Interpretation of the symbols

The pairs compare the successive ionisation enthalpies needed to remove one electron from the indicated gaseous ions:

• $\mathrm{Mn}^{2+}$ means $I_3$ of Mn: $\mathrm{Mn}^{2+} \to \mathrm{Mn}^{3+} + e^-$
• $\mathrm{Fe}^{2+}$ means $I_3$ of Fe: $\mathrm{Fe}^{2+} \to \mathrm{Fe}^{3+} + e^-$
• $\mathrm{Mn}^{+}$ means $I_2$ of Mn: $\mathrm{Mn}^{+} \to \mathrm{Mn}^{2+} + e^-$
• $\mathrm{Fe}^{3+}$ means $I_4$ of Fe: $\mathrm{Fe}^{3+} \to \mathrm{Fe}^{4+} + e^-$

Write relevant electronic configurations

$\mathrm{Mn} : [\mathrm{Ar}]\,3d^5 4s^2$ $\mathrm{Fe} : [\mathrm{Ar}]\,3d^6 4s^2$

Therefore,

$\mathrm{Mn}^+ : [\mathrm{Ar}]\,3d^5 4s^1$ $\mathrm{Mn}^{2+} : [\mathrm{Ar}]\,3d^5$ $\mathrm{Mn}^{3+} : [\mathrm{Ar}]\,3d^4$

$\mathrm{Cr} : [\mathrm{Ar}]\,3d^5 4s^1$ $\mathrm{Cr}^+ : [\mathrm{Ar}]\,3d^5$

$\mathrm{Fe}^{2+} : [\mathrm{Ar}]\,3d^6$ $\mathrm{Fe}^{3+} : [\mathrm{Ar}]\,3d^5$

Check each option

Option A: $\mathrm{Mn}^{2+} < \mathrm{Fe}^{2+}$

This compares removal of an electron from:

• $\mathrm{Mn}^{2+} : 3d^5$
• $\mathrm{Fe}^{2+} : 3d^6$

Removing one electron from $\mathrm{Mn}^{2+}$ breaks the especially stable half-filled $3d^5$ configuration. Removing one electron from $\mathrm{Fe}^{2+}(3d^6)$ gives $\mathrm{Fe}^{3+}(3d^5)$, which is a stable half-filled configuration.

Hence, $IE(\mathrm{Mn}^{2+}) > IE(\mathrm{Fe}^{2+})$

So the given relation $\mathrm{Mn}^{2+} < \mathrm{Fe}^{2+}$ is incorrect.

Option B: $\mathrm{Mn}^{+} < \mathrm{Mn}^{2+}$

Compare:

• $IE(\mathrm{Mn}^+)$: removing $4s$ electron from $[\mathrm{Ar}]3d^5 4s^1$ gives $[\mathrm{Ar}]3d^5$
• $IE(\mathrm{Mn}^{2+})$: removing electron from stable $[\mathrm{Ar}]3d^5$ gives $[\mathrm{Ar}]3d^4$

The second process removes an electron from a half-filled stable configuration, so it requires more energy.

Therefore, $IE(\mathrm{Mn}^+) < IE(\mathrm{Mn}^{2+})$

Option B is correct.

Option C: $\mathrm{Mn}^{+} < \mathrm{Cr}^{+}$

Compare:

• $\mathrm{Mn}^+ : [\mathrm{Ar}]3d^5 4s^1$; removing the electron gives $[\mathrm{Ar}]3d^5$
• $\mathrm{Cr}^+ : [\mathrm{Ar}]3d^5$; removing an electron gives $[\mathrm{Ar}]3d^4$

For $\mathrm{Cr}^+$, an electron is removed from the half-filled stable $3d^5$ configuration, so this needs more energy than removing the outer $4s$ electron from $\mathrm{Mn}^+$.

Hence, $IE(\mathrm{Mn}^+) < IE(\mathrm{Cr}^+)$

Option C is correct.

Option D: $\mathrm{Fe}^{2+} < \mathrm{Fe}^{3+}$

Compare:

• $\mathrm{Fe}^{2+} : [\mathrm{Ar}]3d^6 \to [\mathrm{Ar}]3d^5$
• $\mathrm{Fe}^{3+} : [\mathrm{Ar}]3d^5 \to [\mathrm{Ar}]3d^4$

Removing from $\mathrm{Fe}^{3+}$ breaks the stable half-filled $3d^5$ configuration, so it requires more energy.

Therefore, $IE(\mathrm{Fe}^{2+}) < IE(\mathrm{Fe}^{3+})$

Option D is correct.

Conclusion

The only incorrect relationship is:

$\boxed{A}$