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What the question says. For example, if I knew the specific heat capacities of lithium and oxygen, could I work out the specific heat capacity of lithium oxide with no further information?

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There is not much connection between the heat-capacity of a compound and the heat-capacity of the elements it is made of. Heat capacity is primarily coming from the low frequency vibrations of the lattice (if it is solid), and vibration+rotation+translation if it is liquid or gas. These have no connection with the properties of the lattice etc properties of the given elements.

So short answer: no

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