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

The order of strengths of intermolecular forces is: #"ion-ion > H-bonding > dipole-dipole > London dispersion"#.

Compounds 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"_4"O"# or #"CH"_3"OH"#.

Methanol has strong hydrogen bonds. 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.