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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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To quote from Science Direct on the Fenton process:

The Fenton reaction was first reported by H. J. Fenton (1894) and it is described as the enhanced oxidative potential of H2O2, when iron (Fe) is used as a catalyst under acidic conditions. The reactions involved in Fenton processes are (Haber and Willstätter, 1931):

$\ce{Fe++ + H2O2 -> Fe+++ + .OH + OH-}$ (3.1)

$\ce{H2O2 + .OH -> .HO2 + H2O}$ (3.2)

$\ce{Fe++ + .OH -> Fe+++ + OH-}$ (3.3)

$\ce{Fe+++ + .HO2 -> Fe++ + O2 + H+}$ (3.4)

Also further important comment:

Since the reaction of H2O2 with Fe3+ is significantly slower as compared to Fe2+, the catalyst regeneration phase is considered the most critical stage of the process as it reduces the radical production rate and could end up depleting the iron available for the reaction (Eqs. 4.3–4.4) (Malato et al., 2009).

So, there are, indeed, intermediate reactions with ferric cycling to ferrous.

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