Final answer:
Based on the pattern and values of the ionization energies provided, the element with six valence electrons and a large increase in ionization energy after the sixth electron suggests that the element is sulfur (S).
Step-by-step explanation:
The first eight ionization energies provided for a third-row element are indicative of the energy required to remove each electron successively from an atom. These ionization energies follow a pattern that significantly increases when an electron is removed from a core orbital, following the removal of all valence electrons. In this case, the large increase in ionization energy between the sixth (8,496 kJ/mol) and seventh (27,107 kJ/mol) suggests that the first six electrons are valence electrons, and the seventh is a core electron.
Consulting the periodic table and the given ionization energies, we can match these numbers to the element. Elements from the third row elements that have six valence electrons in their outermost shell include sulfur (S) and phosphorus (P). Comparing the provided ionization energies to known values, sulfur matches the pattern and numbers closely, with a significant increase in ionization energy after the removal of the six valence electrons.
Therefore, the element with the provided first eight ionization energies is likely sulfur (S).