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There are three factors which decide how easily an electron can be removed, in this order of priority:
- Number of shells (distance from the nucleus in effect)
- Effect of shielding
- Nuclear attractive force from protons
First ionisation
Both #"K"# (potassium) and #"Ca"# (calcium) are in Period 4.
To ionise once, we must remove one electron from these two metals.
#"K"# is in Group 1. It, therefore, has one electron in its outer shell (in the #s# orbital). #"Ca"# is in Group 2, and has two electrons in its outer shell (again both in the #s# orbital).
#"Ca"# has the same number of shells as #"K"#, a similar amount of shielding, but more protons in the nucleus. This means there is a stronger attraction between the nucleus and the electron to be removed, meaning more energy is required to remove it (moving it to a potential of #"0 eV"#).
#"Ca"# has a higher first ionisation energy than #"K"#.
In the case of #"K"#, this leaves it with a full third shell, as a #"K"^+# ion. For #"Ca"#, it becomes a #"Ca"^+# ion, with one electron still remaining in the 4th shell.
Second ionisation
In this case, we are ionising the #"K"^+# ion and the #"Ca"^+# ion. These are formed after the first ionisation.
#"K"^+# has the electron configuration #["Ar"]#. This means it has three full shells. #"Ca"^+# has the same electron configuration as an atom of #"K"#, i.e. #["Ar"]4s^1#.
#"Ca"^+# has more shells and shielding than #"K"^+#. This outweighs the fact that #"Ca"^+# has more protons in the nucleus, meaning means there is a weaker attraction between the nucleus and the electron to be removed, meaning less energy is required to remove it.
#"K"^+# has a higher ionisation energy than #"Ca"^+#, so #"K"# has a higher second ionisation energy than #"Ca"#.
Hope this helped :)
The second ionization energy of calcium is higher than its first ionization energy. You are, after all, removing an electron from a positively charged ion; however, the Ca+ ion does not have a noble gas configuration, so it easily loses a second electron to achieve one. If you were to look up calcium's 3rd ionization energy, you would find it to be tremendously higher than the second, just as potassium's second ionization energy is much higher than its first.