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We are looking at the reaction of glucose with potassium permanganate under acidic conditions. This is a complicated reaction but will ultimately produce CO2 and H2O. If you plotted the rate of the reaction versus the concentration of glucose you would get a straight line. This tells us that the rate is directly proportional to the concentration of glucose. So equating the two mathematically:
rate = k*[glucose]
This is the equation of a first order reaction. So we have shown that the reaction of glucose with potassium permanganate is a first order reaction with respect to glucose.
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