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I have a mixture of sodium carbonate, sodium bicarbonate, ammonium carbonate, copper (II) sulfate, hydrogen peroxide, and sulfuric acid. I'm titrating potassium permanganate against this mixture to find out the amount of hydrogen peroxide in it. Will anything else in this mixture react with potassium permanganate? If so, is there another better way of determining the amount of hydrogen peroxide in it? Thank you.

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I would not use potassium permanganate with sulfuric acid present: enter image description here

You'll throw off your result due to the competing reaction..and risk a fire depending on the concentration of sulfuric acid.

If you measure the amount of $\ce{O2}$ produced by the decomposition of $\ce{H2O2}$ you can back-calculate how much $\ce{H2O2}$ was present in the sample. I think (with a decent certainty) that none of the species you list here should produce gas besides the peroxide.

That said I would use a catalyst (like $\ce{MnO2}$) for decomposition instead of a potentially reacting species (like potassium permanganate). You will likely get side reactions that consume the catalyst but they don't produce gas. So simply use excess $\ce{MnO2}$.

Measure the volume of $\ce{O2}$ produced (i.e. wait until the gas volume no longer increases) and use the ideal gas law (as a fair approximation) to get moles of $\ce{O2}$ produced. Once you have that, you can stoichiometrically determine moles of $\ce{H2O2}$ in your sample of known volume, thus concentration of $\ce{H2O2}$ (recalling that $2\ce{H2O2} \rightarrow 2\ce{H2O} + \ce{O2}$).

I've done this before with $\ce{MnO2}$ from batteries, washed thoroughly with water. I capture the gas produced in an upside-down grad-cylinder in a tub/beaker of water, e.g.:

http://www.docbrown.info/page13/ChemicalTests/GasPreparation.htm

P.S. if you manage to / happen to get a pure $\ce{H2O2}$ sample, you can very accurately get concentration (in %) by the following:

$C = \frac{d - 0.9946}{0.00456}$ where $d$ is the density in g/mL at 20deg. C.

That's a standardized function I derived from known and experimental densities of $\ce{H2O2}$

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