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Melting point generally depends on the type of intermolecular forces present in molecules. For example, dipole moments could contribute to a polar structure which would create dipole-dipole interactions. The strength of the double bond and its affect would depend on the size of the atoms involved and the types of intermolecular forces involved.
Melting point and boiling points are determined by how strongly one molecule of a substance is attracted to another molecule of the same substance or another substance, essentially these forces are called intermolecular forces. These stronger these forces, the more energy required to break them and change the substance's phase, resulting in a phase change. The easier way I think about it is you have have a beaker of a solution that just contained the molecule. You wouldn't break the bonds that form the molecule inorder to get a phase shift, you just need to make the molecules less bound to one another. Double bonds are bonds that are present within a molecule and essentially makes up the molecule, so breaking that bond wouldn't result in a phase shift but a breakdown of the molecule.
Melting point has a direct relationship with intermolecular forces. If the strength of a intermolecular force is strong, then the melting point must be high and vice versa with a weaker IMF. For example, molecules that have dispersion would have a lower melting point than for example a hydrogen bonded molecule.
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