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Ionization energy can be defined as the amount of energy required to remove one (or more) electron(s) from a neutral atom to form a positively charged ion. This increases across a period due to the increasing number of valence electrons across a period. For example, an alkali metal such as potassium (K) only has one valence electron, and tends to want to give it away to other, more electronegative molecules. However, a halogen such as fluoride (F) has 7 valence electrons and wants to complete its valence electron octet to reach a more stable state rather than give away an electron and only have 6 remaining. Due to this tendency, much more energy is required to remove an electron from fluoride than potassium, so it is said that the trend of ionization energy increases across a period. Hope this helps!
Its because the atomic radii decrease across a period, so the electrons are in the same shell closer to the nucleus and the increased nuclear charge attracts them closer to the center. Since its closer to the center its harder to remove an electron, and from here you can consider the octet rule where it wants to keep its electrons because it wants to satisfy the octet rule.
Ionization energy as you go across a period because there are more electrons in the valence shell. Since a full shell is the most stable, elements with less valence electrons are more readily able to lose their electrons because they will be closer to a full shell. And since elements with close to a full shell of electrons want to reach a full shell as well, they more readily keep their electrons and try to gain more. Furthermore, ionic radius decreases across a period, requiring more energy to remove an electron.
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