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When iron(III) nitrate is added to hydrogen peroxide as a catalyst, the temperature initially decreases and begins to increase after 30 seconds. Why is this the case? (is there any intermediate reactions of $\ce{H2O2}$ and $\ce{Fe(NO3)3}?)$

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Nitrates tend to lower the temperature on dissolving, having a positive enthalpy change of solution. For example, $\ce{NH4NO3}$ has an enthalpy of solution in water of ~26 kJ/mol at ~300 K, and it is used in self-cooling cold-packs for that reason.

Sorry, I could not locate the enthalpy change of solution for $\ce{Fe(NO3)3}$, so this is not a definitive answer. You could measure that value, though, with some $\ce{Fe(NO3)3}$, some $\ce{H2O}$ and a calorimeter.

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