The element having the greatest difference between its first and second ionization energies, is:

🧠 A Group 1 element gives the biggest jump from $IE_1$ to $IE_2$ — the second electron has to break into the noble-gas core For an alkali metal, the lone valence electron is removed easily ($IE_1$ is small). The second electron must come from the inner noble-gas shell — $IE_2$ shoots up by a factor of ~10. That gap is bigger than for any other group.

🗺️ Compare the candidates K ($[\ce{Ar}]\,4s^1$): $IE_1 \approx 419$, $IE_2 \approx 3052$. Difference $\approx 2633$, a ~7-fold jump. Both metrics huge. Ca ($[\ce{Ar}]\,4s^2$): $IE_1 \approx 590$, $IE_2 \approx 1145$. Both electrons come from the 4s. Modest jump. Ba ($[\ce{Xe}]\,6s^2$): same pattern as Ca — both electrons from the 6s. Modest jump. Sc ($[\ce{Ar}]\,3d^1\,4s^2$): $IE_1$ removes 4s, $IE_2$ removes another 4s. Big jump only at $IE_4$ (after both 4s and 3d are gone).

Only K has the alkali-metal "valence is just one electron" structure, so only K shows the largest $IE_2 - IE_1$ gap.

⚠️ The trap Students sometimes pick Ca because Ca²⁺ is a famous ion and "Ca seems easy". But for Ca, BOTH electrons come from the same 4s subshell, so the $IE_2 - IE_1$ gap is moderate. The big jump for Ca shows up at $IE_3$ (when you finally hit Argon core), not at $IE_2$. K is the one whose noble-gas core sits right between $IE_1$ and $IE_2$.

$\boxed{\text{Answer: (b)}}$