Lemonade-based physics question

[Vort]

While making up some wonderful lemonade a few minutes ago, I formulated a physics question. Give your best answer WITHOUT looking at other answers. Here's the question:

I'm making up a batch of lemonade, and decide I want to use ice so that the lemonade cools down faster in the fridge. However, when all the ice melts, I want the lemonade to be just the right dilution. In other words, I want to add exactly the right amount of water, including ice, to make the lemonade. If I need to add 10½ cups of water, and I want to add part of that (say two cups) as ice, how much water+ice should I add?

Note that ice is less dense than water, so an equivalent amount of ice takes up more room than the water.

[mnn727]

I'd make the ice out of lemonade myself.

[Dravin]

One pint of water weighs close enough to a pound for this application. You want to add 5.25 lbs of H20, so 1 lb of ice, and 4.25 lbs of water.

[Guest]

One pint of water weighs close enough to a pound for this application. You want to add 5.25 lbs of H20, so 1 lb of ice, and 4.25 lbs of water.

Not quite... you have to figure in how much ice is still solid when you drink the lemonade.

[Just_A_Guy]

Freeze two cups of water ahead of time. Once it's frozen, add it to 8 1/2 cups of water and your other ingredients.

(Since water expands when it freezes and contracts as it thaws, freezing two cups ahead of time saves you a bunch of complicated math later.)

[MrShorty]

I would convert that 10 1/2 cups of water to a weight, then use a scale/balance to add the water to the pitcher. 10.5 cups * 240 mL/cup * 1 g/mL = 2.52 kg H2O. Now I don't need to worry about the density difference between ice and liquid water.

I think there's a time element that has not been given. The main question I see is -- how long will this be in the fridge? I want all of the ice to be melted at the time I remove the lemonade from the fridge. If solid ice is still present at serving, the lemonade will be more concentrated than the recipe calls for. So, the real question will be, how quickly will the ice melt in the fridge?

I think we can model this question using Newton's law of cooling. The fridge air will be about 40 F, the temperature of the lemonade + ice will be 25-30 F, depending on how much sugar and other solutes are present. If we had a good estimate for the rate of heat transfer, we could use that to estimate how quickly heat will flow into the lemonade to melt the ice. From there, we could compute how much ice we want (balance will be liquid water at 50 F or whatever our tap water temperature is) so that the last of the ice is just melting as we get ready to pull it out of the fridge.

[Dravin]

Not quite... you have to figure in how much ice is still solid when you drink the lemonade.

No, I don't. Nothing about the question specifies that I need to make allowance for some amount of ice to remain when I drink it.

I'm making up a batch of lemonade, and decide I want to use ice so that the lemonade cools down faster in the fridge. However, when all the ice melts, I want the lemonade to be just the right dilution. In other words, I want to add exactly the right amount of water, including ice, to make the lemonade. If I need to add 10½ cups of water, and I want to add part of that (say two cups) as ice, how much water+ice should I add?

It is asking for the dilution to be correct when all the ice is melted.

Edited November 7, 2013 by Dravin

[Smudge]

I would measure 2 cups of water and freezer that!

[Vort]

ANSWER:

Put ice in the measuring cup, then fill with water until the water line reaches the right position.

Why? Because the floating ice displaces exactly its own weight in water. In other words, when the ice melts, it will occupy exactly the volume of water it displaces. So no special calculation is needed; just ignore the ice sticking up over the surface of the water when you measure.