You have samples of four compounds: dimethyl ether, methane, difluoromethane, and ethanol. You measure the boiling points of the compounds as \(-128^{\circ} \mathrm{C},-52^{\circ} \mathrm{C}\) \(-25^{\circ} \mathrm{C},\) and \(78^{\circ} \mathrm{C},\) but then lose the labels for each sample. Make the following predictions: (a) Which compound boils at \(-128^{\circ} \mathrm{C} ?(\mathbf{b})\) Which boils at \(-52^{\circ} \mathrm{C} ?(\mathbf{c})\) Which boils at \(-25^{\circ} \mathrm{C} ?(\mathbf{d})\) Which boils at \(78^{\circ} \mathrm{C} ?\)

Begin by determining the molecular structures of each compound: 1. Dimethyl ether (\(\mathrm{CH_3OCH_3}\)): An oxygen atom is present, which forms a polar bond with the two carbon atoms. There are no hydrogen bonds present. 2. Methane (\(\mathrm{CH_4}\)): The electrons are distributed symmetrically around carbon, so it is a nonpolar molecule with London dispersion forces. 3. Difluoromethane (\(\mathrm{CH_2F_2}\)): The electronegative fluorine atoms create a polar molecule, but there are no hydrogen bonds. 4. Ethanol (\(\mathrm{CH_3CH_2OH}\)): The hydroxyl group (\(\mathrm{-OH}\)) on the molecule allows for hydrogen bonding.

Generally, the hierarchy of intermolecular forces in relation to boiling points is as follows: hydrogen bonding > dipole-dipole > London dispersion forces. Now, let's determine which compound corresponds to each boiling point: (a) A boiling point of \(-128^{\circ} \mathrm{C}\) is very low and indicates weak intermolecular forces. Methane has the weakest forces, London dispersion forces, so it is most likely the compound that boils at this temperature. (b) A boiling point of \(-52^{\circ} \mathrm{C}\) is still low, but notably higher than \(-128^{\circ} \mathrm{C}\). This indicates a compound with stronger intermolecular forces. Both dimethyl ether and difluoromethane have dipole-dipole interactions, however, dimethyl ether has a slightly larger molar mass and might exhibit stronger dispersion forces. Therefore, this boiling point is likely for dimethyl ether. (c) With a boiling point of \(-25^{\circ} \mathrm{C}\), we can infer that this compound has stronger intermolecular forces than the previous two. The remaining compound with dipole-dipole interactions, difluoromethane, is the likely candidate for this boiling point. (d) Lastly, the boiling point of \(78^{\circ} \mathrm{C}\) is significantly higher than the others, indicating very strong intermolecular forces. Ethanol exhibits hydrogen bonding, the strongest type of intermolecular force among these compounds, and is therefore the compound with this boiling point. In conclusion, we have matched the boiling points to the compounds as follows: dimethyl ether at \(-128^{\circ} \mathrm{C}\), methane at \(-52^{\circ} \mathrm{C}\), difluoromethane at \(-25^{\circ} \mathrm{C}\), and ethanol at \(78^{\circ} \mathrm{C}\).