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I could not find the specific heat capacity of zinc sulfate and saw that people use water's heat capacity instead. Why is that?

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If the substance is dissolved in water, it's very typical (and more correct) to use the heat capacity of water in lieu of the pure salt, acid, base, etc.

The reason is simply that, when in solution with water, there is much more water than the solute. It's an approximation, but a decent one. This is common practice in freshman-chemistry lab courses which I've taught.

It's difficult to find heat capacities for pure salts such as zinc sulfate because the means by which we measure it is typically through a reaction, and typically a salt is dissolved in solvent (e.g. water) in order to perform a reaction. That is to say, reactions of solids are just less common than liquids, gases, and solutions.

That said, the NIST database reports a $c_{p}$ for zinc sulfate of 99.01 Jmol$^{-1}$K$^{-1}$ at 298 K.

Looking at the original paper, I found this plot of $c_p$ as a function of temperature. At room temperature, you can see it is roughly 100 Jmol$^{-1}$K$^{-1}$.

Stuff like this can be easily obtained from citrination.com search engine. zn2so4_c

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