> Oh, I misunderstood your original statement then. All you meant is that normal room temperature fluctuations ensure that the cup of liquid will never ever reach perfect equilibrium. True. However, I am still puzzled by why you even bother to frame the question in terms of "infinitely long duration," or, as in your current example, "an infinitely large room." To make those assumptions seems unrealistic in the first place -- unless the engineer is doing a seat-of-the-pants, super-simplified calculation which required infinite time or infinite space to go to completion. Surely an engineer trying to strive for reasonable accuracy would posit an *enclosed* space for a room, and maybe even a thermal gradient of heat absorption/reflection radiating from the room's walls, plus a possible contribution due to draft current from a forced-air furnace convection, etc. What I'm trying to say is that if engineers go for the practical, wouldn't they look at the real-world situation rather than try to set up for the infinite situation.

AHA! Now we get to the heart of the matter.

My infinitely large room was a simplifying assumption. Yes, it would be more accurate to give the room details as you suggested; I made the assumption of an infinitely large room to allow the ambient room temperature to remain perfectly constant. Any finite room would be slightly cooled by the energy transferred in to melt the ice, and rather than calculate the impact of these changes, I chose to eliminate them. Of course, a good engineer would have said that in a finite room, all those other considerations for temperature change would have a more significant impact on the room temperature than the effect of the melting ice, thereby allowing the room temperature to be essentially unaffected by the melting of the ice-- I just didn't think of that at the time. I probably should not have said "an infinitely large room", as this complicates the matter, and instead gone with "a room of constant temperature". Neither one is particularly realistic as an assumption, but at least the second is a bit clearer. :-)

My final note on this is that the real world situation, though more accurate, would not actually be better for solving the problem (in my professional opinion). The complex model which would result from the full definition of the room's temperature variation would likely have a relatively minor impact on our water-ice system, and would thus have no significant impact on the real problem at hand. Basically, we engineers do like to detail a problem very thoroughly, but we also find it necessary to determine when additional detail is valuable to us, and when it will simply clutter the problem and result in only a 0.000003451% change in our solution. Unfortunately, some engineers (myself included) tend to pick some areas to get picky about (such as the density of water lowering the level in the beaker slightly) and others to ignore (such as the room temperature issue). This tends to lead to a confusing mish-mash of simplifying assumptions, detailed analyses, and pure mathematical truth. Of course, I am used to it, and kind of like it that way. ;-)

> Are you kidding? "Solve the World" equations, situations, and parameters are absolutely FASCINATING. :-)

No argument there. If I had more time for research, I could write a lengthy reserch paper (albeit a very boring one) on this system of water and ice melting in a beaker.

> Wolfspirit

Don Monkey