I'm not sure about all the properties of Potassium Permanganate but if it can be used to determine the excess reagent in a timely manner in the second part of the back titration, why can't they do a direct titration of the Potassium Permanganate in the solution? They wouldn't be using it out of a time concern since it would presumably take the same amount of time for the first and second steps to dissolve and if it were out of concern of solubility, the issue would persist when the acid begins to run out on the second step.
I wondered the same and looked up what a back titration is.
So, basically: They throw in so much KMnO4 that everything in the original solution reacts, and some KMnO4 remains unreacted. Then they titrate the leftover KMnO4 with another reagent with a known concentration to determine how much is left of it. Based on that result and the initial concentration of the KMnO4 solution, they then calculate the unknown concentration of the original solution.
I still don't get why this can't be done by titrating the solution with unknown concentration directly with the KMnO4 solution here, maybe because it would have all kinds of side reactions that make a determination of the concentration with a direct titration with KMnO4 difficult to impossible? In any case, according to what I found explained by Mettler Toledo, it's the method of choice when it's easier to determine the concentration that way than with a direct titration, for example, for carbonate ions in water.
