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I recently did an experiment to determine the activation energy of the decomposition of hydrogen peroxide using iron (III) chloride as a catalyst. I repeated this experiment at different temperatures. I know how to figure out the rate constant and the rate law, but I am not sure if this is a second-order reaction. Is there any way I can figure out the order of this reaction? (because that would impact the units of the rate constant, k, and the rate law)

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Check out this paper. There they studied the decomposition of hydrogen peroxide by iron hydroxide and found that the reaction is a second order reaction for low concentrations of hydrogen peroxide.

In your case, you are working with a soluble iron species which will change the reaction kinetics because the hydrogen peroxide does not need to adsorb to the surface of insoluble iron hydroxide particles. This is assuming you worked under neutral or better slightly acidic conditions, otherwise the iron will precipitate as iron hydroxide.

Therefore, in your case the kinetics should depend on the relative concentration of iron chloride and hydrogen peroxide. As long as the hydrogen peroxide is in excess, I would expect a zero order reaction because the iron will limit the reaction speed. If the iron is in excess, I would expect a second order reaction because you need two molecules of hydrogen peroxide to form molecular oxygen.

How you experimentally determine the reaction order is described on Wikipedia. Briefly, you need to perform the experiment at different concentrations of hydrogen peroxide and measure the initial reaction speed. Then you plot the logarithm of the initial rates against the logarithm of the concentrations. The slope of the resulting line should be your reaction order.

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