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I have an electric kettle at work. On Monday, it's empty and I pour in about 1.5 liters of water. I usually end up drinking about 1 liter per day and refilling 1 liter each morning.

At the end of the week, I rinse it out because in my mind, there is .5 liters of "old" water from the beginning of the week on the bottom. Is that true? Or is the water constantly mixed around from being used and then refilled?

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up vote 4 down vote accepted

Realistically, you're probably very close to having it fully mixed. There are at least three sources of mixing present. Diffusion is a very slow process, so you can ignore it. You will presumably be agitating the water a lot when you pour the new water in, and that might completely mix everything. If that doesn't fully mix the water, there will also be some convection due to the heating you apply when you boil the water for your tea.

Here's what you get if you assume that it's fully mixed each day. On Monday, you add 1.5 L and drink 1.0 L, leaving 0.5 L. On Tuesday you add 1.0 L and mix everything up. You then drink 1.0 L of the mixture, leaving 0.3333 L of Tuesday's water and 0.1667 L of Monday's water. On Wednesday you add 1.0 L, mix everything up, and drink 1.0 L of the mixture. Etc.

After the $n^{\mathrm{th}}$ day, you have $1.5/3^n$ L of Monday's water left, $1.0/3^{n-1}$ L of Tuesday's water left, and $1.0/3^{n-k}$ L of the $k^{\mathrm{th}}$ day's water left. For the particulars you gave, that leaves you with 6.17 mL of Monday's water 12.3 mL of Tuesday's water, 37.0 mL of Wednesday's water, 111 mL of Thursday's water, and 333 mL of Friday's water, left over after you have poured Friday's tea.

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What about agitation caused by the water boiling? –  Anthony X Aug 10 '13 at 18:35
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