I've just made a cup of 10 cl 80 C coffee and want to add 2 cl milk (always kept in the refrigerator that keeps 6 C). I am going to drink the coffe in six minutes and I want my coffee to be as hot as possible at that time. Should I add the milk immediately or at some other point in time? The room temp is 20C.

/Tomas

I'm probably headed down the wrong path here, but I'd say immediately.
Though the coffee is below the boiling point, there probably is some volatility, thus losing some of the coffee as time goes on. The less coffee, the lower the temperature of the coffee/milk mixture.

Let's say nothing evaporates.

/Tomas

I'm not a physics guy, so I was looking up some background knowledge and came across the answer to the problem. Interesting.

I'm PMing you about what this might imply about a jaunt to the beach.

I'd like to think there is a simple answer to this, but I'm not sure there can be. Doesn't it depend on the kind of cup (e.g., whether a good conductor, or whether it has a reflective inner surface, or whether it is transparent, or what the diameter of the cup is)? When do you take the milk out of the fridge, and how much of it do you take out? It may also depend on whether you stir the milk in. Heat can be lost by conduction, convection, or radiation and the laws of heat loss for each of these are different.

If I had to guess, I'd say it makes no difference.

Mike

The loss of radiation is so small, that I think we can neglect it. If it wasn't clear, the milk stays in the fridge until you pour it, so it is always 6 C at the moment you pour it.

/Tomas

Cool the coffee immediately.

By shrinking the differential between the temperature of the coffee and the ambient air, one decelerates the rate of heat transfer over the six-minute time frame; Newton's Law of Cooling.

Not to be confused with JD's Law of Being Cool, which specifically prohibits me from bringing up any of Newton's laws during idle conversation.

I think that is a good explanation. To make it plausable, imagine the extreme case where the coffee has already cooled to room temperature. If you add the milk at that point you will get a mixture _cooler_ than the room. If you add the milk early enough, the mixture can never get cooler than the room.

/Tomas

The coffee/crem mixture is more viscous than the coffee by itself, and will therefore lose heat more slowly. I suspect all those exact measurements are misdirection...

Marc

I'm pretty sure a more viscous mixture _can_ have a lower specific heat than a less viscous, since solids generally have a lower specific heat. But that's just me guessing.

In this example I've calculated with milk and coffee having the exact same specific heat though. But since milk actually has a higher specific heat than coffe, you are right that it's yet another reason to add it early.

But it's not a necessary reason, as calculations show that you should add early _even_ if the specific heat of the mixture doesn't change with the milk.

/Tomas