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What is the maximum number of double bonds a carbon atom can make?

Well, carbon has $4$ valence electrons so... it needs $4$ more to fulfill the octet rule right?

If a double bond yields $4$ electrons, does it mean that the carbon atom can make only one double bond?

OR does it mean it can create up to TWO double bonds (because they would yield $8$ electrons which fulfill the octet rule)?

What about Calcium? I ask because Calcium doesn't follow the Octet rule: it only wants to have 4 electrons, so...

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    $\begingroup$ Calcium generally forms fairly ionic compounds so the concept of a double bond isn't necessarily applicable. In regards to carbon, each double bond gives it two more electrons, on top of those that it already has, so I'll let you figure out which answer that leaves you with. $\endgroup$
    – bon
    Commented Jul 16, 2015 at 19:36
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    $\begingroup$ Rather than focus on what it's gaining, try to draw a Lewis structure of carbon connecting to a carbon on each side. Then try changing the connections between them to different numbers and arrangements of bonds and check whether or not the central carbon has a satisfied octet. By doing this systematically you can probably "prove" the answer to yourself. $\endgroup$
    – Eli Riekeberg
    Commented Jul 16, 2015 at 21:39

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Well known carbon compounds have two sets of double bonds which agrees with the simple idea from the octet rule

Simple bonding theories like the octet rule often don't work because they overly simplistic (just look at the variety of bonding in sulfur compounds).

It is usually better to look at actual examples of real compounds we know about. In the case of carbon we have a whole class of compounds known as allenes which have a single carbon with two double bonds.

So, from observation alone, we can conclude that a carbon with two double bonds is possible.

The situation with calcium is irrelevant as most compounds will not be covalent but ionic (and counting bonds in ionic compounds is pretty much irrelevant).

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[OP ...] carbon has 4 valence electrons so... it needs 4 more to fulfill the octet rule right?

Yes, that is correct.

If a double bond yields 4 electrons, does it mean that the carbon atom can make only one double bond? Or does it mean it can create up to TWO double bonds (because they would yield 8 electrons which fulfill the octet rule)?

The easiest way to do this in your head is to assume that for each bond, one electron "comes" from the carbon atom of interest, and the second "comes" from the bonded atom. So carbon often makes four bonds (which could be two double bonds, as in carbondioxide, in allenes or cumulenes). There are less common cases where carbon has a positive or negative formal charge but it is useful to first become familiar with the common cases (where carbon forms four bonds) before you move on to those.

What about Calcium? I ask because Calcium doesn't follow the Octet rule: it only wants to have 4 electrons, so...

Calcium typically does not form covalent electrons. In many compounds, it occurs as ion with two positive charges. Instead of completing the octet, it forms ions with a completely empty outer shell (so the fourth shell is empty, and the third one is full, with an octet).

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