[firejake308]

The rate of change of temperature is proportional to the difference in temperature. Adding the cold milk later let's you juice a larger temperature differential out of the 2-minute waiting phase. So you should add the milk later.

[adrianmonk]

> the cold milk later

I think what the other person is asking about is: can you be sure that the milk is (as) cold later?

There's a lot of discussion about what happens to the temperature of the coffee during the 2 minutes. What happens to the temperature of the milk during that same time?

Where is the milk stored? Do you grab it out of the refrigerator the moment you add it to the coffee? Or the cold milk sitting out on the countertop getting warmer? If so, how rapidly?

[fc417fc802]

It's a safe bet that freshly brewed coffee is much farther from room temperature than refrigerated milk is. However deriving properties related to that symmetry (or lack thereof) would make an excellent question for an exam in an introductory class.

[FilosofumRex]

The two options are equivalent, since the final (equilibrium) temp of an adiabatic system (coffee + Milk + room) must be the same - ie it's the total amount of heat transferred that matters, and not the rates of heat transfer.

If the system is not adiabatic, ie the room is not big enough to remain constant temp, or equilibrium is not achieved in 2 mins of cooling, then the puzzle statement must be specify all three initial temps to be well poised.

[fc417fc802]

What gave you the idea that this is an adiabatic system? It's a cup of freshly brewed coffee on your kitchen counter. Equilibrium will certainly not be achieved within 2 minutes. Even if it were, different schemes would reach it at different time points.

It is fundamentally a question about rate of energy transfer.

The thing to notice about the symmetric system is that both items will experience the same rate of transfer. However there's presumably more coffee in your coffee than there is milk so it's not actually symmetric. If you adjust the parameters for volume and specific heat to make the final mixed product symmetric then it no longer matters when you do the mixing.