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Step-by-Step Explanation
Step 1: Understanding the concept of ionization enthalpy
Ionization enthalpy (IE) is the energy required to remove one electron from an isolated gaseous atom. The second ionization enthalpy is the energy required to remove a second electron from the already once-ionized atom.
Step 2: Write the ground state electronic configurations
Given atomic numbers:
$V \ (Z = 23) : [Ar] \, 3d^3 \, 4s^2$
$Cr \ (Z = 24) : [Ar] \, 3d^5 \, 4s^1$
$Mn \ (Z = 25) : [Ar] \, 3d^5 \, 4s^2$
$Fe \ (Z = 26) : [Ar] \, 3d^6 \, 4s^2$
Step 3: Configurations after the first electron is removed
When the first electron is removed (first ionization), it generally comes from the $4s$ orbital for these transition metals:
$V^+ : [Ar] \, 3d^3 \, 4s^1$
$Cr^+ : [Ar] \, 3d^5$
$Mn^+ : [Ar] \, 3d^5 \, 4s^1$
$Fe^+ : [Ar] \, 3d^6 \, 4s^1$
Step 4: Removing the second electron
Now for the second ionization enthalpy, one more electron must be removed from these once-ionized species:
$V^{2+} : [Ar] \, 3d^3$
$Cr^{2+} : [Ar] \, 3d^4$
$Mn^{2+} : [Ar] \, 3d^5$
$Fe^{2+} : [Ar] \, 3d^6$
Step 5: Recognizing the special stability of half-filled orbitals
After losing one electron, chromium ($Cr^+$) has the highly stable half-filled $3d^5$ configuration. To remove another electron from this half-filled $3d$ subshell requires more energy compared to removing an electron from a less stable configuration. Thus, the second ionization enthalpy of chromium is especially high.
Step 6: Comparing distances and effective nuclear charge
For $V, Mn,$ and $Fe,$ the second electron is still more likely to come from a relatively higher orbital $(4s)$ or a less stable $3d$ configuration. In chromium's case, removing a second electron means disturbing its stable half-filled $3d^5$ configuration. This stability elevates chromium’s second ionization enthalpy above those of vanadium, manganese, and iron.
Step 7: Final conclusion
Chromium ($Cr$) has the highest second ionization enthalpy among the four given elements because removing an electron from its half-filled $3d^5$ configuration is particularly difficult, requiring more energy. Therefore, the correct answer is Chromium ($Cr$).
