Which of the reactions will need the maximum amount of energy?

The correct option is K⁺ → K⁺⁺ + e^-.

To understand why this reaction requires the maximum amount of energy, we need to analyze the ionization energy required for each process. Ionization energy is the energy required to remove an electron from a gaseous atom or ion. The reactions given represent the following processes:
1. Na → Na⁺ + e^- represents the first ionization energy (IE₁) of sodium (Na).
2. Ca⁺ → Ca⁺⁺ + e^- represents the second ionization energy (IE₂) of calcium (Ca).
3. K⁺ → K⁺⁺ + e^- represents the second ionization energy (IE₂) of potassium (K).
4. C²⁺ → C³⁺ + e^- represents the third ionization energy (IE₃) of carbon (C).

Let's examine the electronic configuration of the starting species in each reaction:
- Sodium (Na) has the electronic configuration: $1s^{2}2s^{2}2p^{6}3s^1$. Removing one electron from a neutral Na atom is relatively easy because it yields a stable noble gas configuration ($2p^6$).
- Calcium ion (Ca⁺) has the electronic configuration: $1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}4s^1$. Removing an electron from Ca⁺ requires energy, but it results in the highly stable noble gas configuration of argon ($3p^6$), making this second ionization relatively favorable.
- Carbon ion (C²⁺) has the electronic configuration: $1s^{2}2s^2$. Removing an electron from C²⁺ involves removing a valence electron from a shell with a principal quantum number of n = 2.
- Potassium ion (K⁺) has the electronic configuration: $1s^{2}2s^{2}2p^{6}3s^{2}3p^6$.

The K⁺ ion already has a stable octet configuration (noble gas configuration of Argon) in its outermost shell (n = 3). Because it is an extremely stable, closed-shell configuration, removing a second electron from K⁺ to form K⁺⁺ requires disrupting this highly stable noble gas core.
Additionally, the positive charge on K⁺ holds the remaining electrons much more tightly due to the increased effective nuclear charge ($Z=19$ protons attracting 18 electrons). Breaking this stable octet configuration requires an exceptionally large amount of energy compared to the other options, making the reaction K⁺ → K⁺⁺ + e^- require the maximum amount of energy.