HE. On Cake and Coffee

Harry Kaplan joins us for discussion of cake and coffee– and leaves us with a counter-intuitive puzzle…


  1. Byon said,

    December 13, 2011 at 12:34 am

    Welcome back, Math Factor!  You’ve been missed. I’m not convinced that hot water will freeze faster than cold water, assuming by freeze you mean “when put in a freezer, will reaches 32F and becomes ice”.  If you mean “drops temperature faster”, then I do buy that.  But take it to the extreme: 2 trays put in the freezer, one at 210F, and the other at 33F.   The later will become ice first, right?

  2. strauss said,

    December 13, 2011 at 8:15 am

    You are correct on all counts! It seems to _me_ (Chaim) like a good way to wreck the freezer!

  3. Jonathan Lundell said,

    December 13, 2011 at 11:15 am

    The discussion on keeping the cup of coffee warm strikes me as more than a little oversimplified. How about evaporative & convective cooling? Thermal conductivity of the containers? Ratio of surface to volume? Heatsink effects (heat transferred to or from the portion of the container that extends beyond the contents)?

    Thought experiment: what happens if the coffee is in a well-insulated lidded travel mug? The result will be hotter if you leave the coffee insulated at more or less the original temperature, and allow the cream to warm up (separately) to room temperature before mixing.

    As a practical matter, I suspect you’d be better off using the time to find something to use as a lid for the coffee mug first, and insulating it second, than worrying about the cream. 

  4. martin said,

    December 13, 2011 at 1:02 pm

    time it takes for 210 tray to reach 32 = time it takes 210 tray to reach 33 + time it takes 33 tray to reach 32 (i guess)
    for coffee/cream problem
    since the cream comes from the fridge, it warms up (in the same way the coffee cools off) if you wait, so there’s a certain symmetry. but i guess there is just so much more coffee than cream that the answer to the problem doesn’t change.

  5. Jonathan Lundell said,

    December 14, 2011 at 12:58 am

    The ΔT of the coffee is greater than that of the cream, too. Figure an ambient 70F, the cream is maybe 35F at the lowest, while the coffee is more like 180F. So (*all* else equal, which it’s not) the coffee is cooling faster than the cream is warming up.

    Still: put a lid on it. 

  6. Mike said,

    December 15, 2011 at 11:04 am

    As I listened to the coffee cooling problem the volume vs surface issue is what I thought of first. assuming you’re going to end up with the same volume of coffee and cream at the end, (assuming 90/10% split for example) the 90% volume of coffee has a smaller surface area to more readily radiate heat from, even through the insulating cup, in fact the open top surface of the coffee is a higher percentage of the total surface area. This means the total “thermal units” in the black coffee radiates more before you return and add cream with it’s negative thermal units than the larger volume of the combined mixture, starting at a lower temperature because the cream has already lowered the total count, radiates through it’s larger surface area. Especially since the change in surface area has all been in the insulated side of the cup. Also the cream in the presumably closed container with a larger thermal mass  hasn’t gained thermal units compared to the 10% in the cup. Unless you pour a proportionately measured tiny cup of cream which you then add to your coffee. Is the change in cooling rates strictly a factor of the temperature gradient vs the radiant surface area? SOOOOOOOO glad you’re back.

  7. Harry Kaplan said,

    December 17, 2011 at 9:22 am

    To Jonathan:
    You clearly know more about the physics of heating and cooling than I ever will.  However, simple experiments show that in reality the immediately-creamed coffee does stay a tad hotter than the beverage made by mixing the (cooled down) coffee and (warmed up) cream after the 20-30 minute interval has passed.  At some point Chaim will put a link here that will make that point among others. That link will give the most relevant numbers (initial temperatures of coffee and cream, masses of both, ambient temperature) that were used in the test, though you won’t see container coefficient of conductivity or the like.  I have no doubt that you could create a situation in which some of the other factors you mentioned might dominate and perhaps change the direction of the result.  The idea was to create a “normal” situation.
    As far as the lidded travel mug is concerned, I’m quite sure that would keep the coffee hotter.  An even better idea would be to throw out the old coffee entirely and make a brand-new pot.  But the question asked in the podcast was pretty specific, and not concerned with the general issue of enjoying the hottest possible coffee at time T!

  8. Carl Sandrock said,

    December 18, 2011 at 11:20 am

    As it happens, I’ve been running this exact experiment as a dynamic modelling problem in my third year chemical engineering class. We’ve run through many different models and many different configurations of liquids and cups.  It all comes to the same conclusion: because the hotter coffee can lose more heat in the same time, it will always be better to do the cooling step first if you want hotter coffee at the end.  This holds for the evaporative cooling argument as well.  We’ve even done an experiment where a thin layer of oil was floated on the top of the coffee to distinguish between the straight convective cooling and the evaporation.  My students always think I’m a bit touched when I give them the problem, then they get drawn in.  

  9. Dave from Knoxville said,

    December 19, 2011 at 4:36 pm

    Like Byon, I am unconvinced about hotter water freezing faster than cooler water. According to my understanding of Newton’s Law of Cooling, the rate of cooling is governed proportionally by the difference between the temperature of the water and the ambient temperature inside the freezer. So let’s say that the water is at 35F, the freezer is at 28F, and it takes 10 minutes to freeze. If we now start with water that is at 70F, while it is true that it initially cools at a much faster rate, it must eventually hit 35F, at which point it will require the same additional 10 minutes to freeze. As Chaim said, it’s cooling faster “but it has farther to go”. He’s exactly right. The graph of the coffee temperature initially at 70F could be overlaid on top of the graph where the coffee was initially 35F, and from that temperature on (assuming ideal conditions) those graphs should be identical.
    On the other hand, I can buy the answer to the other question; dump the cream in initially, and the later the coffee will be warmer than if the cream is dumped in later.

  10. Stephen Morris said,

    December 19, 2011 at 9:39 pm

    The idea that boiled water can freeze faster than room temperature water has been claimed many times experimentally.  

    If it is true then it is to do with physics, not mathematics.  Clearer the hotter water must reach the starting temperature of the cooler water at some point.  It must then cool faster for some physical reason.

    The best explanation I have heard is that the act of boiling drives out gasses and so the two liquids are not the same.

    I would be very interested if anyone wants to do the experiment, whatever results you get. I guess you would want to compare three liquids: tap water at room temperature, tap water that has just been boiled, and boiled tap water that has been allowed to cool to room temperature.  Put them all in the freezer at the same time and see what happens!

  11. Stephen Morris said,

    December 19, 2011 at 10:06 pm

    Here’s a wikipedia entry on hot water freezing faster.


    All rather inconclusive.  One comment is that so many factors can effect freezing rates that it’s difficult to set up a meaningful experiment.  But is that missing the point?  If boiling water makes it liable to freeze faster then we still have hot water freezing faster.  That is still a fun result. 

    A couple of quotes:

    A reviewer for Physics World writes, “Even if the Mpemba effect is real — if hot water can sometimes freeze more quickly than cold — it is not clear whether the explanation would be trivial or illuminating.”
    New Scientist recommends starting the experiment with containers at 35 °C (95 °F) and 5 °C (41 °F) to maximize the effect. 

    So now you have some more info for your own experiment. 

  12. Harry Kaplan said,

    December 21, 2011 at 7:24 am

    My heartfelt thanks to Stephen Morris for turning my glib, unilluminating, and incorrect response to Kyle’s question into a brilliant Mpemba insight!

  13. Patrick Stein said,

    December 25, 2011 at 8:18 pm

    I just listened to this podcast last night while driving.  I was itching to get in here and yell, “No!”  But, I see it’s already been taken care of.

     By Martin with this: “time it takes for 210 tray to reach 32 = time it takes 210 tray to reach 33 + time it takes 33 tray to reach 32″

    And, by Mike with this: “As I listened to the coffee cooling problem the volume vs surface issue is what I thought of first.”

    My cheap, while-driving gedanken experiment: If you used the same quantity of cream as coffee and the cream were as much below room temperature as above room temperature, then Newton’s law of cooling means the cream will warm up at the same rate the coffee cools down.  Any time you mix them, they will become room temperature coffee.

    Taking two cups of coffee at the same temperature and pouring one into a tin can and the other onto a cookie sheet…. both obey Newton’s law of cooling, but with different decay rates.  The surface area to volume is crucial.

  14. John Clay said,

    January 20, 2012 at 9:22 pm

    There is an important factor that has not been considered in the discussion of hot water freezing faster than cold water.  The latent heat required for phase change is actually much larger than the heat required to drop the temperature to the freezing point.  The same initial mass of hot water will freeze faster than the same mass of cold water, but the resulting ice cubes will not weigh the same.  In the low relative humidity freezer, some of the hot water will evaporate, dropping the mass of the resulting liquid and leading to solidification of this control volume more rapidly than for the control sample that started at a colder temperature.

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