electron is very easy to remove (low IE₁), and the resulting K⁺ ion has the highly stable electron configuration of Argon. Removing an electron from this stable core requires a massive amount of energy (very high IE₂).

Related Topics & Formulae

Ionization Energy (I.E.): The minimum energy required to remove the most loosely bound electron from an isolated gaseous atom to form a cation.
Trends: I.E. increases across a period and decreases down a group.

Stable Configurations: Elements tend to lose or gain electrons to achieve the nearest noble gas configuration, which is exceptionally stable. A large jump in ionization energy occurs when the subsequent electron removal would break this stable core.