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In school, for determining whether single displacement reactions occur, we use an activity series. There is one for metals, and one for halogens. Why is there none for non-metals in general? Are halogens the only ones that will participate in single displacement reactions?

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I think, periodic trends is the key word. For the halogens, they are homologue elements from a single group. Therefore, they display very similar overall reactivity and typically either fluorine or iodine is best at something, either because a large size and low electronegativity is good for a process or because of the opposite.

I don’t know which metals are included in your series. But I think that if they are from the first two groups and maybe including aluminium, then the reactivities are still similiar enough to compare along the lines of general trends.

However, once you start going to different non-metals — or worse, to metalliods — things become a lot less predictable. You can no longer draw a line down (or up) a group because you have different groups involved. And you can not draw a line along a period because there will be multiple periods. Quick and dirty methods are bound to fail soon.

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Long story short, metals are relatively uniform and nonmetals are diverse.

Indeed, metals have much in common. Many would react with acids, and nearly all would react with salts of some other metals, so the position of a metal in the activity series tells us a good deal about its possible reactions.

With nonmetals, it is not that simple. True, you can order them by electronegativity, and this is quite useful, but not that useful as it was with metals. While the displacement reactions are not limited to halogens, they do not constitute a majority of all reactions. For example, I expect this to happen: $\ce{K2S + Br2 -> 2KBr + S}$, but this tells us nothing about the oxygen-containing compounds of sulfur. Will bromine replace sulfur in $\ce{K2SO4}$? It won't. How do you know that from electronegativity alone? You don't.

So it goes.

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