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Dr. Lavelle gave a baking example today. If it takes 30 minutes for all the other steps of cookie-making but 45 minutes to bake the cookies, then no matter how much dough you prepare you can only make cookies as fast as the oven will bake them.
Once the reaction reaches the slow step, the slow step is the only part of the reaction that stops the reaction from proceeding, so the slow step determines the overall rate of reaction, as the fast steps proceed fast enough that they are negligible in comparison to the slow step in the rate of reaction.
The rest of the reaction proceeds much faster than the slow step, hence the slow step is the only thing holding every thing back. For example, if you have a bunch of funnels that pour water into each other. The smallest funnel hole will determine how fast the water flows even if the funnels before and after are bigger.
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