How does the number of protons determine an element's qualities?

Its all about valence electrons. These are in the outer shell, and an atom ‘wants’ 8 of them. It will attract other atoms to make its valence shell have 8 electrons. If an atom already has 8 valence electrons, it will be neutral

We order the elements in the periodic table based on this.

I believe that part of it has to do with stability. The greater the number of protons, the more electrons required to have an atom with zero charge. The chemical properties of an atom are related to whether it’s an ion (atom with a positive or negative charge) or neutral.

Is your first question is asking why the proton count determines the element as opposed to some other property?

I suppose the reason why is because it has the most significant physical effect of all the atomic particles. Neutrons don’t have charge, so they cannot affect other particles electromagnetically. Electrons do, but the nucleus of an atom can easily pick up or lose electrons while a nucleus will not lose or gain protons under normal circumstances. Therefore it’s more convenient to categorize elements by the number of protons rather than the number of electrons.

As for how this number translates into the properties we observe, it has to do with the way elements bond with each other in chemical reactions and the resulting geometry of the atoms due to these bonds. For example, graphite, AKA pencil lead, which is a form of carbon, leaves marks on surfaces because the atomic structure of carbon resembles a series of strongly held carbon atoms arranged in flat planes, each plane held together by weaker bonds. When you draw with a pencil, you are essentially removing these weakly bonded planes of carbon.

@jackm Got the answer square on the nose. It’s all about the electrons and how they interact.

And the number of electrons an atom has (and therefore the number it needs to complete a shell) is determined by the number of protons?

@Hobbes

Yes, atoms “want” to be electrically neutral. So the number of protons in turn determines the number of electrons an atom will typically have, as well as how firmly secured those electrons are to the nucleus. It’s these properties that determine the chemical behavior of an atom.

@Hobbes The number of electrons needed to complete a shell is independent of the number of protons. There is a maximum number of electrons that can fit into each energy level (given by the formula 2n^2, although the maximum in an outer shell is 8), and they all want to be full. That is why metals form positive ions and halogens form negative ions. The periodic table treats atoms as electrically neutral, because that is the ‘ground’ state where they are not ions.

The nature of a thing is determined by it’s posotive and not its negative.