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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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Yes, one can calculate the specific heat capacity of the compound from its constituent elements that is just some of its constituent element's specific heat capacity. As it is known that specific heats of elements remain unchanged when they enter into compounds. However, the density/volume/mass may change and so the heat capacity.

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    $\begingroup$ While the empirical Neumann-Kopp method for estimating the heat capacities of compounds works reasonably well in many cases, it fails miserably in many more. $\endgroup$
    – Jon Custer
    Jul 9 '21 at 15:29

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