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The following is an answer for US 9th / 10th grade students or UK GCSE students. Advanced chemistry students will understand that there are some errors in the description but they are of no significance for students studying basic chemistry. So with apologies to the purists...

All atoms have electrons that orbit the nucleus. The electrons orbit in layers with two electrons in the first orbit, 8 in the next orbit, 8 in the third orbit and so on. Different elements have different numbers of protons in the nucleus and they normally have the same number of electrons orbiting the nucleus.

Ionic bonds

Let's take Sodium as one example. It has 11 protons which means it also has 11 electrons. The electrons are arranged with 2 in the lowest orbit, 8 in the next orbit and finally, one left over which is in the third orbit.

Another example is Chlorine. Chlorine has 17 protons and 17 electrons. These electrons are arranged with 2 in the lowest orbit, 8 in the next orbit and then 7 in the third orbit. The third orbit has just one space left for another electron.

Atoms are very happy when they have their electron orbits completely filled. Sodium has one electron in an orbit and the atom doesn't like it much. Likewise, the Chlorine atom has a hole in the third orbit and it's not keen on having a hole. So, the Sodium atom wants to get rid of its spare electron and the Chlorine really wants to find a spare electron to fill the hole. When the two atoms are brought together, the spare atom belonging to the Sodium atom finds its way to the hole in the Chlorine atom's electron orbit. Now both atoms have filled electron orbits and they are both happy, as long as they stay close to each other.

That's the way an ionic bond works, with one atom giving up one or more electrons and another atom collecting one or more electrons. It is this need to stay close to each other that gives ionic bonds their strength.

Covalent bondsCovalent bonds normally happen when both atoms have a few electrons in their outer electron orbits. The atoms still want to have their electron orbits filled but neither of them have the right number of electrons to be able to give or take the way ionic bonds do. The two atoms will share their electrons between each other. In this way, one atom can fill its electron orbit with a few electrons shared from the other atom. Likewise, the second atom can share a few electrons from the first atom. Each atom feels like it has filled electron orbits so they are now happy atoms. All the atoms have to do is to stay together so they can keep sharing their atoms. Covalent bonds are strong but because the atoms aren't attracted to each other as much as ionic bonds, covalent bonds are often not as strong as ionic.
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