Why are carbonyl bonds stronger than those of alkenes?
I will use homolytic bond dissociation energies here to quantify bond strength. Other ways to quantify it exist.
I've read several times that carbonyl bonds are shorter and that you need to apply a higher amount of energy to break them. Why is that so? Is it due to the electronegativity of oxygen, which pulls the bonding electrons near it, thus attracting the carbon as well?
There is a pattern of bonds between different atoms being stronger than bonds between identical atoms. If you take a table of bond dissociation energies to calculate the differences
$$\ce{2 BDE(X-Y) - BDE(X-X) - BDE(Y-Y)},$$
where BDE stands for bond dissociation energy in kJ/mol, you get the following picture for single bonds:
| Single |
H |
C |
N |
O |
| C |
44 |
|
|
|
| N |
183 |
101 |
|
|
| O |
344 |
224 |
93 |
|
| F |
539 |
469 |
248 |
79 |
So for example, two C-H bonds are approximately 44 kJ/mol stronger than a C-C bond and a H-H bond.
For double bonds, you get (taking the bond in dioxygen as a double bond):
| Double |
C |
N |
| N |
210 |
|
| O |
498 |
298 |
Bond dissociation energies are from http://ch301.cm.utexas.edu/thermo/#thermochemistry/enthalpy-bonds.html.
There is a trend as you turn two homo-atomic bonds into two "mixed" heteroatomic bonds: The higher the electronegativity difference, the higher the gain in energy. This is a rough calculation because it is based on average bond dissociation energies. To make a more rigorous case, you would have to go to specific examples, e.g. comparing a given amount of ethane and hydrazine with twice the amount amino methane:
$$\ce{H3C-CH3 + H2N-NH2 -> 2 H3C-NH2}$$
This only shows the correlation, not the cause. In the most direct way, the reason why two C=O double bond is more stable than a C=C bond and a "O=O" bond is because that's what we get when we calculate an approximation of the solution of the Schrödinger equation. And we can test it experimentally.
Trying to explain it with partial ionic interactions, partial charges, or electronegativity is a slippery slope because these are all constructs we have come up, rather than measurable quantities.