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By removing an electron, there is now a larger ratio of protons to electrons meaning that the electrons that are left are more attracted to the positive nucleus. This attraction is harder to overcome, therefore it takes more energy to remove the next electrons.
In certain cases, such as the alkali metals in Group 1A, this idea is very easily imagined. Since these metals all have only 1 valence electron in their ground state, it should be fairly easy (low energy requirement) to remove that one electron. However, after this electron is lost the atom is left with a electron structure identical to a noble gas, which are inert and very hard (high energy requirement) to ionize.
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