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The first and second electron have an energetic relationship, meaning that there is some e-/e- repulsion between the 2. When the first is removed, that repulsive force disappears, leaving the second electron more tightly bound to the nucleus. Additionally, since the nucleus has less electrons to hold on to, the effective nuclear charge on each electron increases with the removal of the first.
One reason for this is that once you remove the first electron from at atom, it is now a positively charged ion. Because of this greater positive charge in the nucleus, the electrons will be more attracted to the nucleus and therefore more difficult to remove. Additionally, subsequent electrons may be in lower energy shells and therefore closer to the nucleus. Electrons closer to the nucleus are shielded less and by other electrons because there are less electrons between the outer electron and the nucleus. the electron will again experience a greater positive attraction and be more difficult to remove.
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