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Previously, I was trying to understand why $\ce{Al2O3}$ has a lower melting point than MgO. An answer on quora said

Do not confuse the character of the bond between the metal and oxygen with the bonds between molecules in the compound. The polar character of the bonds between aluminum and oxygen is indeed less than between magnesium and oxygen but this describes the energy needed to separate the metal from oxygen. When a solid melts it doesn’t break down the bonds between oxygen and aluminum or magnesium. The intermolecular bonds (that is bonds between one $\ce{Al2O3}$ and another $\ce{Al2O3}$) are in fact weaker than the bonds between two magnesium oxide molecules. This results in the greater melting point of magnesium oxide in spite of the less covalent character of the bonds within the molecule.

But if $\ce{Al2O3}$ is ionic and forms a giant lattice, which bonds that are "between one $\ce{Al2O3}$ and another $\ce{Al2O3}$" are broken?

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Right, the text you are mentioning is not clear enough. It is not correct to say that the molecule $\ce{Al_2O_3}$ exists and is separated from the next $\ce{Al_2O_3}$. Such a molecule does not exist. A sample of $\ce{Al_2O_3}$ is made of a superposition of $\ce{Al^{3+}}$ ions and $\ce{O^{2-}}$ ions. It is not made of a superposition of molecules $\ce{Al_2O_3}$. Now the fact that MgO melts at a higher temperature than $\ce{Al_2O_3}$ is due to the stronger attraction between $\ce{Mg^{2+}}$ and $\ce{O^{2-}}$ ions. For some reason, the attraction between $\ce{Al^{3+}}$ ions and $\ce{O^{2-}}$ is weaker.

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    $\begingroup$ Actually, it is well known that in the liquid phase Al is predominantly 4-fold coordinated with O. Oxide melts are not simple, generally speaking. Now, this does not imply that there are specific, identifiable, fixed molecules floating around - each atom prefers a surrounding amenable to it, which constrains the available liquid configuration. $\endgroup$ – Jon Custer Jan 1 at 16:56
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    $\begingroup$ A substantial part of this answer is beating around the bush and can be replaced by a single mention that $\ce{Al2O3}$ is a formula unit for the network solid and not a molecular formula. I doubt you can directly compare magnesium and aluminium oxides in terms of $\ce{M-O}$ bonding like that. Both have different crystal structures (trigonal vs cubic) and coordination environments, and it's not clear how you draw a conclusion about the "attraction" (whatever this means) between metal and oxygen atoms. Sorry, but to me this looks like an answer for the sake of answering. $\endgroup$ – andselisk Jan 4 at 14:57
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The quote you give clearly summarises the situation. In a solid (or liquid) there has to be some significant interaction between individual molecules otherwise there would only be gasses. This inter-molecular interaction is not that forming chemical bonds, i.e. a solid is not a super big molecule, but is additional to these and is due to the nature of the molecules and has several different forms.

The general term is inter-molecular forces, sometimes also call van-der-Waals forces and these are generally much weaker than a chemical bond. These forces arise from charge-charge (Coulomb), charge-dipole, dipole-dipole, dipole--induced-dipole and induced-dipole--induced dipole (also called dispersion or London forces) interactions between molecules.

You can find a summary in many phys. chem. textbooks but for all the gory details see 'Intermolecular and Surface Forces' by J. Israelachvili, publ Academic Press.

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