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What happens when blue litmus paper is dipped into chlorine water?

Answer
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Hint: Chlorine gas is a strong oxidizing agent and also an acidic gas. Chlorine gas in water is a bleach, bleaching is an oxidation process, if you are considering the color change only refer to it as bleaching.

Complete answer:
If you observe carefully when the blue litmus goes into the chlorine water, you may notice that just before the bleaching process happens, the blue litmus turns red. This happens because before bleaching can occur, chlorine gas needs to dissolve in the water absorbed by the paper. This produces disproportion reaction, chlorine forms a +1 and 1 oxidation state in different compounds at the same time
Cl2+H2OHClO+HCl (hypochlorous acid and hydrochloric acid) with the hypochlorous acid the oxidation state of chlorine is +1 and with hydrochloric acid 1. Formation of hydrochloric acid blue litmus paper turns red, but due to the reaction:
HClOHCl+[O] (nascent oxygen)
HClO is the bleach in the reaction, because the formation of nascent oxygen takes place when it dissociates, which is an oxidizing agent. Thus, it bleaches the litmus paper and turns white.

Note:
This effect is seen better if you dissolve chlorine in water and place a single drop of solution onto the end of a strip of dry blue litmus paper using a glass rod. As the solution diffuses along the paper, you see a red colorization at the front of the moving solution but that is soon discolored as the chlorine solution bleaches the litmus.
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