How do I rank the following compounds from lowest to highest boiling point: calcium carbonate, methane, methanol (CH₄O), dimethyl ether (CH₃OCH₃)?

The order of boiling points is: ##”CH”_4 < "CH"_3"OCH"_3 < "CH"_4"O" < "CaCO"_3## The order of strengths of intermolecular forces is: ##”ion-ion > H-bonding > dipole-dipole > London dispersion”##.
with stronger intermolecular forces have higher boiling points.
The strongest intermolecular force in each of the compounds is:
##”CaCO”_3## — ion-ion attractions.
##”CH”_4## — London dispersion forces
##”CH”_3″OH”## — hydrogen bonding
##”CH”_3″OCH”_3## — dipole-dipole attractions
##”CaCO”_3## is an ionic compound. It has the highest boiling points
Next comes methanol, ##”CH_4O”## or ##”CH_3OH”##.
Methanol has strong . It will have the next highest boiling point.
Dimethyl ether, ##”CH”_3″OCH”_3##, is a polar molecule.
The ##”C-O”## bond dipoles reinforce each other, so the molecule has a dipole moment.
Dipole-dipole forces are not as strong as hydrogen bonds, so dimethyl ether has a lower boiling point than methanol does.
Finally, the ##”C-H”## bonds in methane are nonpolar, so the molecule is also nonpolar.
It has only weak London dispersion forces,
##”CH”_4##, has the lowest boiling point.
The order of boiling points is: ##”CH”_4 < "CH"_3"OCH"_3 < "CH"_4"O" < "CaCO"_3## Here's a good video on ordering compounds according to their intermolecular forces and boiling points.

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