What is Homogeneous and heterogeneous catalysis?

Example 42.4 (Iodide ions and hydrogen peroxide)

Hydrogen peroxide decomposes very slowly on its own: $\ce{2H2O2 -> 2H2O + O2}$. With a little potassium iodide dissolved in it, it froths at once. The iodide ion opens a path in two steps:

Added, the two steps give back $\ce{2H2O2 -> 2H2O + O2}$: the iodide ion, used in the first step and returned in the second, is the catalyst; the ion $\ce{IO-}$ is an intermediate.

Example 42.5 (Iron(III) ions)

Iron(III) ions also catalyse the decomposition, through the couple $\ce{Fe^{3+}}$/$\ce{Fe^{2+}}$ (Chapter 35): they first oxidise hydrogen peroxide, $\ce{2Fe^{3+} + H2O2 -> 2Fe^{2+} + O2 + 2H+}$, then the iron(II) ions formed are oxidised back by more hydrogen peroxide, $\ce{2Fe^{2+} + H2O2 + 2H+ -> 2Fe^{3+} + 2H2O}$. The pale orange colour of the solution turns greenish while the gas is given off, then comes back when the hydrogen peroxide is used up: a sign that the catalyst takes part and is regenerated.