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.