Compare the amount of thermal energy needed to melt a solid and the amount of thermal energy needed to freeze the same liquid

They have the same magnitude but differ on the sign.

Explanation:

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Melting enthalpy is related with the necessary energy a solid sample needs to be converted to liquid; in general, such energy is positive as the initial solid needs to gain energy to become liquid as liquid molecules are arranged farther than solid molecules. On the contrary, freezing enthalpy is related with the energy a liquid sample needs to be converted to solid, in such a way, it is negative since that energy must be withdrawn to promote the liquid molecules to get closer and consequently turn solid. Summing this up, they have the same magnitude but differ on the sign.

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To melt a solid, energy must be input into the solid to give the molecules high energy to disrupt the lattice bonds and change phase. To return the liquid to solid, the same amount of energy used in melting is required to be removed from the liquid, ideally. This is assuming there is no loss of energy in the process.