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I have come across 4 ways of defining oxidation:

  • loss of electrons
  • increase in oxidation number
  • gain of oxygen
  • loss of hydrogen

From these definitions, I have an idea of what I think 'oxidation' and 'reduction' really refer to, but I would appreciate if someone could verify or tell me if I am going wrong.

I think oxidation and reduction simply refer to the degree to which the electrons of an atom 'dissociate' or 'distance' themselves from the nucleus of an atom. We define this extent of distancing to be 0 when an atom is bonded to another atom of the same element.

Thus oxidation really is 'loss of electrons', and reduction is 'gain of electrons', but the electrons do not have to be completely gained or lost. It is more like, oxidation is the valence electron shell moving further from the nucleus and reduction is the valence electron shell coming in closer to the nucleus. I.e. in a chemical reaction when the atoms surrounding a particular atom, say A1, are changed, the terms 'oxidation' and 'reduction' give a measure as to whether the atoms surrounding A1 have an overall effect of being more electronegative than the atoms that were bonded to A1 previously (in which case A1 has been oxidised) or less electronegative (A1 has been reduced).

Then the increase in oxidation number coincides with these definitions.

When it comes to REDOX in terms of hydrogen and oxygen transfer, I think this is only valid for organic chemistry. In general, loss of gain of hydrogen is to a carbon atom or an oxygen atom, or maybe nitrogen and sulfur too. As these are more electronegative than hydrogen, a gain in hydrogen means that these atoms can attract more electrons towards themselves, whereas a loss of hydrogen means that they lose electrons that they could be attracting towards themselves.

In terms of oxygen, in most of these organic systems it is gain of oxygen to carbon, sometimes nitrogen, which are less electronegative than oxygen. So by the same idea as above with hydrogen, a gain in oxygen corresponds to decrease of electron density being held around the atom, i.e. oxidation.

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You are quite right, those rules of thumb are not always valid.

To be precise, the "adding hydrogen = reduction" rule only works for elements more electronegative than hydrogen, and the "adding oxygen = oxidation" rule only works for elements less electronegative than oxygen. For example, sodium is less electronegative than hydrogen, and as such adding hydrogen to sodium ($\ce{Na -> NaH}$) is in fact an oxidation of sodium from 0 to +1.

However, I wouldn't be so quick to limit their application to "organic chemistry". The elements that are less electronegative than hydrogen are just the metals, and the only element more electronegative than oxygen is fluorine.

Therefore, adding oxygen to chlorine is an oxidation, although the oxides or oxoanions of chlorine are classified under inorganic chemistry. Likewise, nearly the entirety of the p-block obeys these rules (except for Group 13). Think about germanium, arsenic, etc.. those are undoubtedly not "organic chemistry".

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