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I was reading an article about a chemical reaction, and I came across the phrase:

The oxygen atom at this point has three bonds and has a net positive charge

How can this happen? Oxygen has 2 missing electrons in the valence shell. Therefore it can only form 2 bonds at the most, if both are sigma bonds.

Does it mean the 3rd bond is not covalent? Can it happen with a hydrogen or an ionic bond?

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If you think that's strange, try a carbon with three, a carbon with five, a nitrogen with four, or a boron with four. As perhaps you're not used to this, you'll see later that counting bonds is not always simple, or even a sensible thing to try. – Nij 49 mins ago

Consider the auto-ionization of water :

$\ce{ 2H_2O->H_3O+ + OH-}$

The first oxygen has three bonds, the second only has one.

You can think of the reaction taking place by a lone pair on the oxygen of one water molecule ripping off the proton only of the hydrogen of another water molecule to form a covalent bond between them using just the lone pair. The electron of the hydrogen is left behind and stays with the oxygen of the other molecule.

If you calculate the formal charges on each oxygen you will see the first one has a positive charge and the second one has a negative. The formal charge is just the valence number of electrons minus the number of bonds minus non-bonding electrons (using the lewis structure) and is a useful book keeping method to think about where the electrons go/are and what are the most stable structures.

Formal Charge Calculation

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By the way, how does a lone pair create a bond between a hydrogen atom? Wouldn't only one hydrogen be required? Since there is only one hydrogen required? If both electrons in the lone pair covalently bond to the hydrogen, the hydrogen would have 3 electrons, which would be unstable, right? – Michael Mshtc 6 hours ago
    
Yes, if the lone pair and the electron on the hydrogen were in the bond then the hydrogen would have three electrons, but that doesn't happen. The lone pair on the oxygen basically rips just the proton of the hydrogen off the other molecule leaving the electron that was already on that hydrogen behind. – Cliff Stamp 6 hours ago
    
@MichaelMshtc This type of covalent bond, where both bonding electrons come from the same atom, is called a coordinate bond or sometimes a dative covalent bond. It's far more common than you'd guess based on its coverage in intro chem courses. – Jason Patterson 3 hours ago
    
+1 for hydronium. How could anyone who opened a chemistry book not know that one?!? – MaxW 2 hours ago

Oxygen has $6$ electrons in its outermost shell.

However, if you give it a positive charge by removing an electron, then it would only have $5$ electrons in its outermost shell.

Therefore, it can now from $3$ bonds while carrying a positive charge.

You can usually see trivalent oxygen in intermediates of reactions, because you can protonate a bivalent oxygen to make it trivalent, and then cleave one of the original two bonds to leave yourself with new molecules.

A bivalent oxygen is also easily protonated because it has a lone pair, making it a nucleophile.

Trivalent oxygen intermediates in organic reactions are also resonance-stabilized because the positive charge can be shared by the carbon in immediate vicinity.

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Apart from hydronium ion $\ce{H3O+}$, there are some other molecules or ions which possess three oxygen bonds. All pictures taken from Wikipedia

  1. Carbon monoxide

enter image description here

  1. Ozone

enter image description here

  1. Nitrosonium ion

enter image description here

  1. Transition metal aqua complex

enter image description here

Here each oxygen has three single bonds, one with the central metal and other two with tho hydrogens.

  1. Transition metal oxo complex

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Namely vanadyl chloride, a tungsten oxocarbonyl

enter image description here

trismesityliridium oxide

And there is many more.....

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