Explanation:

The third ionization energy is the energy required to remove a third electron from an atom. It is generally higher than the first and second ionization energies because removing a third electron requires breaking into a more stable, deeper energy level.

To determine which of the given transition elements is expected to have the highest third ionization enthalpy, we need to consider their electron configurations and the trends in ionization energies across the period.

The electron configurations of the given transition elements are as follows:
a) Iron (Z=26): 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁶
b) Chromium (Z=24): 1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹ 3d⁵
c) Vanadium (Z=23): 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d³
d) Manganese (Z=25): 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁵

Trends in Ionization Energies:
1. Ionization energy generally increases across a period from left to right due to increasing effective nuclear charge and decreasing atomic radius.
2. Within a given energy level (e.g., 3d), removing an electron from a half-filled or fully-filled subshell requires more energy due to increased stability.

Analysis:
In this case, we compare the electron configurations of the given transition elements to determine the expected highest third ionization enthalpy.

- Iron (Z=26): The electron configuration of iron shows that it has one unpaired electron in the 3d subshell. Removing a third electron from the 3d subshell requires breaking into a more stable, half-filled subshell. Therefore, iron is expected to have a relatively high third ionization enthalpy.

- Chromium (Z=24): The electron configuration of chromium shows that it has a half-filled 3d subshell. Removing a third electron from the 3d subshell requires breaking into a fully-filled subshell. This is more stable than iron's configuration, so chromium is expected to have a higher third ionization enthalpy than iron.

- Vanadium (Z=23): The electron configuration of vanadium shows that it has three unpaired electrons in the 3d subshell. Removing a third electron from the 3d subshell requires breaking into a more stable, half-filled subshell. Therefore, vanadium is expected to have a relatively high third ionization enthalpy.

- Manganese (Z=25): The electron configuration of manganese shows that it has a half-filled 3d subshell. Removing a third electron from the 3d subshell requires breaking into a fully-filled subshell. This is more stable than vanadium's configuration, so manganese is expected to have a higher third ionization enthalpy than vanadium.

Conclusion:
Based on the analysis, it can be concluded that the transition element with the highest expected third ionization enthalpy is manganese (Z=25).