Ionization energy concept

• The first ionization energy ($I_1$) is the energy required to remove the first electron from a neutral atom.
• The second ionization energy ($I_2$) is the energy required to remove the second electron from the singly charged cation.

The greatest difference between $I_1$ and $I_2$ occurs when, after removal of the first electron, the atom attains a stable noble gas configuration. Then removing the second electron becomes extremely difficult.

Check each option

A: Ba ($Z=56$)

Electronic configuration: $[Xe]6s^2$

• After first electron removal: $Ba^+:[Xe]6s^1$
• After second electron removal: $Ba^{2+}:[Xe]$ Here, $I_2$ is greater than $I_1$, but the huge jump comes after the second ionization, not between first and second.

B: Ca ($Z=20$)

Electronic configuration: $[Ar]4s^2$

• After first electron removal: $Ca^+:[Ar]4s^1$
• After second electron removal: $Ca^{2+}:[Ar]$ Again, the very large jump is after the second ionization, i.e. between $I_2$ and $I_3$, not between $I_1$ and $I_2$.

C: Sc ($Z=21$)

Electronic configuration: $[Ar]3d^14s^2$

• First two electrons are removed from $4s$.
• So $I_1$ and $I_2$ are not drastically different. No exceptionally large jump between $I_1$ and $I_2$.

D: K ($Z=19$)

Electronic configuration: $[Ar]4s^1$

• After first electron removal: $K^+:[Ar]$ Now $K^+$ has a noble gas configuration.
• Removing the second electron means taking one electron from the stable inner-shell noble gas core.

Therefore: $I_2 \gg I_1$ and the difference $I_2 - I_1$ is the greatest for potassium.

Conclusion

The element with the greatest difference between first and second ionization energies is: $\boxed{K}$

So the correct option is D.