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The first ionization energy of chloride is significantly greater than that of sulfur. I suggest that this is very expected as the effective nuclear charge of chloride is higher and additionally, sulfur has even more electron-electron repulsion due to advent of paired electrons in 2p orbitals. However I am not too sure on why the 2nd ionization energies are almost the same.

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The second ionization of chlorine leaves $[\text{Ne}]3s^23p^3$ where the $3p$ subshell is half-filled with one electron per orbital and gives a favorable exchange energy. With sulfur this relatively favorable configuration is achieved instead after the first ionization, the second ionization of sulfur breaks the half-subshell and is thus less favorable.

The effect of the half-filled subshell can be seen elsewhere, as well. The chart below (reference) shows trends in first ionization energies. Counting backwards from the third peak (argon) we see that phosphorus exceeds sulfur violating the usual rules based on effective nuclear charge. The first ionization of sulfur gives the half-filled $3p$ subshell after removing the electron, whereas the first ionization of phosphorus would have to break that configuration. The half-filled shell effect beats effective nuclear charge here.

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