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What are the consequences of using only distilled water to rinse a buret before a titration?
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If you rinse the buret with distilled water, when you add the NaOH to it, there may still be water left in the buret, thus, diluting the concentration of the NaOH solution. This would give you a lower molarity of NaOH than if you rinsed the buret with NaOH solution because mixing NaOH with NaOH (if they are both identical in concentration) would not change the concentration.
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That was a little confusing, but I think you're asking, if you're doing a titration with NaOH solution and you rinse the buret with DI H2O, how that would affect your result. The answer is, when you rinse the buret, some of the liquid you rinsed it with remains in there. If you rinse it with DI H2O, the water left in the buret contains no NaOH and therefore the molarity of your NaOH solution is lightly lower than recorded. You would measure the acid to be slightly stronger than it actually is. By rinsing the buret with the solution you intend to titrate with, you ensure that any liquid remaining in the buret won't affect the overall molarity.
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